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Universl^ MicrdMlms International 300 N. ZEEB RD„ ANN ARBOR. Ml 48106

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F a s a n e l u , F l o r e n c e D o w d e l l

THE CREATION OF SHEAF THEORY

The American University PhJD. 1981

University Microfilms Internstionel 300 X . Zeeb Road. Ann Arbor. M I 48106

Copyright 1981 by Fasanelli, Florence Dowdell All Rights Reserved

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. THE CREATION OF SHEAF THEORY

BY Florence Dowdell Fasanelli

Submitted to the

Faculty of the Department of Mathematics, Statistics and Computer Science and the School of Education of The American University

in Partial Fulfillment of the Requirements for the Degree

of Doctor of Philosophy in Mathematics Education

Signatures of Committee: "1 -1 V J J ^ _____ Chair : ^ ;v? ^ Dean of the College^

n r i Member : Member :

Member : 1981 The American University Washington, D.C. 20016

5 %'\

THE AMEHICiH ÜÎTIVEBSITY ÊIBRARY

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. THE CREATION OF SHEAF THEORY BY Florence Dowdell Fasanelli

ABSTRACT A mathematical theory is often created by combining the resources of several branches of mathematics to solve a prob­ lem previously inaccessible in one (or more) branches. Establishing a combination as a theory is accomplished by

1 , the discovery of new axioms and theorems; 2, the putting together of new techniques for working between the branches (which however must remain autonomous); and 3, the systemati­ zation of an abstract structure of axioms, theorems and techniques by its extension to a general application. The creation of a new theory is often recognized by a change in the language employed to describe the concepts involved. This dissertation is an analysis of the creation of one such theory in the history of mathematics, recognizing 1953 as the year in which sheaf theory changed from an instrument of specific application used by Cartan to solve a long out­ standing problem (second Cousin problem) to an instrument of general application, extended by Serre's exposition beyond the original use (in analysis to algebra). The historical development of sheaf theory is discussed including the proliferation of its techniques in courses, texts, seminars

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and papers which were conducted, prepared and written by the pioneer mathematicians in this undertaking (1940-1972). The introduction contains the necessary definitions

translated from Serre's "Faisceaux algébriques cohérents," called FAC, (1955). In Chap:er I the historical background in several complex variables begins with the work of Riemann, Weierstrass, Mittag-Leffier, Poincare, Appell, and Cousin.

Recognition of Cousin's error by Gronwall is discussed, followed by a chronological interpretation of the work of Henri Cartan, Behnke, Thullen, and Oka and an introduction of Ideal by Ruckert, and faisceau by Leray. The solutions of

the Cousin problems by Oka and Cartan are analyzed. In Chapter II the concern is the development of the terminology, primarily in the séminaire Cartan from 1948-1963. Abstraction of the theory is considered in Chapter III. In 1950 Leray published his paper on homology and cohomology of Lie algebras which gives applications of sheaves. By 1951 the definition of a sheaf had taken on a topological form (Lazard). In Stein's paper (1951) everything related back to convexity. In May 1952 general properties of coherent sheaves were developed (Serre and Cartan). Meanwhile at Princeton, Kodaira worked with Weyl on the Riemann-Roch theorem, and applications were developed in algebraic geometry using coherent sheaves. In 1953 Serre transferred the theory of sheaves to abstract varieties using the Zariski topology; as a consequence sheaves became a major instrument of general application. By 1954 Serre had developed an exposition of

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the purely algebraic theory and gave the first systematic application of cohomological algebra to abstract algebraic geometry. The remainder of Chapter III deals with the work of

Grothendieck, who developed the formal analogy between the theory of the cohomology of a space with coefficients in a sheaf and the theory of derived functors of modules, general­ ized by using categories. Thus the theory developed did not need the restriction hypothesis on the spaces which was

required in FAC. Building on these ideas and that of adjoint

functors, Godement published the first book which gave a complete exposition of sheaf theory (1958). Chapter IV contains a discussion of applications to

rings, to categories and to topology. In Chapter V the educational implications of a historical study of theories are related to the creation of sheaf theory.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PREFACE The story of analysis of several complex variables from Bernard Riemann to Henri Cartan-Kiyoshi Oka is an exposition of the manner in which one discipline— analysis— has given rise to another, sheaf theory. The view taken here is that progress in mathematics frequently derives from attempts to deal with problems for which existing techniques are inade­ quate. Original quotations are employed extensively so that the contemporary judgments, when superimposed upon those of

the preceding eras, do not obliterate the atmosphere in which the mathematics had developed. The essential story of the creation of sheaf theory takes place in three countries : France, Japan and the United States. The action takes place over 100 years, spanning two World Wars, which play an interesting part in the scenario. Several institutions are important: The séminaire Cartan from its inception in 1948 until its completion 16 years later; Bourbaki; the Institute for Advanced Study at Prince­ ton; the Mathematisches Institut at Munster. There mathema­ ticians were able to work in a productive atmosphere. The National Science Foundation made grants of at least $250,000 for papers discussed here. The Wars separated the Japanese from the West, yet during the Wars all countries produced significant mathematics even when the mathematicians were in confinement. As the

IX

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. course of the solutions was developing new ideas came from unexpected sources outside the speciality and each one of these will be treated here with a briefer and less detailed history. In 1950 Jean Leray and Henri Cartan introduced the concept of a sheaf, using specific language to define and formalize ideas that had been developing since the early 1930s.. Sheaves play an important role in many branches of mathematics in 1981. This unifying concept has made it possible to state a problem in one area of mathematics but to solve it in another. Insights can be gained into the way mathematical ideas are promulgated, and the effect mathema­ ticians have on one another, by tracing the brief history of the development of sheaf theory. I hope to have given a flavor of the times, to have shown interchanges between math­ ematicians, to have noted the mathematics of the several disciplines and to have shown how what was originally a faisceau finally became a sheaf. It was in a seminar in Mary Gray's home that I first learned about sheaves. I would like to thank her for that introduction, her advice in my graduate studies, her patience, kindness and understanding. Judy Sunley and Richard Holzsager generously helped me to change the emphasis of my study. Ronald Maggiore and Daniel Antonoplos have welcomed me into mathematics education. For continuous encouragement, I want to thank Margaret and Peter Matelski, Antonia and James Fasanelli.

iii

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS

PREFACE ...... ii

INTRODUCTION ...... 1

What Sheaves Are ...... Chapter I. SHEAVES AND FUNCTIONS OF SEVERAL COMPLEX VARIABLES ...... 24 II. DEVELOPMENT OF THE TERMINOLOGY...... 64

III. ABSTRACTION OF THE T H E O R Y ...... 109 IV. APPLICATIONS...... 151

A. Sheaves and ring t h e o r y ...... 151 B. Sheaves and categories...... 166 C. Sheaves and topology...... 198 V. IMPLICATIONS FOR MATHEMATICS EDUCATION...... 201 BIBLIOGRAPHY ...... 209

IV

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. INTRODUCTION In light of the conclusions of this thesis, particularly that it was in the year 1953 that sheaf theory changed from an instrument of specific application to a major instrument of general application, I have chosen to state what sheaves are by translating the definitions from a memoir written by

Jean-Pierre Serre in the years 1953-1954 and published as "Faisceaux algébriques cohérents" [192] in the Annals of Mathematics, March 1955 (the article had been received

October 8 , 1954). Serre discovered that the sheaf theoretical

concepts which he and Henri Cartan had originally worked out (in the Séminaire Algébrique Topologie organized by Cartan at L'École Normale Supérieure) and applied to analytic geometry could be applied to algebraic geometry over arbitrary fields. Thus his paper is historically important and, furthermore, the definitions in it follow those of the séminaire Cartan. Dr. I. M. James held a seminar at Oxford in 1958 when he was Savilian Professor of Geometry in which R. G. Swan gave a course of lectures. A student participant R. Brown took notes which Swan later revised and which were published under Swan's authorship by the University of Chicago [202]. Much of the material came from the 1950-1951 "Séminaire Cartan" [28] and A. Grothendieck's "Sur quelques points d'algèbre homologique". The Tohoku Mathematical Journal, August

1957 [87]. More definitions were added at that time and

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. will also be included in this introduction in brackets. For example : [A PROTOSHEAF (according to James) is a structure over a space X. Let F be the union of groups F^, one for each

X E X. Let p: F -»■ X be defined by p(f) = x if f e F^. Since the F^ are disjoint (accomplished if necessary by making an isomorphic copy of F^), note that F is just a set with no topology so there is no question of continuity involved. This construction leads to the picture]

X

§ 1. Operations on Sheaves 1. Definition of a sheaf. Let X be a topological space. A SHEAF OF ABELIAN GROUPS on X (or simply a SHEAF) is formed by

(a) A FUNCTION x -»■ F^ WHICH MAKES CORRESPOND TO

EVERY X E X AN ABELIAN GROUP F X , (b) A TOPOLOGY ON THE SET F, UNION OF THE SETS F^. If f is an element of F^, we will put ir(f) = x; the mapping TT is called the PROJECTION of F on X; the subset of F x F

formed by the pairs (f,g) such that ir (f ) = 77(g) will be noted F + F. These definitions being made, we are able to state the two axioms to which the given (a) and (b) are subjected:

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (I) FOR EVERY f e F,. THERE EXISTS A NEIGHBORHOOD V OF

f AND A NEIGHBORHOOD Ü OF 7r(f) SUCH THAT THE RESTRICTION OF tt

TO V IS A HOMEOMORPHISM OF V ON U. (Said in another way, tt ought to be a local homeomorphism) (II) THE MAPPING f ^ -f IS A CONTINUOUS MAPPING OF F INTO F; AND THE MAPPING (f,g) f + g IS A CONTINUOUS MAPPING OF F + F INTO F. One observes that, even if X is separated (which we have

not supposed), F is not necessarily separated, as shown in the example of the sheaf of germs of functions. G being an abe­ lian group, we put F^ = G for every x e X; the set F may be identified by the product X x G, and, if one provides it with the product topology of the topology of X by the discrete to­ pology of G, one obtains a sheaf, called the CONSTANT SHEAF

isomorphic to G and often identified with G. (See diagram on

next page.)

[A sheaf is TRIVIAL if it is isomorphic to the constant sheaf. James] 2. Sections of a sheaf. Given a sheaf F on the space

X, and U a subset of X, one names as SECTION of F over U a

continuous mapping s: U -»■ F such that tt o s is the identity mapping of U. One has therefore s(x) e F^ for every x e U.

The set of sections of F over U will be designated by r(U,F);

axiom (II) produces as a consequence that r(U,F) is an abelian

group. If U C V, and if s is a section over V, the restric­

tion of s to U is a section over U; whence a a homomorphism

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. f e-i I V

TT

* X ----- ^ X

ir(f) fibre parameterized by points of X

F ~ X X G

X

r(u,s (V,S)

■i----- — 5- X X

If a sheaf is a generalized product a section is a gen­ eralized function.

Example of a Sheaf.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. p^: r(V,F) -»■ r(ü,F). [THE SUPPORT Of S, written ISI = {x e U: s(x) 0}. Cartan] If U is open in X, s(U) is open in F, and when U traverses an open basis of X, the s(U) traverse an open basis of F; this is another way of expressing axiom (I).

We notice a further consequence of axiom (I); For every f e F^, there exists a section s over a neighborhood of

X such that s(x) = f, and two sections possessed of this property coincide in a neighborhood of x. Said in another

way, F^ is the INDUCTIVE LIMIT of the r(U,F) following the filtering order of the neighborhood of x.

3. Construction of sheaves. We suppose given for every open U C X, an abelian group F^, and, for every pair of

open sets U C V, a homomorphism F^ F^, such that the transitive property o $ ^ is verified every time

U C V C W. [Swan calls this structure a STACK. Also called a PRESHEAF] The collection of the (F^,permits us to define a

sheaf F in the following manner: (a) One puts F^ = lim F^ (inductive limit following the filtered order of the open neighborhoods U of x). [F^ is called the STALK]. If x belongs to the open set U, one has therefore a canonical homomorphism F^ F^. (b) Let t e Fy and designate by [t,U] the set of $^(t) for X varying over U; one has [t,U] C F, and one takes for F the topology engendered by the [t,U]. So, an element f e F^

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. admits for a base of neighborhoods in F the sets [t,U] where

X E U and $^(t) = f . One verifies at once that the given (a) and (b) satisfy

axioms (I) and (II), said in another way, that F is indeed a sheaf. We will say that it is the SHEAF DEFINED BY THE SYSTEM (Fy,$y). If t e Fy the mapping x -»■ $^(t) is a section of F over U whence the canonical homomorphism i:F^ r(U,F).

PROPOSITION 1. IN ORDER THAT i:F^ r(U,F) BE INJECTIVE

IT IS NECESSARY AND SUFFICIENT THAT THE FOLLOWING CONDITION BE VERIFIED: IF AN ELEMENT t e F^ IS SUCH THAT THERE EXISTS AN OPEN COVERING {U. } OF U WITH (t) = 0 FOR EVERY i, THEN t = 0. Ui PROPOSITION 2. GIVEN U OPEN IN X, AND SUPPOSE THAT

i:F^ 4. r(V,F) IS INJECTIVE FOR EVERY OPEN V C U. IN ORDER THAT i:Fy r(U,F) BE SUBJECTIVE (HENCE BIJECTIVE) IT IS NECESSARY AND SUFFICIENT THAT THE FOLLOWING CONDITION BE VERIFIED:^ FOR EVERY OPEN COVERING {U.} of V, AND EVERY SYSTEM Ui u {t. }, t. E F„ , SUCH THAT O U. (t. ) = O U. (t.) FOR 1 i J i J J

EVERY PAIR (i,j) THERE EXISTS t e F^ WITH (t) = t^ for . 2 ^ every i.

According to Tennison "The basic question in sheaf theory is to what extent does a map of sheaves which is locally surjective have surjective section maps." [207, p. 31]. 2 A germ summarizes the local behavior of a function at a point.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PROPOSITION 3. IF F IS A SHEAF OF ABELIAN GROUPS ON X,

THE SHEAF DEFINED BY THE SYSTEM (r(U,F),p^) IS CANONICALLY

ISOMORPHIC TO F. EXAMPLE. Given an abelian group G, and take for F^ the

set of functions on U with values in G; we define F^ 4- by the operation of RESTRICTION of a function. One obtains in this manner a system (F^,Py) whence a sheaf F, called the

SHEAF OF GERMS OF FUNCTION with values in G. One is able to identify the sections of F on an open U with the elements of

^u- 4. Collation property of sheaves. Given F on a sheaf

on X, and U a part of X; the set ir ^ (U) C F, endowed with the topology induced by that of F forms a sheaf on U, called the sheaf INDUCED by F on U, and noted F(U). PROPOSITION 4. LET U = {U^} BE AN OPEN COVERING OF X AND FOR EACH i e I, GIVEN F^ A SHEAF ON U^^; FOR EVERY PAIR

(i,j) LET e . . BE AN ISOMORPHISM OF F. (U. 0 U.) ON F. 13 3 1 3 1 (U. n U.); WE SUPPOSE THAT ONE HAS 0..O 9., = 0., AT EVERY 1 3 13 3*^ POINT OF U. U. for each system (i,j,k). 1 n 3 n u, K THEN THERE EXISTS A SHEAF F, AND, FOR EACH i e I, AN ISOMORPHISM of F(U^) ON F^, SUCH THAT 0^^ = n^OnT^ AT EVERY POINT U^ D U^. MOREOVER, F AND THE ARE DETERMINED BY AN ISOMORPHISM BY THE PRECEDING CONDITION. One says that the sheaf F is obtained by collation of the sheaves F^ by means of the isomorphisms

5. Extension and restriction of a sheaf. Given X a topological space, Y a closed subspace of X, F a sheaf of X,

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. we will say that F IS CONCENTRATED ON Y if one has F^ = 0 for

every x e X-Y. PROPOSITION 5. IF THE SHEAF F IS CONCENTRATED ON Y,

THE HOMOMORPHISM p^: r(X,F) 4- r(Y,F(Y)) IS BIJECTIVE.

PROPOSITION 6 . GIVEN Y A CLOSED SUBSPACE OF A SPACE X

AND G A SHEAF ON Y, WE TAKE F^ = G^ IF x e Y; F^ = 0 if X / Y AND LET F BE THE SET SUM OF THE F^. ONE CAN PUT ON F ONE AND ONLY ONE STRUCTURE OF A SHEAF ON X SUCH THAT F(Y) = G. One says that the sheaf F is obtained by PROLONGING THE SHEAF B BY 0 BEYOND Y.

6 . Sheaves of rings and sheaves of modules. A sheaf of rings A is a sheaf of abelian groups A^, x e X, where each A^ is given a structure of a ring such that the mapping

(f,g) 4- f g is a continuous mapping of A + A into A. We will suppose that each A^, possesses a unit element, varying continually with x.

If A is a sheaf of rings satisfying the preceding condi­

tion, r(U,A) is a ring with unit element, and r(V,A)

4- r(U,A) is a unitary homomorphism if U C V. Inversely, if one is given the ring A^ with unit element, and the unitary

homomorphisms A^4-a ^ satisfying $^o$y = the sheaf A defined by the system (A^,$^) is a sheaf of rings. For example, if G is a ring with unit element, the sheaf of germs of functions with values in G is a sheaf of rings. Let A be a sheaf of rings. A sheaf F is called a SHEAF of A-MODULES if each F^ is given the structure of a unitary A^-module (on the left, in order to fix the terminology),

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. varying "continuously" with x, in the following sense: if A + F is the subset of A x F formed by the pairs (a,f) such

that 7r(a) = ir(f) , the mapping (a,f) -»■ a.f is a continuous

mapping of A + F into F. 7. Sub-sheaf and quotient sheaf. Let A be a sheaf of

rings, F a sheaf of A-modules. For every x e X, let be a subset of F^. One says that G = UG^ is a SUB-SHEAF of F if:

(a) G^ IS A SUB-A^-MODULE OF F^ FOR EVERY X e X,

(b) G IS AN OPEN SUBSET OF F. Condition (b) can be expressed:

(b') IF X IS A POINT OF X, AND IF s IS A SECTION OF F OVER A NEIGHBORHOOD OF x SUCH THAT s(x)eG^, ONE HAS s (y) eG^

FOR EVERY y SUFFICIENTLY CLOSE TO x. It is clear that if the conditions are satisfied G is a

sheaf of A-modules.

Let G be a sub-sheaf of F, and put H^ = F^/G^ for every

X in X. We take H = UH^ of the quotient topology for the topology of F; one easily sees that one obtains thus a sheaf

of A-modules, called a QUOTIENT SHEAF of F by G, and noted

F/G. One is able to give another definition, utilizing the

procedure of N°3: if U is open in X, we put H^ = r(U,F)/

r(U,G), being the homomorphism defined by passing, to the

quotient p^: r(V,F) r(U,F); the sheaf defined by the system (Hy,0y) is none other than H. One or the other definition of H shows that, if s is a

section of H in the neighborhood of x, there exists a section t of F in the neighborhood of x, such that the class of t(y)

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 0

mod Gy is equal to s(y) for every y in the neighborhood of x. This is no longer true globally, in general: if U is open in X, one has only the exact sequence;

0 4- r(u,G) 4- r(u,F) 4- r(u,H)

the homomorphism r(U,F) 4- r(U,H) not being in general

surjective (cf. 24).

8 . Homomorphisms. Let A be a sheaf of rings, F and G two sheaves of A-modules. An A-HOMOMORPHISM (or more an A-linear homomorphism, or simply a homomorphism) of F into G

is given, for every x e X, by an A^-homomorphism F^ 4. G^,

such that the mapping $: F 4- g defined by the is contin­ uous. This condition can also be explained by saying that,

if s is a section of F over U, x 4- (s (x) ) is a section of G over U (section that one denotes $(s), where $ o s). For

example, if G is a sub-sheaf of F, the injection G 4- f, and

the projection F 4- F/G, are the homomorphisms. PROPOSITION 7. LET 0 BE A HOMOMORPHISM OF F INTO G. FOR EVERY x IN X, LET N BE THE KERNEL OF $ ; AND LET I BE X X X THE IMAGE OF * . THEN N = VN IS A SUBSHEAF OF F, I = Vl X X X IS A SUBSHEAF OF G AND $ DEFINES AN ISOMORPHISM OF F/N ON I. The sheaf N is called the KERNEL of $, and noted Ker ($); the sheaf I is called the IMAGE of $, and noted Im ($); the sheaf G/I is called the COKERNEL of $, and noted Coker ($). A homomorphism $ is called INJECTIVE, or biunivoque, if each of the is injective, which is equivalent to Ker ($) = 0; it is called SURJECTIVE if each of the is surjective, which is equivalent to COKER (4>) = 0; it is called BIJECTIVE

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 1

if it is at the same time injective and surjective, in which case Proposition 7 shows that it is an isomorphism of F onto

G, and $ ^ is also a homomorphism. All the definitions rela­ tive to the homomorphisms of modules can be transposed in the same way to sheaves of modules; for example, a sequence of

homomorphisms is called EXACT if the image of each homomor­ phism coincides with the kernel of the following homomorphism. If $: F 4- G is a homomorphism, the sequences:

0 4- Ker ($) 4- F 4- Im ($) 4-0

0 4- Im ($) 4- G 4- Coker ($) 4-0

are exact sequences. 9. Direct sum of two sheaves. Let A be a sheaf of

rings, F and G two sheaves of A-modules; for every x e X, we form the module F + G , the DIRECT of F and G ; an X X SUM X X element of F^ + G^ is a pair (f,g) with f e F^ and g e G^. Let H be the set sum of the F^ + G^ when x traverses X; one can identify H by the part of F x G formed by the pairs (f,g)

such that ir(f) = 7r(g). If one takes for H the topology Y induced by that of F x G, one verifies immediately that H is

a sheaf of A-modules; one calls it the DIRECT SUM of F and

G, and one denotes it F + G. Every section of F + G on an

open U C X is of the form x 4- (s(x), t (x) ) where s and t are

the sections of F and G on U; in other terms r(U, F + G) is

isomorphic to the direct sum r(U, F) + r(U, G). The definition of the direct sum extends itself by recurrence to a finite number of A-modules. In particular.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 2

the sheaf direct sum of p sheaves isomorphic to the same sheaf F will be noted F^. 10. Tensor product of two sheaves. Let A be a sheaf of rings, F a sheaf of right A-modules, G a sheaf of left A-modules. For every x e X, we put H = F H G , the tensor ^ X X X product being taken on the ring A^ (cf. for example, H. Cartan and S. Eilenberg, Homological Algebra [36]). PROPOSITION 8. THERE EXISTS ON THE SET H ONE AND ONLY ONE SHEAF STRUCTURE SUCH THAT, IF s AND t ARE THE SECTIONS OF

F AND OF G ON OPEN U, THE MAPPING X 4- s(x) S t(x) e H^ IS A SECTION OF H OVER U. THE SHEAF H THUS DEFINED IS CALLED THE TENSOR PRODUCT (ON A) OF F AND OF G AND ONE DENOTES IT F H G; IF THE A^ ARE

COMMUTATIVE IT IS A SHEAF OF A-MODULES.

All the usual properties of the tensor product of two modules are transposed to the tensor product of two sheaves of modules. For example, the whole exact sequence:

F4-F'4-F"4-0

gives rise to an exact sequence

F 8^ G 4. F' 8, G 4. F' 8.^ G 4- Q A A A

From the canonical isomorphisms one has

F 8^ (G^ + G^) =: F 8^ G^ + F 8^ Gg, F 8^ A = F,

and (supposing A^ commutative in order to simplify the notations):

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F G « G F, F (G a^ k ) % (f a^ G) a^ k .

11. Sheaf of germs of homomorphisms of a sheaf in another sheaf. Let A be a sheaf of rings, F and G two sheaves of A-modules. If Ü is open in X, let be the group of

homomorphisms of F(U) in G(U) (we will say equally "homo­ morphism of F into G over U" in place of "homomorphism of F(U) into G(U)"). The operation of restriction of a homo­

morphism is defined by h ^; the sheaf defined by the system (H^, is called the sheaf of germs of homomorphisms

of F into G, and noted Hom^ (F,G). If the A^ are commutative, Hom^ (F,G) is a sheaf of A-modules. An element of Hom^ (F,G), being a germ of homomorphism of F into G in the neighborhood of x, defines without ambi­ guity a A^-homcmorphism of F^ into G^; whence a canonical

homomorphi sm

p; Hom^ (F,G)^ Hom^ * X

But contrariwise to what happens for the operations studied

at present, the homomorphism p is not in general a bijection; in n° 14 we will give a sufficient condition for it to be.

§2. Coherent sheaves of modules In this paragraph, X designates a topological space, and

A a sheaf of rings on X. One supposes that every A^, x e X,

is commutative and possesses a unit element varying continually with x. All the sheaves considered through n° 16

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14

are sheaves of A-modules and all the homomorphisms are A-homomorphi sms. 12. Definitions. Let F be a sheaf of A-modules, and let s^,...,Sp be sections of F over an open U C X. If one

makes correspond to every family f^,...,f^ of elements of i=p A the element E f.*s.(x) of F , one obtains a homomorphism x X

(}>: A^ 4- F defined over the open U. The kernel R(s^,... ,8^) of the homomorphism (j) is a sub-sheaf of A^, called the SHEAF OF

RELATIONS between the s^^; the image of ({> is the sub-sheaf of F engendered by the s^^. Inversely, every homomorphism $: A^ 4- F defines the sections s. ,... ,s of F by the formulas :

s^(x) — (j)^(l,0,...,0),..., Sp(x) — 4>^(0,...,0,1).

DEFINITION 1. A SHEAF OF A-MODULES F IS SAID TO BE OF FINITE TYPE IF IT IS LOCALLY GENERATED BY A FINITE NUMBER OF

ITS SECTIONS. PROPOSITION 1. LET F BE A SHEAF OF FINITE TYPE. IF

S^,...,Sp ARE SECTIONS OF F DEFINED OVER A NEIGHBORHOOD OF A

POINT X E X, AND GENERATING F^, THEY GENERATE F^ FOR EVERY y

SUFFICIENTLY NEAR X. DEFINITION 2. A SHEAF OF A-MODULES F IS CALLED COHERENT IF; (a) F IS OF FINITE TYPE

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(b) IF s^,...,Sp ARE SECTIONS OF F OVER AN OPEN U C X, THE SHEAF OF RELATIONS BETWEEN THE IS A SHEAF OF FINITE

TYPE (ON OPEN U). One notes the LOCAL character of definitions 1 and 2. PROPOSITION 2. LOCALLY, EVERY COHERENT SHEAF IS

ISOMORPHIC TO THE COKERNEL OF A HOMOMORPHISM : A 4- A^. PROPOSITION 3. EVERY SUB SHEAF OF FINITE TYPE OF A COHERENT SHEAF IS A COHERENT SHEAF.

13. Principal properties of coherent sheaves. a 3 THEOREM 1. LET O 4-F 4-G 4-H 4-OBEAN EXACT SEQUENCE OF HOMOMORPHISMS. IF TWO OF THE THREE SHEAVES F, G, H ARE COHERENT THE THIRD IS ALSO. COROLLARY. THE DIRECT SUM OF A FINITE FAMILY OF COHERENT SHEAVES IS A COHERENT SHEAF. THEOREM 2. LET i() BE A HOMOMORPHISM OF A COHERENT SHEAF F INTO A COHERENT SHEAF G. THE KERNEL, THE COKERNEL, AND THE IMAGE OF

14. Operations on the coherent sheaves. We are going to show that the direct sum of a finite number of coherent sheaves is a coherent sheaf. We are going to demonstrate the analogous results for the functors 8 and Horn. PROPOSITION 4. IF F AND G ARE TWO COHERENT SHEAVES, F 8^ G IS A COHERENT SHEAF. PROPOSITION 5. LET F AND G BE TWO SHEAVES, F BEING

COHERENT. FOR EVERY X e X, THE MODULE Hom^(F,G>^ IS ISOMORPHIC TO Horn (F ,G ). ax X X

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PROPOSITION 6. IF F AND G ARE TWO COHERENT SHEAVES,

Hom^(F,G) IS A COHERENT SHEAF. 15. Coherent sheaves of rings. The sheaf of rings A can be regarded as a sheaf of A-modules; if this sheaf of

A-modules is coherent, we will say that A is a coherent sheaf of rings. As A is evidently of finite type, that signifies that A satisfies the condition (b) of Proposition 2. Otherwise

said: DEFINITION 3. THE SHEAF A IS A COHERENT SHEAF OF RINGS IF THE SHEAF OF RELATIONS BETWEEN A FINITE NUMBER OF SECTIONS OF A OVER OPEN U IS A SHEAF OF FINITE TYPE ON U. EXAMPLES (1) If X is a complex analytic variety, the sheaf of germs of homomorphic functions on X is a coherent sheaf of rings, after a theorem of Oka (cf. [20], expose XV) (2) If X is an algebraic variety, the sheaf of local rings of X is a coherent sheaf of rings. When A is a coherent sheaf of rings, one has the following result: PROPOSITION 7. IN ORDER THAT A SHEAF OF A-MODULES BE COHERENT IT IS NECESSARY AND SUFFICIENT THAT, LOCALLY, IT BE

ISOMORPHIC TO THE COKERNEL OF A HOMOMORPHISM (j) : A^ 4- A^. PROPOSITION 8. IN ORDER THAT A SUB-SHEAF OF A^ BE COHERENT, IT IS NECESSARY AND SUFFICIENT THAT IT BE OF FINITE TYPE. PROPOSITION 9. LET F BE A COHERENT SHEAF OF A-MODULES.

FOR EVERY X £ X LET I BE THE IDEAL OF A FORMED FROM a £ A XXX

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SUCH THAT a*f = 0 FOR EVERY f e F THE FORM A COHERENT SHEAF OF IDEALS (CALLED THE ANNIHILATOR OF F). In effect I^ is the kernel of the homomorphisin:

More generally the TRANSPORTER F;G of a coherent sheaf G into a coherent sub-sheaf F is a coherent sheaf of ideals (it is the annihilator of G/F).

§ 3. Cohomology of a space with values in a sheaf'*' In this paragraph, X designates a topological space, separated or not. By a COVERING of X, we always intend an opening covering. 18. [following the numbering in FAC] Cochains of a covering. Let U = {UU}^ ^ ^be a covering of X. If s (ip,...,ip) is a finite sequence of elements of I, we put:

"s = % ... ip ' % n ... n

Let F be a sheaf of abelian groups on the space X. If p is an integer >_ 0, one calls a p-COCHAIN OF U WITH VALUES IN F a function f which makes correspond to every sequence s = (i ,...,i ) of p + 1 elements of I, a section If P f_ = f. . of F over U. . . The p-cochains form an ® ip iQ ip abelian group, noted C^ (U,F); it is the product group nr(Ug,F) the product being extended to all the sequences

^This is the method by which one obtains relations between the local and global properties of a space.

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s of p + 1 elements of I. The family of the (U,F), p = 0,1,..., is noted C(U,F). A p-cochain is also called a cochain of degree p. A p-cochain f is called ALTERNATING if: (a) f. = 0 each time that two indices i , ...,i 1q ...ip u p

are equal. (b) f. . = E f. if o is a permutation of ^aO ***^ap ^ ^0 •••^p the set {0,...,p} (e^ designates the signature of a). The alternating p-cochains form a sub-group C'^ (U,F)

of the group (U,F); the family of the C'^ (U,F) is noted

C(U,F). 19. Simplicial operations. Let S(I) be the simplex having for the set of vertices the set I; a simplex (ordered) of S(I) is a sequence s = (i^,...,ip) of elements of I; p is

called the dimension of s. Let K(I)= E K (I) be the com- p=0 P plex defined by S(I): by definition, K^d) is the free group having for base the set of the simplexes of dimension p of

S(I) . If s is a simplex of S(I), we denote by |s[ the set of the vertices of s. A mapping h: (I) (I) is called a SIMPLICAL ENDOMORPHISM if (i) h is an homomorphism. (ii) For every simplex s of dimension p of S(I), one g * g I has h(s) = Zg,Cg «S' with e Z, the sum being extended the simplexes s' of dimension q such that |s'|C|s|.

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20. The complexes of cochains s. We apply the preced­ ing to the simplicial endomorphism

3: Kp^i(I) ^ Kp(I),

defined by the usual formula:

i=P+l 4 3(io^,...,ip_i_l) - ( 1)

the sign signifying, as usual, that the symbol above which

it is found ought to be omitted. Thus we obtain a coboundary homomorphism

^3: C^(U,F) ->■ cf*l(U,F)

denoted d; by definition, one has therefore:

(df)io...ip^i = " T (-i)ip (fig.-.ii.-.ip+i, j-0

pj designating the restriction homomorphism

p j : r (Uig ...ij... ,F) ->■ r (Ui^... i^^^ / F) .

Since 3 « 3 = 0, one has d*d = 0. Thus C(U,F) has the properties of a cobordism operator (which makes it a complex.) The g - th group of cohomology of the complex C(U,F) will be noted H^(U,F). One has PROPOSITION 1. H°(U,F) = F (X,F).

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PROPOSITION 2. THE INJECTION OF C'(U,F) INTO C(U,F) DEFINES AN ISOMORPHISM OF H^^(U,F) ONTO H^^U,F) FOR EVERY g > 0. COROLLARY. H^fUfF) = 0 for q > dim(ü)

21. Passage from a covering to a more refined covering. A covering U = is called more refined than a covering

V ={Vj^}jif there exists a mapping t ; I -»■ J such that

U ^ C V^ for every i e I. If f e C^(V,F) , we put:

py designates the restriction homomorphism defined by the

inclusion of U. . into Vt. t. . The mapping f -»■ -rrf 1. o ***1 q 1 o * # * 1 q is a homomorphism of C^(V,F) into C^(U,F) defined for every q ^ 0, and commuting with d, that which defined the homomorphi sm

T*: h ‘5( V , F ) H:S(U,F) .

PROPOSITION 3. THE HOMOMORPHISMS t *: H^^V,F) ^ h‘^(U,F) DEPEND ONLY ON Ü AND V NOT ON THE CHOSEN MAPPING. Thus, if Ü is more refined than V, there exists for every integer q ^ 0 a canonical homomorphism of H^(V,F) into H^(U,F). In the following this homomorphism will be noted

ct(U,V) .

22. Cohomology groups of X with values in the sheaf F . The relation "U is more refined than V" (which we will desig­ nate Ü < V) is a PREORDERING relation between coverings of X;

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moreover, this relation is FILTERING, for if U = V = {V.}. _ are two coverings W = {U. A V.},. .. is a ] ]GJ 1 J llfJ/EZXu covering which is at the time more refined than U and more

refined than V. We will say that two coverings Ü and V are equivalent if one has U < V and V < U. Every covering U is equivalent to a covering U' where theset of indices is a part of V(x); in effect one can take for U' the open SETS of X belonging to

the FAMILY U. DEFINITION. ONE CALLS THE qth COHOMOLOGY GROUP WITH

VALUES IN THE SHEAF F, NOTED h ‘^(X,F) , THE INDUCTIVE LIMIT OF

THE GROUPS h ‘^(U,F) DEFINED FOLLOWING THE FILTERING ORDER OF

THE CLASSES OF COVERINGS OF X WITH RESPECT TO THE HOMOMOR­ PHISMS a(U,V). In other words, an element of H^(X,F) is none other than a pair (U,x) with x e H^(U,F), agreeing to identify two pairs (U,x) and (V,y) if there exists W, with W < U, W < V, and o(W,U) (x) = a (W,V) (y) in H^(W,F). With every covering U of X is thus associated a canonical homomorphism

o(U) : h ‘^(U,F) HS(X,F) . PROPOSITION 4. H°(X,F) = T(X,F).

23. Homomorphisms of sheaves. For every q ^ 0 H^(X,F) is an additive covariant functor of F. Suppose that F is a sheaf of A-modules. The sections of sheaf A on X together define an endomorphism of F, for each H^(X,F). It follows that the H^(X,F) are the modules on the ring r(X,A).

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24. Exact sequence of sheaves; general case. Let OL 3 0->A-»-B->C->0bean exact sequence of sheaves.] If U is a covering of X, the sequence

a g 0 C(U,A) -»■ C(ü,B) -)■ C(ü,C)

is evidently exact, but the homomorphism 3 is not in general

surjective. We will designate by Cq (U,C,) the image of that homomorphism; it is a sub-complex of C(U,C) where the cohomology groups will be noted (U,C). The exact sequence of complexes:

0 -V C(U,A) -> C(U,B) C q (U,C) ->■ 0

gives rise to an exact cohomology sequence:

...h ‘^(U,B) 4. H^(U,C) 2 H9+1(U,A) 4. H9+1(U,B) + ...,

where the cobordism operator d is defined in the usual fashion. PROPOSITION 5. THE SEQUENCE

...h ‘^(X,B) 2 H^(X,C) 2 H9+1(X,A) “ HS*1(X,B) -> ...

IS EXACT. (d designates the homomorphism obtained by passage to

the limit of part of d: H q (U,C) 4- H^^^(U,A)). PROPOSITION 6. THE CANONICAL HOMOMORPHISM H^(X,C)

4- h ‘^(X,C) i s BIJECTIVE f o r q = 0 AND INJECTIVE FOR q = 1.

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COROLLARY 1. ONE HAS AN EXACT SEQUENCE

0 4- H°(X,A) 4- H°(X,B) 4- H°(X,C) 4- H'(X,A) 4- H'(X,B) 4- H'(X,C).^

COROLLARY 2. IF H'(X,A) = 0 THEN r(X,B) 4- r(X,C) IS

SURJECTIVE.

^The obstructions to pulling back global sections of C are measured by elements of H'(X,A).

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SHEAVES AND SEVERAL COMPLEX VARIABLES

There appear to be three eras of major importance in the development of the theory of several complex variables as it pertains to the creation of sheaf theory. First, problems were introduced in the 1880s' which were studied

for about thirty years. The second period began in the late 1920s when interest was revived following the First World War, and continued as solutions were attained over the next twenty-five years. With the solutions in hand in 1950, the subject flourished and became modernized and expanded in the

third (and present) stage.^ The theory of several complex variables had its begin­ nings in the late nineteenth century as mathematicians gener­ alized the concepts and the methods of one complex variable. Difficulties were encountered, for what had been clear for

one variable was muddy for two or more variables. Karl Weierstrass, a professor at the University of Berlin, proved the following theorem in the course of one of his lectures 1213, p. 97] in 1876 (as summarized by Osgood [177, p. 562]):

Lipman Bers [9, p. iii] stated "Every account of the theory of several complex variables is largely a report of the ideas of Oka." According to Robert Gunning and Hugo Rossi [93, p. vii] "...the impressive work of many others during the past decade [1955-1965] and at the present time [1965] is based on this foundation." 24

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Gegeben sei eine unendliche Punktmenge a ^ , , lim a = Jedem Punkte an derselben werde eine n-»-oo ^ natürliche Zahl zugeordnet. Dann gibt es stets eine ganze transzendente Funktion G(z) , welche irti Punkte a^ (n = 1, 2,...) eine Wurzel y^ ter Ordnung besitzt und sonst nirgends verschwindet. Im übrigen wird die allgemeinste derartige Funktion r{z) durch die Formel gegeben. r(z) = e^^^^ G(z), wo G(z) eine spezielle Funktion von der genannten Beschaffenheit und g(z) eine ganze (rationale oder transzendente) Funktion ist. Furthermore, «, g (x) G(z) = IT f (z) = TT (1 - — )e , n=l ^ n=l ^n

+ ••• + n n n n This theorem has as a consequence the important and well known corollary:

Gegeben seien zwei Punktmengen: a^, f a^ m • m f lim a = b,, b_,..., lim b = »; dabei soil nur n=» n=“ kein Punkt der ersten Menge mit einem Punkte der zweiten Menge zusammenfalien. Ferner sei jedem Punkte a , b n “ eine natürliche Zahl y bzw. zugeordnet. Dann gibt es stets eine eindeutige Funktion welche im Punkte a ter ter einen Nullpunkt y und in b einen Pol v n n n Ordnung besitzt, und sich sonst im Endlichen analytisch verbalt und nicht verschwindet. Die allgemeinste derartige Funktion f(z) wird durch die Formel gegeben:

WO g(z), G^(z), Gg(z) ganze Funktionen sind. Dabei liegen die Wurzeln von G^(z) und (z) in den Punkten a^ bzw. b^ und weisen dort je die Multipizitat y^ bzw. auf.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 26 Gustav Mittag-Leffler, a former student of Weierstrass,

extended the theorem on partial fractions of rational func­ tions to transcendental functions [177, p. 565]:

Vorgelegt sei eine unendliche Punktmenge a^, a2 ,•••, lim a = “, sowie, dem Punkte a^ entsprechend, ein n=oo ^ beliebiges Polynom in l/(z - a^):

1 ~ z - a ■*■••• (z - a ) ^n' n n n

Dann gibt est stets eine eindeutige Funktion f(z), welche im Punkte a^ einen Pol mit dem Hauptteil g^ besitzt und sich sonst im Endlichen analytisch verhalt. Eine solche Funktion wird durch die unendliche Reihe gegeben:

f(2) = I ISnlr-è-lT* - n— J. n wo Yjj(z) ein geeignetes Polynom in a bedeutet. Die allgemeinste derartige Funktion F(z) erhalt man dann, indem man F (z) = f (z) + G (z) setzt, wo G(z) eine beliebige ganze Funktion ist. Arbitrary meromorphic functions have two standard

representations. Weierstrass' theorem concerned factoriza­ tion of numerator and denominator, whereas Mittag-Leffler's representation is by partial fractions. (See also [11, p. 74])

Henri Poincaré, Répétiteur an d'École Polytechnique, (apparently beginning his extraordinary interest in problems of n dimension which eventually won for him the King of Sweden prize) proved a theorem analogous to that mentioned above of Weierstrass. In March 1883, Poincaré demonstrated

(by the theory of. harmonic functions) that for two complex

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variables and two regions containing all points at a finite distance in the Xg planes, every meromorphic function is expressible as the quotient of two entire functions. His paper "Sur les fonctions de deux variables" [180] was published by his good friend Gustav Mittag-Leffler in his journal Acta Mathematica. Poincaré stated [180, p. 97]

Voici quel est le problème. Je considère une fonction de deux variables F(X,Y) et je suppose que dans le voisinage d'un point quelcon- N que X^, Y^, on puisse la mettre sous la forme g-, N et D étant deux séries ordonnées suivant les puissances de de X - X q et Y - Y q et convergentes lorsque les modules de ces quantités sont suffisamment petits. Je suppose de plus que, lorsque les modules de X - X q et Y - Y q restent assez petits, les deux séries N et D ne peuvent s'annuler à la fois que pour des points isolés. Je dis G (X Y} que cette fonction peut se mettre sous la forme g (x "'y T' G et G^ étant des séries ordonnées suivant les puissan­ ces de X et Y et toujours convergentes. Ainsi autour du point X^, Y- il existera par hypothèse une region R. où la fonction F pourra se mettre sous la forme de même

autour d'un autre point X-, Y.., il existera une région ^1 R^ on F pourra s'écrire Mais si les deux régions Rq et R^ ont une partie commune, pourra ne pas être la continuation analytique de N q . Tout ce que nous savons, c'est que dans la partie commune aux deux régions, le

rapport ^1 ne devient ni nul ni infini. 1 "o 2. On sait la partie réele u d'une fonction, d'une variable imaginaire x + iy, satisfait à 1 'equation .2 ,2 — ^ ^ = 0, de sorte que l'étude des fonctions d'une dx^ dy^ seule variable se ramène à l'étude d'une attraction

1 9 Now called a polydisc in C .

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s'exerçant en raison inverse de la distance. On a vu dans les derniers numéros des Mathematische Annalen, quel parti M. Klein [115] su tirer de considérations physiques qui sont au fond tout à fait analogues. De même si nous posons; X=x+iy Y=z+it la partie réelle u d'une fonction quelconque de X et Y satisfera à l'équation;

dx^ dy'^ dz^ dt^ de sorte qu'à ce point de vue l'étude des fonctions de deux variables se ramè à celle d'une attraction s'exerçant dans l'espace à quatre dimensions en raison inverse du cube de la distance. M. Kronecker à déjà fait voir (Monatsberichte 1869) que la considération d'une pareille attraction peut être utile au géomètre qui veut étudier les fonctions de plusieurs variables. Je n'emploierai pas cependant le langage hypergéométri- que; je me bornerai à lui emprunter quelques expres­ sions. Ainsi 1'ensemble des points x,y,z,t qui satisfont à l'inégalité:

(1) (x - Xq )^ + (y - yg)^ + (z - Zq )^ + (t - s'appellera une région hypersphérique dont le centre sera X q , Y^, Zq , tg et le rayon R. L'ensemble des points qui satisferont à l'égalité:

(x - Xq )^ + (y - yg)^ + (z - Zq )^ + (t - tp)^ = R^ formeront une surface hypersphérique. The change occurs when going from one to two variables

[180, p. 99]: II y a toutefois une différence essentielle entre cette théorie et celle des fonctions d'une seule variable. Pour que u soit la partie réelle d'une fonction de X et Y, il ne suffit pas qu'il satisfasse à l'équation A u = 0. Il doit en outre satisfaire aux équations sui­ vantes :

A,u = A + 0 4,u = 3!|+â!|=0 ^ dx^ dy^ ^ dz^ dt^

^3^ %3z " ixât ^ ^ iÿSt °

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 9 J'appelerai fonction potentielle toute fonction u gui satisfait à 1'équation A u = 0.^ Je supposerai que ma fonction potentielle est holomorphe pour toutes les valeurs de x, y, z, t, sauf pour certaines valeurs exceptionnelles qui formeront des points singuliers, ou même des lignes et des plages singulières. Toute fonction potentielle qui n'aura aucune singu­ larité à distance finie sera dite entière. Toute fonction potentielle entière qui reste constamment inférieure à une quantité donnée se réduit à une constante. A paper concerning Mittag-Leffler's representation and its extension to two variables appeared alongside Poincaré's

article in volume 2 of Acta Mathematica. "Sur une classe de

fonctions de deux variables indépendants" [4] was written by Paul Appell, professor at the University of Paris and a lifelong friend of Poincaré [68, pp. 1-61]. He extended the

above theorem of Mittag-Leffler for a particular class of uniform functions:

une fonction uniforme F(x,y) des variables indépendantes X et y n'ayant à distance finie d'autres points singu­ liers que ceux des fonctions et telle que la différence F(x,y) - f^(x,y) soit régulière en tous les points singuliers de f^(x,y) à l'exception de ceux de ces points singuliers gui peuvent appartenir à certaines autres des fonctions [4, p. 71]. using methods of Weierstrass.

Poincaré called the solutions of Au = 0 "fonction potentielle." In 1897 [181] he introduced "harmonic fonc­ tion" in a long paper on potential theory in which he proved the convergence of a series in more general situations than had been previously proved.

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Meanwhile, Weierstrass showed that his theorem— any meromorphic function may be expressed as the quotient of two entire functions^— could be extended to a function that has a finite number of essential singularities at a finite 2 distance. As we have seen, Mittag-Leffler generalized this result, proving that a function meromorphic in an arbi­ trary region can be expressed as the quotient of two functions each analytic in the region [168]. Picard, Mittag-Leffler and others investigated special cases in which the number of essential singularities becomes infi­ nite, and in 1884 Mittag-Leffler published the following generalization [177, p. 569]. Vorgelegt sei eine isolierte unendliche Punktmenge {a}; a^, ag,..., dereh Ableitung mit {C} bezeichnet werde. Jedem Punkte a^ werde ferner ein Polynom in l/(z - a^) :

' willkiirlich zugeordnet. Dann gibt es stets eine eindeutige Funktion f(z), welche in jedem den beiden Mengen {a} and {c} nicht angehorigen Punkte z analytisch ist und im Punkt a^ einen Pol mit dem Hauptteil g^ (l/(z - a^)) besitzt. Die Funktion f(z) wird durch eine unendliche Reihe von der Gestalt

n=l n

^The term "meromophic" was introduced in 1875 by Briot and Bouquet [17]. 2 It is difficult to date Weierstrass' theorems as he was more interested in lecturing than in publishing and many of his results were not published until much later according to E.T. Bell [8, p. 492].

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definiert, wobei Yj^(z) eine rationale Funktion von z ist. Wird die Ebene durch die Menge {c} zerstuckelt, so stellt f(z) Stiicke verscheidener monogerer analytischer Funktionen vor. He further generalized Weierstrass' theorem [177, p. 574];

Vorgelegt sei eine isolierte unendliche Punktmenge {a}: a^, ag,...; ihre Ableitung werde mit {c} bezeichnet. Jedem Punkte a derselben werde eine n natürliche Zahl zurgeordnet. Dann gibt es stets eine eindeutige Funktion F(z), welche in jedem von den Punkten der Menge {c} verscheidenen Punkte z analytisch ist und iiberdies im Punkte an eine Wurzel y n - ter Ordnung besitzt, sonst aber nirgends verschwindet. Die Funktion F(z) wird durch ein unendliches Produkt von der Gestalt definiert:

F(z) = I II - ^ 2 - ^ ) “'' e - °n’, n=l ^ ^n

1 ^n . a c k

WO c^ einen bestimmten Punkt der Menge {c} und m^ eine geeignete natürliche Zahl bedeutet. Wird die Ebene durch die Menge {c} zerstuckelt, so stellt F(z) Stücke verschiedener analytischer Funktionen vor. Pierre Cousin, a student of Poincaré and Appell,pub­ lished the fourth part of his thesis (at Caen, October 28, 1893), for which he gained the Doctor of Science degree from

the University of Paris in 1894 [46]. In this last part of of his thesis he extended the Heine-Borel theorem to the case where a finite set of covering intervals can be se­ lected from an uncountable infinite set. In the same thesis Cousin proposed to establish an extension to several vari­

ables of Mittag-Leffler's theorem of 1884, acknowledging

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 2 the earlier work of Poincaré. Cousin, building on results

of Appell, also extended Mittag-Leffler's theorem of 1882 (which also extended Weierstrass' previous work) proving three analogous theorems for n variables. As a consequence he proved that in a polydisc [46, p. 2] "si une fonction de n variables complexes n'admet que singularités non essentielles à l'intérieur de n cercles avant pour centres les n origines et dont chacun a un rayon fini où infini, cette fonction est le quotient de deux séries entières par rapport aux n variables, convergentes à l'intérieur des n cercles." In order to accomplish his proof he proceeded by two stages, each with two parts. First the theorems are derived for any regions (s^y...,s^) interior to (S^,...,S^). For example [46, p. 23]: "Théorème I. Soient et deux aires connexes prises sur les plans respectifs des deux variables x et y; soient s^ et s^ deux aires connexes à contour fermé simple ou complexe, complètement intérieures respectivement à et S^." This is done for n equals 2 and then for any n. Second, a limiting process is used to extend the region from (s^,...,s^) to (S^,...,S^). These processes are separate for circular regions and more general regions [46, p. 21]: Soit F(x,y,...,z,t,u) une fonction de n variables complexes x,y,...,z,t,u, monotrope et sans espace la­ cunaire à l'intérieur d'une région S. Je suppose que pour chaque point (a,b,...,c,d,e) intérieur a S, on connaisse une fonction f^ , ^ ^(x,y,...,z,t,u) monotrope et sans espace lacunaire, définie à l'inté­ rieur d'un cercle a,b,...,c,d,e, „ j ^ intérieur à S, et ayant pour centre le point (a,b,...,c,d,e), et équiva­ lente à F(x,y,... ,z,t,u) à l'intérieur de r, , ^ ,q â/Jj / • # e / Cfd/C

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Une condition nécessaire à laquelle doivent satis­ faire les fonctions f^ . ^ ^(x,y,...,z,t,u) ci f D f m • m fCfU/0 correspondent aux différents points de S, est la sui­ vante; si (a',b',...,c',d',e') est un point assez voisin de (a,b,...,c,d,e) pour être intérieur au cercle â/iDf # • • fC/d/0„ J les deux fonction fa,b,...,c,d,e(X'y'''"'='t'U) et fa',b',...,c',d',e'(x,y,...,z,t,u) doivent être équivalentes au point (a',b',...,c',d',e'); il faut en effet qu'en ce point les deux fonctions soient équivalentes à une même troisième F(x,y,..., Z/tfU). Je me propose de montrer que si, sans se donner la fonction F(x,y,...,z,t,u), on se donne pour chaque point (a,b,...,c,d,e) intérieur à S, une fonction f, . . „ ^ ^(x,y,...,z,t,u) monotrope et sans espace & f D f # • * f C f C l / 0 lacunaire à l'intérieur de a,b,...,c,d,e ^ et si les fonctions données f^ . _ , (x,y,...,z,t,u) satis- â . / D / * * # fC f Q / 0 font à la condition qui vient d'être expliquée, il existe une fonction F(x,y,...,z,t,u) monotrope et sans espace lacunaire, définie à l'intérieur de S, et qui, en chaque point intérieur à S est équivalente à la fonction donnée correspondant à ce point. J'aurai ainsi établi la condition nécessaire et suffisante à laquelle doivent satisfaire les fonctions données, pour qu'il existe une fonction F(x,y,...,z,t,u) définie à l'intérieur de S, et équivalente en chaque point intérieur à S a la fonction donnée en ce point. Il est clair que le théorème bien connu de M. MITTAG-LEFFLER, relatif aux fonctions d'une variable complexe, n'est qu'un cas particulier de celui que je viens d'énoncer. De cette extension du théorème de M. MITTAG-LEFFLER se déduira d'une façon immédiate le théorème de M. POINCARE. His proof of the extension of Mittag-Leffler's theorem and hence of the extension of Weierstrass' theorem depended on. the lemma [46, p. 19]: II existe une fonction ü(x,y) régulière en tout point (x,y) intérieur à (r,S) et telle que en un point quelconque intérieur à S son quotient par la

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fonction Up(x,y) qui correspondent à ce point est régulier et différent de o au point considéré. In 1899 Poincaré modified the method of his 1883 theorem and extended the results to n variables in another paper published in Acta Mathematica "Sur les propriétés du potentiel et sur les fonctions Abéliennes" [182]. The end of the first era in analysis of several com­ plex variables came when two papers were published which revealed more clearly the distinctive properties of several complex variables. In 1906, Fritz Hartogs, Privatdozent at München, published his 1903 dissertation "Zur Theorie der

analytischen Funktionen mehrerer unabhangigen Veranderlichen, insbesondere über die Darstellung derselben durch Reihen, welche nach Potenzen einer Veranderlichen fortschreiten [95]" in which he studied the conditions of convergence of the entire series of two variables: Bei einer Potenzreihe zweier Veranderlichen

00 B(x,y) = a/^^x^y^ in ausser der Konvergenz der y, v=o Doppelreihe selbst vornehmlich die der Zeilenreihe

CO / CO y^ sowie die der Diagonalrenreihe

v= 0 v=o X X — y y von Interesse. y X=o y=o In 1908 he published "Uber die aus den singularen Stellen einer analytischen Funktionen mehrerer Veranderlichen bestehenden Gebilde" [96]. Analytic continuation was one

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of the basic concepts of the theory of one complex variable which Weierstrass created and lectured about at the Univer­ sity of Berlin from 1857 to 1890. Hartogs discovered re­ sults about analytic continuation and natural boundaries which are true for two variables but false in one variable! For two variables and z^ consider the union A of two compact sets |z^J 1, jzgl ± 1 and z^ = 0, \z^\ ^ 1/2. Every function holomorphic in A can be extended to a holomorphic function in the compact polycylinder |z^| £ 1, jzgl £ 1« For n = 1, every domain is the total domain of existence of some holomorphic function [96]. In 1911 E. E. Levi generalized Hartog's results to the case of meromorphic functions, and introduced the notion of pseudo-convexity [15]. The characterization of domains of holomorphy came to be known as the Levi problem. In a talk "On the expressibility of a uniform function of several complex variables as the quotient of two functions of entire character" [82] given to the American Mathematical Society on October 25, 1913, T. H. Gronwall discussed

Cousin's, memoir. He pointed out that Cousin had general­ ized Mittag-Leffler's theorem in a rigorous way, but that

the proof of Cousin's main theorem was not valid. Thé difficulties arose in his proof of the generalization of Weierstrass' theorem. Gronwall explained the nature of the error. He showed that Cousin's proof would be valid if all, or all but one, of the regions are simply connected. He

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provided a counter-example when any of the regions are multiply connected. The proof of the extension of Weierstrass' theorem depended on the following lemma. Quoted from Gronwall it is a variation in language on Cousin's Lemma 7 [82, p. 53]: 2. THE DOMAIN OF VALIDITY OF COUSIN'S PROOFS OF THEOREM B To abridge the notation, we shall write x for the system of n - 1 variables x^, Xg,...,^^ _ ^ and S for (S^, S2 ,...,S^ j) Î x^ will be denoted by y and S^ by S'. A simply connected part Z of S we define as a system of regions (Z^y Z2 /...,2^ _ where, for V = 1,2,...,n - 1, every interior or boundary point of the simply connected region Z^ is interior to or on the boundary of S^. The boundaries of S^, S2 ,...,S^ _ Z^, 22''"''^n - 1' S' are assumed to be regular, that is, each is to consist of a finite number of pieces of analytic curves without singular points. We now assume S' to be subdivided, by a finite number of pieces of regular curves into a finite number of simply connected regions R^, R 2 ,...,Rp, ... . When R^ and Rp are adjacent regions, we denote by their common boundary, or, should this consist of several pieces, any one of these. If any is a closed curve, we cut it at three points, thus obtaining three pieces such that no two of them taken together from a closed curve. The direction of & np_ is that which leaves the interior of the region R^ to the left, so that and I are the same curve described in opposite directions, pn Finally, let T^^ consist of all points in the y-plane interior to at least one circle with center on & np_ and sufficiently small radius r, this r being constant not only for different points on but also for all the various curves Inp The proof of Theorem B now depends on the following lemma: Let a function Up(x,y) be given for every region R^, uniform and holomorphic in (S,Rp), boundaries included.

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and such that for any two adjacent regions and R^, the quotient u (x,y) u^(x,y) “ 9np(*'y) is holomorphic and different from zero in (S,T^p). Then there exists a function G(x,y) holomorphic in (S,S'), uniform in (Z,S'), where Z is any simply con­ nected part of S, and such that in (S,Rp) (boundaries included, except those y which are end points of an &^p and lie on the boundary of S') the quotient G(x,y) Up(x,y)

is holomorphic and different from zero. When S is simply connected, we may evidently let Z coincide with S. In his formulation of the lemma (i.e., §7; proof in §6) Cousin makes no distinction between Z and S, so that, when S is multiply connected (that is, one at least of S^, ^ is multiply connected) he tacitly assumes the function G(x*y) to be uniform in (S,S'), while the uniformity is proved only in (Z,S'). This constitutes the gap in Cousin's proofs. Gronwall, an instructor in mathematics at Princeton, ex­ pected to continue this work, stating at the end of his paper [82, p. 64]: In a subsequent paper, it will be shown that this representation as the quotient of two functions of entire character with common divisor is possible for any function f(x,y), meromorphic everywhere at finite distance except at the points defined by G(x,y) = 0, where G(x,y) is an entire function. The common divisor cannot in general be removed except when G(x,y) is irreducible. However, there are no further publications on this mate­ rial.^

^Gronwall became editor of the Annals in 1913.

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Up until this time, communication of the problems suffered no impediments, and they were worked on by a number of people, including William Fogg Osgood, professor of mathematics at Harvard, noted for his commitment to rigor in current research. But international communication broke down during the dire circumstances of the First World War, and energies were sapped. Moreover, in France, where mathe­ maticians served at the front, the very production of mathematics were altered.^ After Hartogs there was no

progress for about twenty years. The second period began after 1926, when Gaston Julia, professor at the University of Paris, published "Sur les families de fonctions analytiques de plusieurs variables" in Acta Mathematica [109]. His purpose was [109, p. 54]: "... 1'étude des points où une famille de fonctions de plu­ sieurs variables, holomorphes dans un domaine D cessait

J. Dieudonné wrote [52, p. 140] "the 1914 war, we can very well say, was extremely tragic for the French mathema­ ticians.... In the great conflict of 1914-1918, the German and French governments did not see things in the same way where science was concerned. The Germans put their scholars to scientific work, to raise the potential of the army by their discoveries and by the improvement of inventions or processes, which in turn served to augment the German fighting power. The French, at least at the beginning of the war and for a year or two, felt that everybody should go the front; so the young scientists like the rest of the French, did their duty at the front line. This showed a spirit of democracy and of patriotism.that we can only re­ spect, but the result was a dreadful hecatomb of young French scientists. When we open the war time directory of the École Normale, we find enormous gaps which signify that two thirds of the ranks were mowed down by the war."

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. d'être normale, devait donner des résultats analogues à ceux qu'ont obtenus M. M. Hartogs et E. E. Levi...." Two centers became active, building on the work of Levi and Hartogs. One was in Paris where Julia and his student Henri Cartan [41] explored "Julia problems."^ The other was in Munster under Heinrich Behnke. His earliest student, Peter Thullen, became his collaborator and helped him to develop the theory of domains and envelopes of holomorphy. The theory of several complex variables has had other major centers where mathematicians carried out their investigations. Besides the Mathematische Institut at Munster under Behnke since 1907, work proceeded at the Institute for Advanced Study at Princeton since 1932 under Solomon Bochner, in Hiroshima under Kiyoshi Oka, and in Rome under Francesco Severi. Work done at each of these locations will be described below. In 1932, Henri Cartan, then maître de conférence a Faculté des Sciences de Strasbourg, and Peter Thullen put their ideas together and published "Zur Theorie der Singularitaten der Funktionen mehrer komplexen Verander­

lichen. Régularités — und Konvergenzberiche" , [34]', giving a characterization of a domain of holomorphy by holomorphic

Beginning in 1934 Gaston Julia held a seminar which consisted in studying in a systematic way [52, 141] "the great new ideas which were coming from all directions. This is when the idea of drawing up an overall work which, no longer in a seminar, but in book form, would encompass the principle ideas of modern mathematics. From this was born the Bourbaki treatise."

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convexity. Both Cartan and Thullen [199] had published ear­ lier papers that led them to this collaboration. In 1931 Cartan had shown for the first time [35] "Qu'un domaine d 'holomorphie possède certaines propriétés de 'convexité' par rapport aux fonctions holomorphes. Cette notion de 'convexité' s'est depuis lors montrée féconde et elle est devenue classique." He proved [35] "La 'convexité' est non seulement nécessaire pour que D soit un domaine 1'holomor­ phie, mais qu'elle est suffisante pour certains domaines d'un type particulier (par exemple les domaines cerclés). Qu'elle soit suffisant dans le cas général a été démontré peu après par P. Thullen." In 1932 they proved that if a domain D is convex with regard to polynomials or to the rational functions in x,y then the theorem Cousin tried to prove about the existence of a meromorphic function with specified poles is true for D.

Systematization of the theory of several complex variables occurred in 1935 when Behnke and Thullen published Theorie der Funktionen mehrerer Komplexer Veranderlichen [7]. The bibliography is a complete listing of all articles in the theory from 1905 to 1934. Sixty authors made contri­ butions. Their intention was expressed as "Die vorliegende Schrift steht in ihrem Aufbau zwischen einem Lehrbuch und einem enzyklopadischen Bericht [7, p. Vorwort]."^ At that time

In the forward to the first edition [7, p. Vorwort]. 'Nun dart keinswegs übersehen warden, dass mehrere allgemein verbreitete Gesamtdarstellungen vorliegen. Als altestes Lehrbuch ist hier zu nennen: FORSYTHE:

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there were three major unsolved problems: the problems raised by Cousin, by Levi and Runge. The systematization of Behnke and Thullen seems to 2 have revived or organized endeavors for solutions. The mathematical efforts were as international as the politics of the 1930s. Among the leaders were Henri Cartan, a Frenchman who turned 30 in 1934, and Kiyoshi Oka, a 33 year old Japanese. Oka received his doctoral degree at Kyoto University in 1923 and then studied in Paris until 1932.

When he returned to Japan he developed new ideas and originated new methods which eventually solved the out­ standing problems. These will be discussed below in the order in which he solved them.

Cartan wrote in his note [20] "Les problèmes de Poincaré et de Cousin pour les fonctions de plusieurs vari­

ables complexes," published in Comptes Rendu in 1934, that the question remained of knowing when a function of n complex variables, meromorphic in a domain D, can be put into the form of a quotient of two holomorphic functions

Theorie of functions of two variables. Cambridge 1914, sodann der Enzyklopadieartikel von BIEBERBACH, abge- schlossen in Jahre 1922. Wenige Jahre spater (1924, in zweiter Auflage 1929) erschien dann das Werk von OSGOOD; Lehrbuch der Funktionentheorie Band II, eine Darstellung, der wohl die meisten von uns ihre Einfiihrung in diesen Stoff verdanken. Schliesslich hat F. Severi 1931 einen Bericht verfasst, der die rasche Entwicklung der Untersuchungen lebhaft widerspiègelt." 2 "By this time the peculiarities of several complex variables were well exposed and the central difficulties stated" 190, p. vii].

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 2 in D. The classification "Premier problème de Cousin" and "Deuxième problème de Cousin" possibly originated in this paper [20, p. 1285],^ Premier problème de Cousin — On suppose que le do­ maine considéré D est recouvert à l'aide d'une infinité dénombrable de domaines partiels intérieurs à D, et que, dans chaque D^, on a défini une fonction méromorphe f^; on suppose en outre que, chaque fois que deux domai­ nes et Dj ont une partie commune , la différence f^ - fj est holomorphe dans D^j. On se propose de trouver une fonction F, méromorphe dans D, et telle que, dans chaque D^, la différence F - f^ soit holomorphe. Deuxième problème de Cousin — Mêmes hypothèses que les f^ sont remplacées par des ^ : (f>j est supposé holomorphe et jamais nul. On se propose de trouver une fonction $, holomorphe dans D, et telle que, dans chaque D^, le quotient $ : <|)^ soit holomorphe et non nul.

Cartan pointed out that if the second theorem is true for a domain, the theorem of Poincaré is true for that domain, but not conversely. The theorem of Poincaré is true for all the circular domains [15, p. 8],

{z|(|z^ - 1,...,I - z^^O)|)E E where E is a set

in the (|z^ - z ^ | ,...,|z^ - space

In a footnote in [22, p. 1] written in 1940, H. Cartan referred the reader back to this 1934 paper for terminology.

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and the domains of Hartogs

{ (z,w) |z e D, r(z)

tions

|w| = R(z)

jw| = r(z)

when the theorems of Cousin are not true. The theorem of Poincaré is true for the hypersphere as well as the theorems of Cousin. The hypersphere and the polycylinder are special circular domains. The polycylinder: {z| |z^J< r^^, = r = r this is the ball of radius r in the ^1 ^2 n maximum norm. The polycylinder is the direct product of

the n discs

{z- z^I< r^} X ... X {z^l

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The hypersphere: {z| |z^|^ + ... + |z^J^ < r}. (This is the ball of radius r in the euclidean norm.) For n = z.

Zll

For complex dimension 1 both hypersphere and polycylinder coincide with the circle. For higher dimension they take with equal right the place of the circle, but they cannot ever be mapped holomorphically onto each other. In his proof Cartan wanted to use the integral of André Weil, whereas Cousin had used the Cauchy integral (unsuccessfully): Ce résultat, qui dépasse de beaucoup celui de Cousin, s'obtient par une méthode analogue à la sienne; mais il faut se servir de 1'intégrale d'André Weil (pour les fonctions de plusieurs variables) tandis que Cousin utilisait seulement l'intégrale classique de Cauchy (pour les fonctions d'une variable). "L'intégrale d'André Weil" was in a note in Comptes Rendu [215, p. 1304-1305]. There Weil referred to the earlier paper of Cartan, "Sur les domaines d'existence des fonctions de plusieurs variables complexes" [19] of 1931, referred to above1 H. Cartan, dans un intéressant Mémoire, a souligné 1'importance, pour la théorie générale des fonctions de plusieurs variables complexes, de l'extension à ces fonctions des résultats connus sur le développement en séries de polynômes, dans un domaine donné, des fonc­ tions d'une seule variable. Il a donné aussi, dans ce même Mémoire, une condition nécessaire à laquelle doit satisfaire un "domaine d'holomorphie" A pour que toute

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fonction holomorphe dans A y soit développable en série de polynômes: il faut que A soit "convexe par rapport à la famille des polynômes." Or cette condition est non seulement nécessaire, mais aussi suffisante. C'est ce qui résulte en effet du théorème suivant, que je me borne à énoncer dans le cas de deux variables, et qu'on vérifie sans difficulté; Soient X^, X^, ..., X^ n polynômes en x, y, et soit D le domaine (s'il existe) qui est défini par les inéga­ lités (1) lx^(x,y)l

2 * ' ^ ç ç j ( ? f n 7 X, y ) f ( Ç , Ti ) dÇdri (2*1)2 JL/ Jjo. [Xi(S,n)-Xi(x,y)][Xj(S,n)-Xj(x,y)] " (i/j) ij si (x,y) est intérieur à D: = 0 si (x,y) est extérieur à D [215, p. 130]. But the mathematics was not available to Cartan as he wrote later in his "Analyse des Travaux;" "J 'avais vu que problème additif pouvait se résoudre in utilisant l'inté­ grale d'André Weil, mais comme à cette époque il manquait certaines techniques permettant d'appliquer l'intégrale de Weil au cas générale des domaines d'holomorphie, je renonçai à publier ma démonstration." [35, p. 9J. Nevertheless he discovered that although for two variables the first problem of Cousin did not have a solution for a domain that was not a domain of holomorphy, for three variables the situation is quite different. In "Sur le premier problème de Cousin

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 6 "[21] published in Comptes Rendu in 1938, he gave [21, p. 538] "Le premier exemple d'ouvert qui n'est pas domains

d'holomorphie et dans lequel cependant le problème additif de Cousin est toujours résoluble; il s'agit de C^ privé de l'origine." Other new concepts entered into the endeavors. In 1933 Walther Ruckert^ in Heidelberg had taken the concept of ideal from polynomial rings and interpreted it in the ring of functions on a fixed domain. He was studying the problem of limits of convergence of several complex variables [186,

p. 259]: In der Eliminationstheorie der konvergenten Potenzreihen mehrerer komplexen Veranderlichen werden die gemeinsamen Nullstellen eines Systems in Nullpunkt verschwindender Potenzreihen (1) P^(x^,X2,...,x^) (i “ 1,2,3,...,m) untersucht. Das Hauptergebnis der Theorie ist das folgende Theorem von Weierstrass; Die gemeinsamen Nullstellen des Systems (1) in der Umgebung des Nullpunktes bilden eine endliche Anzahl irreduzibler analytischer Gebilde, die durch den Null­ punkt gehen. Die bisheringen Beweise dieses grundlegenden Eergebnisses stiitzen sich vorwiegend auf Hilfsmittel aus der Funktionentheorie. Die Gebilde werden aber in der folgenden algebraischen Darstellung gewonnen. (2) f(w) s 0)^ + ojP " ^

+ ... + a^(^^,^2/***/— 0

Riickert's sources for "Zum Eliminationsproblem der Potenzreihenideale" were B. L. van der Waerden Moderne Algebra, Springer, Berlin, Bd I (1930)u. Bd II (1931); E. Noether "Abstrakter Aufbau der Idealtheorie in alge­ braischen Zahl-und Funktionenkorpern," Math. Annalen 96 (1927), S. 34 u. 35; E. Noether, "Idealtheorie in Ringbereichen," Math. Annalen 83(1921),.s. 24-66," [186, p. 261, p. 265]."

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sei eine irreduzible algebraische Gleichung mit konvergenten, im Nullpunkt verschwindenden Potenzreihen von 5^, ... f als Koeffizienten; es sei ferner

D{C^,?2 ' •••/ 5]^) die Diskriminante von f(w) und

(3) D ( ^2 / * • • » ~ ( i — 1,2,3,...,s) ein System ganzer rationaler Funktionen von w wieder mit konvergenten, im Nullpunkt verschwindenden Potenz­ reihen von ^2 ' f als Koeffizienten; dann bestimmen die sich fur die Umgebung des Nullpunktes im Raume der Variabeln 52'"'''^k (2) und (3)

ergebenden Werte der ^2 ' " ' ' irreduzibles analytisches Gebilde von der Dimension K im Raume von n = k + s Veranderlichen. In dieser Arbeit wird gezeigt, dass eine sachgemass Behandlung des Eliminationsproblemes bis zu der algebraischen Darstellung (2) und (3) der Gebilde nur formale Methoden, also keine funktionentheoretischen Hilfsmittel benotigt. Als solche Methoden erweisen sich die allgemeine Idealtheorie und die allgemeine Korpertheorie. Zunachst geben wir der Eliminationsaufgabe eine zweckentsprechends Formulierung. Es ist klar, dass mit den Potenzreihen des Systems (1) auch jede Reihe von der Form (4) P(x^,X2, ,x^) = m ^ (x^,X 2 ,...,x^) 'P(x^,X2 ,...,x^) 1— 1 mit irgendwelchen (x^,X2 »•..,x^) aus dem Bereiche aller konvergenten Potenzreihen in n Veranderlichen innerhalb einer bestimmten Umgebung des Nullpunktes in den gemeinsamen Nullstellen von (1) verschwindet. Die Gesamtheit (4) ist aber gerade ein Ideal in diesem Bereich. Man wird daher ein Ideal zugrunde legen und nach den gemeinsamen Nullstellen eines Ideals im Bereich der konvergenten Potenzreihen in n Veranderlichen fragen. Der Aufbau der Arbeit ist dann der folgende: In §2 entwickeln wir die Arithmetik im Berich der konvergenten Potenzreihen in n Veranderlichen; wir bezeichnen diesen Bereich mit I n . Der §3 handelt von der Idealtheorie in I . Es wird der Basissatz fur n Ideale bewiesen, wodurch der Zerlegungssatz der all- gemeinen Idealtheorie auf die Ideale von I^ anwendbar wird. Jedes Ideal ist aufspaltbar in endlich viele Primarideale mit zurgehorigen eindeutig bestmmten

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 8 Primidealen. Das Eliminationsproblem allgemeiner Ideale ist hiermit auf das Eliminationsproblem von Primidealen zurückgeführt. In 14 führen wir die formale Elimination eines Primideals p aus durch Restklassen- bildung in I nach p und gewinnen so das zu p gehorige Gebilde in algebraischer Darstellung. Damit ist das aufgestellte Ziel erreicht. Es bleibt in §5 noch zu zeigen, dass in der Tat die durch die algebraische Darstellung gelieferte Punktmannigfaltigkeit ein- schliesslich der Grenzstellen genau alle gemeinsamen Nullstellen des zurgehorigen Primideals ausmacht und ferner ein irreduzibles analytisches Gebilde ist. Als Answendung folgt in §6 das Theorem von Weierstrass. Dieser Aufbau lehnt sich an eine Arbeit von B. L. van der Waerden an. die von dem Nullstellenproblem bei Polynomidealen handelt. Cartan used this idea of Riickert's In his 1940 paper "Sur les matrices holomorphes de n variables complexes" [22]. This paper's importance was in part recognized by its dedication to his father Elie Cartan on his 70th birthday. Cartan realized at the time that "un certain lemma sur les matrices holomorphes inversibles joue un rôle décisif dans ces Cousin questions" [35, p." 9], as he had indicated.

Dans ce travail il sera question de matrices à p lignes et p colonnes dont les éléments sont des fonctions holomorphes de n variables complexes x^, ..,,x^ dans une certaine région A de 1'espace de ces variables; le déterminant de ces matrices sera supposé différent de zéro en tout point de A. Lorsque p = 1, on retombe sur le cas d'une fonction holomorphe et non nulle sur A. Or, relativement à ce cas particulier, on connaît un théorème dont Cousin a mis en évidence le le rôle important dans la recherche des fonctions holo­ morphes admettant des zéros donnés. Voici ce théorème, que, pour simplifier, j'énonce dans le cas d'une seule variable complexe x. Étant donnés, dans le plan (x), deux domaines A' et A" limités par des courbes régulières et dont l'intersection A est simplement connexe, toute fonction f(x) holomorphe et non nulle dans A et sur sa frontière peut être mise sous la forme du quotient d'une fonction holomorphe et non nulle dans A" (frontière comprise) par une fonction holomorphe et non null dans A' (frontière comprise).

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Ce théorème se démontre facilement en considérant le logarithme de f (x) dans A, et en se servant du fait que toute fonction holomorphe dans A est la différence de deux fonctions holomorphes dans A" et A ' respective­ ment. Nous nous proposons de généraliser le théorème précédent au cas d'une matrice holomorphe de détermi­ nant non nul. Cette généralisation n'est pas triviale, car le procédé qui consisterait à prendre le logarithme de la matrice donnée (ce qui d'ailleurs ne peut se faire sans précaution) ne conduirait pas au but, l'exponen­ tielle d'une matrice A ne jouissant pas de la proprite A B A + B fondamentale e e = e . L'énoncé précis du théorème sera donné au §4 (théorème I); on s'y est affranchi notamment de la restriction relative aux courbes "régulières" limitant les "domaines" A ' et A" Notre théorème semble susceptible de jouer un rôle important dans l'étude globale des idéaux de fonctions holomorphes. Remarquons à ce propos que le "deuxième problème de Cousin" se rapporte à l'étude globale des idéaux qui ont, au voisinage de chaque point, une base formée d'une seule fonction holomorphe. En dehors de particulier, on n'a pas encore abordé, semble-t-il, l'étude globale des idéaux. C'est ce que nous ferons systématiquement dans un Mémoire ultérieur. Ici, nous nous bornerons à quelques applications immédiates de notre théorème I; elles pourront servir ensuite de point de départ pour une théorie systématique. Qu'il me soit permis d'adresser mes vifs remerciements à M. H. Villat qui a bien voulu accepter de publier ce travail dans le Volume de son Journal dédié aux deux savants français E. Borel et E. Cartan [22, p. 1]. On page 9 Cartan stated his fundamental theorem: THÉORÈME I. - Soient, dans l'espace de n variables complexes, deux polycylindres compacts A' et A" qui ont mêmes composantes dans les plans de toutes les variables sauf une, et dont l'intersection A' n A" = A est simplement connexe. Toute matrice A holomorphe et inversible sur A peut être mise, sur A, sous la forme A = A' l'A", A étant une matrice holomorphe et inversible sur A ', et A" une matrice holomorphe et inversible sur A".

and then asked A quelle condition deux idéaux I' et I" de bases finies sur A ' et A" respectivement admettent-ils une base unique, holomorphe sur la réunion A ' U A"?

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He resolved this question with

THÉORÈME II. A', A" et A ayant la même signification qu'au théorème I. considérons, sur A ' et A" respective­ ment, deux idéaux I' et I" de bases finies. Pour que I' et I" admettent une même base holomorphe sur la réunion A' U A", il faut et il suffit que I' et I" engendrent le même idéal sur l'intersection A. He made various applications and proved [22, p. 22] "Si des fonctions f^,...,fp sont holomorphes et n'ont pas de zéro commun sur un polycylinder compact et simplement connexe A,

il existe p fonctions c^,...,Cp holomorphes sur A telles que

=1*1 + ••• * V p ' Meanwhile tlie three problems which were unsolved at the time Behnke and Thullen wrote their book were being intensively studied by Kiyoshi Oka, then an assistant pro­ fessor at Hiroshima Bunrika University.^ In 1936 he published "Domaines convexes par rapport aux fonctions rationelles [172]," in the Journal de Science Hiroshima University, which included a "beautiful induction on poly­

nomial polyhedra" [93, p. 63], and a second memoir "Domaines d'holomorphie" in 1937 [171]. After many unsuccessful attempts to generalize the theorem that Cousin had patched together Oka succeeded in his third memoir "Deuxième problème de Cousin" [172], published in 1939. Shortly thereafter the outbreak of World War II complicated the international situation for mathematicians. As would be expected it isolated them so that Oka, a Japanese, knew only what his French colleagues

^Oka later was professor at Nara Women's University. .

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had done prior to Pearl Harbor, December 7, 1941, and his French colleagues lost track of him.^ But the situation in France remained a vigorous one for mathematicians despite the fall of France in 1940. Bourbaki developed when mathe­ maticians had the need to share ideas, and the organization carried on through the war. In Japan Oka continued work on the same problems, producing "Domaines d'holomorphie et domaines rationnellement convexes" [173] in 1941 as well as "L'intégrale de Cauchy" in the Japan Journal of Mathematics [175]. In 1942 he published "Domaines pseudoconvexes," [176], Some ideas were apparently developed in personal correspondence with other 2 Japanese mathematicians. However he lost touch with the work Cartan was doing. In 1944 Cartan published a memoir "Idéaux de fonctions analytiques de n variables complexes" [23] in Annales de Science École Normale Supérieur which clarified the rela­ tions among all the problems. There he stated the Cousin problems in terms of ideals and modules and concluded [23, p. 187] : Nous signalerons seulement deux des problèmes essentiels à résoudre.

Kunihiko Kodaira, a young Japanese mathematician whose work is discussed below, was also "cut off from western mathematics" by the "turbulent conditions of the world," as Hermann Weyl remarked when presenting him with the Fields Medal in 1954 [231]. 2 In a footnote on page 127 Oka wrote "L'auteur I'a écrit aux détails en japonais à Prof. T. Takagi en 1943."

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PREMIER PROBLEME. - Les modules dérivés (ponctuels) d'un système fini quelconque de fonctions holomorphes forment-ils un système cohérent? On peut voir facile­ ment qu'il suffirait de résoudre ce problème dans le cas où les fonctions f^,...,f^ sont à valeurs complexes (c'est-à-dire de dimension q = 1). Et l'on voit aussi qu'il suffirait de résoudre le problème suivant: I désignant un idéal de base finie, et f une fonction holomorphe (à valeurs complexes), considérons, en chaque point X , l'idéal I^ des fonctions qui appartiennent à I^ et sont divisibles par f; ces idéaux forment-ils un système cohérent? Autrement dit, si a désigne un point particulier, une base finie de I^ engendre-t-elle I^ en tous les points x suffisamment voisins de a? Si ce problème pouvait être résolu, des déductions que nous n'exposerons pas ici montrent que le théorème III du paragraphe XI s'étendrait à tous les systèmes cohérents de modules ponctuels, sans exception. D'une façon précise, les trois énoncés suivants seraient vrais: "Tout système cohérent de modules ponctuels, sur un polycylindre compact et simplement connexe, est engendré par le module associé à ce système." "Deux modules (sur un polycylindre A conpact et simplement connexe) qui engendrent en chaque point de A des modules identiques, sont identiques." "Tout module, sur un polycylindre compact et simple­ ment connexe, a une base finie." DEUXIÈME PROBLÈME. - Soit V une variété analytique au voisinage d'un point a, I^ l'idéal de cette variété au point a (c'est-à-dire l'idéal des fonctions, holo­ morphes au point a, qui s'annulent identiquement sur V dans un voisinage de a.) Une base finie de I^ engendre-t-elle, en tout point x de V suffisamment voisin de a, l'idéal I^ de la variété V au point x? Si ce problème pouvait être résolu par l'affirmative, ainsi que le 'premier problème,' l'énoncé suivant serait exact: "Toute variété analytique sur un polycylindre A compact et simplement connexe, peut être obtenue comme l'ensemble des zéros communs à un nombre fini de fonctions holomorphes sur A." Ce résultat n'à été démontré ici que dans le cas particulier des variétés analytiques considérées au paragraphe X. Jean Leray, winner of the 1938 Prix Internationaux de Mathématiques de Malaxa, was professor in Nancy until 1941 and then in Paris. In 1944 while a prisoner in Oflag XVII

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he wrote "Sur la forme des espaces topologiques et sur les points fixes des representation" [140]. Three articles appeared on algebraic topology. In the first Leray provided

a brief history indicating the background people for his ideas — Hopf, Brouwer, and Lefschetz as well as E. Cartan (Betti numbers) and De Rham. In summary he introduced [144, p. 1419] "une notion beaucoup plus maniable, celle de couver­ ture, qui appartient à la topologie algébrique; cette notion et celle d'intersection de complexes, que je crois origina­ les, fournissent une définition de l'anneau d 'homologie extrêmement directe." In 1946 Leray published two notes, "L'anneau d 'homo­ logie d'une représentation" [143] and "Structure de l'anneau d'homologie d'une representation" [144], In the first of these papers he stated [143, p. 1366]T 1. Définitions préliminaires. - Un faisceau 8 de modules (ou d'anneaux) sera défini sur un espace topologique E par les données que voici: 1° à chaque ensemble fermé F de points de E est associé un module (ou un anneau) qui est nul quand F est vide; 2° à chaque couple d'ensembles fermés, f et F, de points de E. tels que f C F, est associé un homomorphisme de gp dans g^, qui transforme un élément b^ de g^ en son intersection b^'f par f; si f C f C F et si bpEgp on doit avoir (bp*f)*f = bp*f. Le faisceau g est dit normal quand il possède les deux propriétés suivantes: 1° si b^Eg^, il existe un voisinage fermé V de F et un élément b.^ de g.^ tel que bp = b^-F; 2® si bpEgp, si f C F et si bp'f = 0, alors f possède dans F un voisinage fermé v tel que bp*v = 0. Exemple : Les classes d'homologie à p dimensions des ensembles fermés de points d'un espace E constituent un faisceau que nous

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nommerons faisceau d'homologie de E; si E est normal, ce faisceau est normal.^ In this way "faisceau" entered mathematics. In the second paper he derived applications and gave more details of the structure, going on to study the local properties of a continuous mapping and the global coho­ mologies. These papers are discussed in Chapter 2. At the end of World War II France was a center of creative mathematics, which is not surprising historically. Cousin problems remained of interest. According to Roger Godement "Une demonstration du théorème de Rham par A. Weil, datant, ait joué un role important dans l'évolution de la situation" [77]. He is apparently referring to the extract of a letter André Weil, who was then living in Sâo Paulo, wrote to G. De Rham. De Rham on November 8, 1946, asked that it be published, and consequently it appeared as "Sur la théorie des formes différentielles attachés à une variété analytique complexe" [220].

In 1948-1949 the first Séminaire Cartan was held at L'École Normale Supérieure. The participants included H. Cartan, J.-P. Serre, J. Cerf, P. Samuel, J. Dixmier and J. Frenkel. The subject was algebraic topology. At that time "faisceau" theoretic terminology was developed. These seminars will be discussed in Chapter 2. "Faisceau" there generalized the idea of chain and cochain.

^S. Eilenberg translated "faisceau" as "bundle" in his review in Math. Reviews 1946.

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Meanwhile, Oka, not knowing of Cartan's 1944 paper which presented all the material for a solution, had systematically continued his own work, bringing into the theory of analytic functions brilliant new ideas based primarily on the earlier work of Cartan. In 1948 he solved the first Cousin problem, using concepts of congruence and ideal taken from a field of polynomials to analytic functions, which he knew about from Cartan's 1940 paper. The problems he then encountered required for a solution both this idea of Cartan's and the consequent theorems Cartan had proven. Oka wished to extend them to the "ramified domain" and pointed out that this technique was also useful for the less complicated domains. The mémoire "Sur quelques notions arithmétiques" {176] was published in 1950 in volume 78 of the Bulletin de la Société mathématique de France. It began with a statement that Cousin problems C^ and C^ arise when one wants to go from given local to global conditions [176, p. 2].

PROBLÈME (C^). - Étant donné un système fini de fonctions holomorphes (F^, ..., F^) et une fonction holomorphe $ (x) au voisinage d'un ensemble fermé borné E, telle que = o[mod(F)] en tout point de E, trouver p fonctions A^(x) holomorphes au voisinage de E, de façon que 4» = 2 A.F. identiquement, i ^ ^ PROBLÈME (Cg). - Soit donné, au voisinage d'un ensemble formé borné E de l'espace (x), un système fini de fonctions holomorphes (F^y ..., F^); supposons atta­ ché à chaque point P de E un polycylindre (y) autour de P et une fonction $(x) holomorphe dans (y); supposons que, pour tout couple de polycylindres (y), (y) d'inter­ section (ô) non vide. On ait la relation ^(x) s (x)

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[mod(F)] entre les fonctions correspondantes en tout point de (6). On se propose de trouver une fonction $(x) holomorphe au voisinage de E, telle que (x) = (j) (x) [mod(F)] en tout point P de E. Two systems are equivalent in a domain [176, p. 3]: Considérons alors deux systèmes finis de fonctions holomorphes dans un domaine D de l'espace (x), soient (f^, ..., fp) et g). Supposons les deux relations (1) (j)^ E o [mod(f)], fj 5 o [mod(ij))] (i = 1, ..., q; j = i, ..., p) globalement pour D. Nous dirons alors que les systèmes (f) et (ij)) sont equivalents dans D, et nous écrirons les relations (1) sous la forme (f) - (*); le seul premier groupe des relations (1) s'écrira (4>) C (f). Nous dirons que deux systèmes de fonctions (f) et ((f) sont équivalents en un point P de D, s'ils sont équivalents dans un voisinage de P: de même pour (<()) C (f) en un point P. This equivalence suggests problem E [176, p. 3] : PROBLÈME (E). - Les notations E, (y), (S) ayant la même signification que dans le problème (Cg), on suppose donné, pour chaque (y), un système fini de functions holomorphes (f), de manière que dans chaque intersection non vide (ô) la relation (f) (f) ait lieu en tout point de S entre les systèmes (f) et (f) correspondant aux polycylindres (y) et (y') d'intersec­ tion (ô). On se propose de trouver un système fini de fonctions holomorphes (F) dans un voisinage de l'en­ semble E, de manière que (F) ~ (f) en tout point P de E. In 1940 Cartan proved the following theorem [22]; THÉORÈME DE H. CARTAN. - Considérons dans l'espace (x) deux ensembles cylindriques fermés bornés A', A" qui ont mêmes composantes dans les plans de toutes les variables sauf une, et dont l'intersection A' H A" = Aq soit non vide et simplement connexe; supposons donné, au voisinage de A', un système fini de fonctions holomor- phones (f), et, au voisinage de A", un système fini (f"), de manière que la relation (f) ~ (f") ait lieu globalement au voisinage de Aq. Dans ces conditions, on

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peut trouver un système fini de fonctions holomorphes (f) au voisinage de la réunion A = A' U A", de façon que if) ~ (f^^^) au voisinage de A^^^ (i =1, 2). However this theorem can not always be applied. If can be solved for the closed polycylinder, Cartan*s theorem can be reapplied with the result that C^ is the same as E. The

relation between C^ and E is considered for closed bounded cylinders, and Oka concluded that if C^ is solvable so is

C 2 . Thus the only problem needing to be solved was C^^. Next he utilized the notion of ideals he got from Cartan, not knowing of their earlier source as he stated in a foot­ note on page 1 "Ou l'auteur a exposé le Mémoire VII, sans connaître l'existence de premier de ces deux Mémoires de Cartan et du Mémoire du Rückert" [176, p.5]î Considérons dans l'espace (x) un domaine D et un ensemble (I) de fonctions holomorphes dans D; supposons vérifiées les deux conditions suivantes: 1° si f E (I) et si a est une fonction holomorphe dans D, alors af G (I); 2® si f e (I) et f" G (I), alors f + f" G (I). Oka made the important extension of this definition

[176, p. 5]; Considérons dans l'espace (x) des couples (f,ô), où 6 est un domaine et f une fonction holomorphe dans ô. Considérons un ensemble (I) de couples (f,6); au lieu de dire que (f,6) G (I), nous dirons parfois que f 6 (I) pour 6. Supposons que cet ensemble (I) satisfasse aux deux conditions suivantes : 1® si (f,6) G (I) et si a est une fonction holomor­ phe dans un domaine (connexe ou non) ô', alors af G (I) pour 6 0 6'; 2® si (f,6) G (I) et (f',6') e (I)/ alors f + f + f G (I) pour 6 0 6'. Nous dirons alors que (I) est un idéal holomorphe de domaines indéterminés, ou, plus brièvement, un idéal de domaines indéterminés, ou enfin, tout simplement, un idéal. [176, p. 6]

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58 After studying the properties of an ideal of indeterminate domain he defined pseudobase : Considérons dans l'espace (x) un idéal (I) de domaines indéterminés. Un système fini (F) de fonctions holomorphes F^, ..., F^ dans un domaine D sera dit une pseudo-base finie de (I) dans D s'il possède les pro­ priétés suivantes: 1° une quelconque des fonctions F^ appartient à (I) en tout point de D; 2® pour tout point P de D, toute fonction f gui appartient à (I) en P est telle que (f) C (F^, ..., F^) au point P. This definition motivated two problems : PROBLÈME (I). - Étant donné, dans l'espace (x), un idéal (I) de domaines indéterminés et un ensemble fermé borné E, trouver une pseudo-base finie de (I) dans un voisinage de E. and a special case, PROBLÈME (J). - Étant donné dans l'espace (x) un idéal (I) de domaines indéterminés et un point P, trou­ ver une pseudo-base finie de (I) au point P. Betifeen the two problems there is the relation that J plus C^ solves I as follows : Given in a space (x) an ideal (I) of indeterminate domain and a closed bounded polycylinder E. Assume J is solvable for the ideal (I) and for every part P of E and that C^ is solvable for a closed bounded cylinder. Then the problem I is solvable for the ideal (I) and E. However J does not always have a solution. So Oka studied two categories of ideals of indeterminate domain so problem J can be solved for closed bounded cylinders, but only one is in this paper. The other is the geometric ideals of indeterminate domain (which correspond to the ideals of polynomials attached to the algebraic varieties).

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In part 3, Oka gave an example of an ideal which will

solve J, called (I^), and reformulated J as k ; PROBLÈME (K). - Étant donné une equation fonction­ nelle linéaire homogène dans un domaine D de l'espace (x), et un point P de D, trouver une pseudo-base finie de l'idéal (I^^) au point P. After (I^) is considered as a pseudobase so are (Ig),..., (Ip), and problem X is posed [176, p. 9]: PROBLÈME (X). - Pour tout ensemble fermé E contenu dans le domaine D, trouver, pour chacun des idéaux (I^), (Ig), (Ip), une pseudo-base finie dont les fonc­ tions appartiennent globalement, au voisinage de E, a l'idéal correspondant. With the following definition in hand problem L is stated [176, p. 10]; Nous appellerons solution formulaire de l'équation fonctionnelle (I) pour le domaine D ' [connexe ou non, contenu dans le domaine D où les fonctions F^, ..., Fp de l'équation (I) sont holomorphes], tout système de fonctions (i = I, ..., p; j = I, ..., N) holomorphes dans D!, et satisfaisant aux conditions suivantes: 1® toute solution (A^, A^, ... Ap) de l'équation (I), en un point quelconque P de D, admet une représen­ tation de la forme (3) , ou les sont des fonctions holomorphes convenables (respectant naturellement les idendités indiquées) au voisinage de P; 2° inversement, tout système de fonctions C^j, holomorphes en un point P de D', et satisfaisant aux idendités indiquées, définit, au moyen des formu­ les (3), une solution de l'équation (I) au point P. Le système (ir) d'une solution formulaire sera dit le noyau de la solution formulaire. Ainsi, nous venons de prouver: LEMME 1. - Si le problème (X) relatif à une équation fonctionnelle linéaire homogène est résoluble, le problème suivant (L) est aussi résoluble pour cette équation fonctionnelle : PROBLÈME (L). - Étant donné une équation fonctionnelle linéaire homogène dans un domaine, et un ensemble fermé borné E dans ce domaine, trouver une solution formulaire de l'équation au voisinage de E.

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Problem was found to rely on the "solution formulairy" of the linear homogeneous functional equation by a lemma which required the integral of Cousin. Lemma 3 was another application of Cartan's theorem. LEMME 3. - Dans la configuration géométrique du lemme 2, supposons que l'intersection Aq soit simplement connexe et jouisse de la propriété que tout problème (C^) soit résoluble au voisinage de Aq . Considérons p fonctions holomorphes F . non identiquement nulles au voisinage de A, et le problème (A) correspondant; supposons les problèmes (A) résolus au voisinage de A ' et de A". Alors le problème (A) est aussi résoluble au voisinage de A. In section IV ail these problems were reduced to the local problem K, i.e. C^, C^, E and L are always solvable in a neighborhood of a closed polycylinder provided K is solva­ ble. Now in solving K the following theorem is proved

[176, p. 13]: THÉORÈME DU RESTE. - Considérons, dans l'espace (x^, ..., x^, y), un domaine de la forme [D, (C)], où D est un domaine (univalent et fini) de l'espace (x), et (C) un cercle du plan y. Envisageons dans [D, (C)] une fonction holomorphe F(x, y) telle que, pour tout point (x^) de D, l'équation F(x^, y) = 0 ait A racines dans (C), A étant un entier fini (indépendant) de (x®). Alors toute fonction f(x, y) holomorphe dans [D, (C)] peut se mettre sous la forme f(x, y) = fg(x, y) + *(x, y)F(x, y), où et sont holomorphes dans [D, (C)], fQ étant un polynome en y, de degré au plus égal à A - I (identique­ ment nul si A = o); en outre, une telle décomposition est unique. Oka pointed out that the condition underlined above is inevitable. He stated "Ceci est un phénomène curieux que l'on rencontrera dans le champ de fonctions analytiques.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 61 quand on quitte la portion d'une seule variable [176, p. 16]," He used Weierstrass' preparation theorem to prove Théorème du reste. Oka's 1948 paper was published in 1950 along with a

paper of Cartan. As previously stated, he did not know of the 1944 paper of Cartan when he wrote this Memoir, but the editor had apparently informed him and Cartan of each other's papers. Both Cartan and Oka had been in the dark as to what each other was doing. Oka wrote [176, p. 1] ...nous allons donc examiner le Mémoire-la en compa­ rant avec les Mémoires-ici; Le Mémoire VII consiste des deux parties, dont la première montre que les problèmes (Cj), (Cg), et (E) se réduissent au seul problème (K); ce qui est déjà indiqué par Cartan, sans démonstration; mais avec toutes les préparations. Dans la deuxième partie, l'auteur a d'abord préparé le théorème du reste pour résoudre le problème (K); ce théorème est déjà exposé et utilisé par Rückert. As soon as Cartan knew of the theorem of Oka he took up the study of the Cousin problems as a whole, introducing systematically a notion of "faisceau" and "coherent faisceau." Cartan took "ideals of indefinite domain" from Oka's 1948 paper, borrowed the term "faisceau" which originated in Leray's 1946 paper, unified them and simplified them in a new paper entitled "Idéaux et modules des fonctions analytiques de variables complexes" [27]. This paper of September 1949 and Oka's Memoir VII of September 1948 were then published together in 1950. Cartan wrote [27, p. 33] 4. Nous allons introduire, avec K. Oka [177], la notion de faisceau de modules. Nous empruntons le mot de "faisceau" à la Topologie algébrique, où il a été introduit par J. Leray [144] en théorie de 1'homologie; c'est à dessein que nous utilisons ici le même mot, pour désigner une notion analogue. D'ailleurs, ici

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comme en Topologie algébrique, la notion de faisceau s'introduit parce qu'il s'agit de passer de données "locales" a l'étude de propriétés "globales." Définition - Les entiers n et q étant donnés, un faisceau F est une fonction qui, à chaque sous- ensemble ouvert non vide X C C^, associe un module q-dimensionnel dans X, noté F^, de manière que, si X C Y, le module engendré par F^ dans X soit contenn dans F^ [27, p. 35] Exemple 2. - Soient p fonctions f^ (l

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THÉORÈME D'EXISTENCE ET D'UNICITÉ. - Soit F un faisceau cohérent dans un domaine d'holomorphie D. Il exist dans D un module fermé et un seul qui engendre, en chaque point x de D, le module F^; c'est le module des fonctions (holomorphes dans D) qui appartiennent à F^ en chaque point de D. THÉORÈME 6 ter. - Soient p fonctions f^,...,fp holomorphes dans un domaine d'holomorphie D (et a valeurs q-dimensionnelles). Le module des systèmes de fonctions scalaires c^y...,Cp holomorphes dans p, et tels que Z c.f. soit nul dans D , engendre, en chaque i 1 1 point X de D, le module des relations entre les f^ en ce point. C'est 1'unique module fermé dans D, qui jouisse de cette propriété [27, p. 61]. Cartan also realized that Oka's work had a far wider basis than just accomplishing these solutions. The Séminaire Cartan of 1951-1952 and 1953-1954 were devoted to developing this theory. For his work. Oka became the first mathemati­ cian to receive the Ashake Cultural Prize (1954).

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DEVELOPMENT OF THE TERMINOLOGY In 1948, Henri Cartan, Professeur à la Faculté des Sciences de Paris and Chargé de mathématiques à l'École Normale Supérieure, established his annual Séminaire de l'École Normale Supérieure, which continued until 1963-1964. Of the 16 Séminaires, notes have been published of the following 14, seven of which are discussed here.

1948-49 Topologie algébrique 1949-50 Espaces fibres et homotopie 1950-51 Cohomologie des groupes, suites spectrales, faisceaux 1951-52 Fonctions analytiques de plusieurs variables 1953-54 Fonctions automorphes 1954-55 Algèbres d'Eilenberg-Maclane 1955-56 (en collaboration avec C. CHEVALLEY) Géométrie algébrique 1956-57 Quelques questions de Topologie 1957-58 Fonctions automorphes (avec la collaboration de I. SATAKE et R. GODEMENT) 1958-59 Invariant de Hopf, d'après J. F. ADAMS 1959-60 (en collaboration avec J. C. MOGRE) Périodicité des groupes d'homotopie des groupes classiques 1960-61 Déformation de structures complexes; fondements de la Géométrie analytique (par A. GROTHENDIECK) 1961-62 et 1962-63 Topologie différentielle 1963-64 (en collaboration avec L. SCHWARTZ) Théorème d 'Atiyah-Singer In the 1948-1949 Séminaire Henri Cartan on Algebraic Topology exposés were made by Henri Cartan, Jean Cerf, Jean Dixmier, Jean Frenkel, Pierre Samuel, and Cartan's student Jean-Pierre Serre. The first nine seminars were on simplicial homology theory, singular homology theory, and the Cech-Alexander cohomology theory of locally compact

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spaces. Exposé X made by Frenkel was an introduction to the Steenrod theory of local coefficients. XI by Cartan and Serre, "Produits Tensoreils" contains a proof of the funda­ mental theorem on the homology groups of product spaces. We will not concern ourselves with these first eleven here. Cartan wrote Exposé XII on the seminar given March 3, 1949 "Faisceaux et Carapaces; définitions et exemples." He began [25, p. 1]: "La notion de faisceau est due à LERAY que il utilisée pour étudier les propriétés homologiques d'une application continue." The definitions were made on the following spaces [25, p. 2]. Nous n'envisagerons que les espaces localement compacts. Un recouvrement d'un tel espace E est dit localement fini si toute partie compacte ne rencontre qu'un nombre fini d'ensembles du recouvrement. L'espace E est dit paracompact si, pour tout recouvrement ouvert de E, il existe un recouvrement plus fin, localement fini, par des ouverts. Pour que E soit paracompact, il suffit que E possède un recouvrement localement fini par des ensembles compacts. De plus, si E est localement compact et connexe, E est paracompact si et seulement si E est réunion dénombrable de parties compacts. Un espace localement compact et paracompact est normal. Il y a une notion de faisceau pour chaque espèce de structure algébrique: structure de groups, plus parti­ culièrement de groupe abélien: d 'anneau de module sur un anneau commutatif (donne une fois pour toutes); d'algèbre (sur un anneau ou un corps donne une fois pour toutes); de group gradué à dérivation, d'anneau gradué à dérivation de module ou d'algèbre gradué à dérivation. Pour fixer les idées, nous formulerons le plus souvent les définitions en parlent de groupes, mais on pourra y sustituer toute autre structure algébrique. DEFINITION: Soit E un groupe topologique localement compact (pas nécessairement paracompact). Un faisceau sur E est défini par la donnée: 1) d'une application X -»■ A^ qui, à chaque partie fermée non vide X de E, associe un groupe A^;

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Examples of faisceaux were given. In particular there were "faisceau dit de 'Steenrod'," "faisceau des chaînes singulières," and "faisceau des cochaînes singulières," "chaînes et cochaînes différentiables pour une variété," "faisceau des cochaînes de Cech-Alexander." Cartan also gave accounts of the next four seminars: XIII "Faisceaux (Suite): Etude locale;" on April 4, XIV "Faisceaux fins;" on May 5, XV "Produits tensoriels de faisceaux." There he proved [25, XV 6] Un lemme fondamental Le lemme suivant joue un rôle décisif dans toute la théorie. Le processus de la démonstration est basé sur un mode de raisonnement dû à LERAY [140]. Lemma 3. Soient A et F deux faisceaux gradués à dérivation, dont les operateurs de dérivation sont de degré +1. On suppose: 1) A ou F est sans torsion, et F est fin; 2) H (A ) est nul en tout point x (H(A) est localement nul)* 3a) les degrés de A sont bornés enférieurement (par exemple > 0),. ou les degrés de F sont bornés supé­ rieurement; 3b) les degrés de A sont bornés supérieurement, ou les degrés de F sont bornés intérieurement (N.B: il n'y a pas de correspondance entre les "ou" de 3a) et de 3b)). Dans ces conditions, le faisceau H (A à F) est nul. On May 9 in the seminar XV entitled "Théorèmes fonda­ mentaux de la théorie des faisceaus," Cartan proved [25, XVI, p. 3] THÉORÈME 1. Soient A un faisceau à dérivation, et A un sous-faisceau A. Supposons: 1) A et A' fins; 2) en tout point s de l'espace, 1'homomorphisme canonique H(A^) H(A^) est un isomorphisme sur; 3) l'opérateur de dérivation de A étant supposé de de­ gré + 1, les degrés de A sont bornés intérieurement.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 Dans ces conditions, 1'homomorphisme de faisceaux H (A') H (A) est un isomorphisme sur. De plus, si l'espace est de dimension finie, la conclusion reste valable sans 1'hypothèse 3). In the 1st exposé, XVII, no authors are given. The theory developed in the previous exposés was used to study "Homologie et cohomologie des variétés, particularly for a topological variety [XVII, p. 1]: Dans tout ce qui suit, l'espace topologique E est une variété de dimension n , c'est-à-dire: chaque point de E possède voisinage ouvert homéomorphe a R^. On ne fait aucune hypothèse de différentiabilité ou de trian- gulabilité. On suppose seulement que E est paracompacte chaque fois qu'il est question de faisceaux à supports non compacts; par exemple, si on parle d'homologie singulière de deuxième espèce, ou de cohomologie (tout court, c'est-à-dire à supports non nécessairement compacts). There were six duality theorems as follows [25, XVII, p. 2]: THÉORËM 1. Sur une variété E de dimension n, le groupe d'homologie singulière de dimension p (a coeffi­ cients dans D est canoniquement isomorphe au groupe de cohomologie de deuxième espèce (supports, compacts) pour la dimension n - p (coefficients dans r). Si en outre E est paracompacte, le groupe d'homologie singulière de deuxième espèce de dimension p (à coefficients dans r) est canoniquement isomorphe au groupe de cohomologie pur la dimension n - p coefficients dans r'). (Dans le das d'une variété compacte, orientable, c'est la une formulation du "théorème de dualité de Poincaré)" THÉORÈM 2. Soit F une partie fermée non vide d'une variété E de dimension n. On a un isomorphisme canoni­ que du groupe d'homologie singulière relative (E mod E - F) à coefficients dans r, sur le groupe de cohomologie de deuxième espèce ^ ' (F) à coefficients dans r, sur le groupe de cohomolgie ^(F) à coefficients dans r. (Dans cet énoncé, il s'agit de la cohomolgie de dech-Alexander de l'espace localement compact F). THÉORÈM 3. Le groupe d'homolgie singulière de deuxième espèce de la variété connexe E, pour la

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 68 dimension n (n'étant la dimension de E) est isomorphe au groupe Z dans chacun des deux cas suivants: - E est orientable, r est isomorphe au faisceau constant Z; - E est non orientable, r est isomorphe au faisceau des entiers tordus. THÉORÈM 4. Le groupe H^(E) est isomorphe à Z dans chacun des deux cas suivants: - E est orientable, et r' est le faisceau constant Z; - E est non orientable, et r' ist la faisceau des entiers tordus. Le "théorém de dualité d'Alexander-Pontrjagin" THÉORÈM 5. Si F est une partie fermée non vide de la variété E de dimension n, si les groupes d'homologie singulière H p (E) et H p i , (E) (coefficients dans r) sont nuls, il y a isomorphisme canonique de H _ , (E - F) n ' sur le groupe de cohomologie de deuxième espèce H (F), à coefficients dans r. THÉORÈM 6. ("Jordan-Brouwer)." Si F est une variété de dimension n - 1, ayant k composants connexes, plongée dans une variété connexe E de dimension n, de manière que F soit une partie fermée de E, le nombre des compo­ santes connexes de E - F est égal a k + 1, tout au moins si H^(E) est nul (coefficients entiers). The theorem is then shown without triangulation. Products were defined [25, XVII, p. 8]: Prenons pour A une carapace fondamentale, pour B le faisceau des chaînes singulières (a coeff. entiers, pour fixer les idées). Alors H (A) a H (B) -> H(B) montre que la cohomolgie de Cech-Alexander opère dans 1'homologie singulière; en explicitant les degrés: un élément de degré p de la cohomolgie définit un endomorphisme de 1'homologique singulière, qui abaisse le degré de p unités. On a des opérations analogues pour 1'homologie singulière de deuxième espèce. Ces operations se rattachent au "cap-product" classique (voir par ex. EILENBERG, Singular homology Theory, Ann. of Math., 45, 1944, pp. 407-447; voir p. 433) [57]. The theory of intersections is also discussed [25, XVII, p. 9]

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A et B sont tous deux des faisceau de chaines singu­ lières; si E est orientable, on prend pour A et B le même faisceau des chaînes singulières à coeff, entiers; sinon, A et B seront à coeff. entiers ou entiers tordus (on pourra prendre A = B, et aussi A ^ B). En désignant par H (A) le faisceau dérive de A pour la dimension n ... on obtient un homomorphisme canonique H(A) S

H(B) H [H^ (A) a B]; en explitant les dimensions;

Hp(A) a Hg(B) - Hp + g ^ „[Hj^(A) B B] Cette application définit l'intersection de deux classes d'homologie, de dimensions p et q; c'est une classe d'homologie de dimension p + q = n. In 1950 Jean Leray, then Professeur d'équations diffé­ rentielles et fonctionelles au Collège de France, published the lectures he had given there on the cohomology theory of a topological space and of a continuous mapping of one space into another. The article "L'anneau spectrale et L'anneau filtré d'homologie d'un espace localement compact et d'une application continue" [149] was published in Journal mathé­ matiques pures et appliquées. Leray said that the ideas developed during the Colloque de Topologie Algébrique, Paris 1949, as well as from Cartan's notes in Comptes Rendu in 1948, and his student J. L. Koszul's thesis. "Homologie et cohomologie des algebres de Lie" .[130] Leray's article is divided into three chapters. In Chapter I he defined the fundamental algebraic ideas: ... anneau filtré, gradué, différentiel, canonique (anneau gradué ayant une différentielle homogène de degré 1): il définit l'anneau gradué GA d'un anneau filtré A, l'anneau d'homologie HA d'un anneau différentiel A, l'anneau filtré d'homologie HA et l'anneau spectral d'homologie HA d'un anneau différentiel-filtré A.

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Without explanation Leray used "homology" where other

authors used "cohomology" 1149, p. 7]. "Le présent article est le premier exposé détaillé de la notion, que résume "L'anneau d'homologie d'une représentation, d'anneau d 'homologie spectrale et filté d'un espace ou d'une appli­ cation relatifs à un faisceau; c'est H. Cartan qui substitua le terme filtré au terme sous-valué que j'utilisais primiti­ vement. "

Leray's introduction included the origin of his ideas

[149, 5]; II est superflu de que ce calcul est l'outil essen­ tiel de la Géométrie différentielle et que son application à la Topologie des variétés est due a E. Cartan et de Rham; c'est Alexander qui le premier en appliqua le formalisme à la Topologie des espaces abstraits. La définition d'un anneau différentiel, quand cet anneau n'est pas supposé gradué, est due à Koszull. Referring to his "Sur la forme des espaces topologiques et, sur les points fixes des representations [140]" he modified his original definition of complexes [149, p. 9].

Un complexe n'a pas nécessairement de base; une multiplication est définie dans un complexe. Cette seconde modification, due à H. Cartan permet de définir commodément la multiplication dans les anneaux d'homologie d'un espace ou d'une application.

Furthermore he continued [149, p. 9]î Je définis l'anneau d'homologie d'un espace localement compact à l'aide de couvertures, comme dans [25]; mais cette définition est considérablement simplifiée par 1'emploi de couvertures fines; j'ai proposé cette notion au Colloque de Topologie algébrique [25], en même temps que H. Cartan proposait une notion assez voisine [23]. Quand l'espace n'est pas compact, j'adopte un point de vue de H. Cartan [23]; la notion d 'application propre lui est due.

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Je tiens à le remercier de m'avoir si utilement tenu au courant de toutes ces belles mises au point qu'il a faites de la théorie de la cohomologie des espaces. Le raisonnement fondamental que répètent avec diver­ ses variantes de mon article [25] équivaut à l'emploi de la proposition, c'est-à-dire a la considération d'un anneau spectral indépendant de son indice r; c'est l'analyse de ce raisonnement fondamental qui me condui­ sit à envisager des anneaux spectraux, puis filtrés. Le présent article perfectionne et développe donc les Chapitres I, II, IV de [25]; mais il est sans relation avec la théorie des équations qu'exposent les Chapitres III, V et VI de [25] et qui était 1 'object essentiel de [25]. In Chapter II he combined topological and algebraic ideas: Un faisceau (différentiel, filtré) B est constitué par un anneau (différentiel, filtré) attaché à chaque partie fermée de l'espace et par un homomorphisme de l'anneau attaché à dans l'anneau attaché à F quand F^ C F; par exemple les anneaux d'homologie des parties fermées de l'espace constituent un faisceau: le faisceau d'homologie de l'espace; on définit la conti­ nuité d'un faisceau, le transformé ÇB d'un faisceau par une application continue Ç, le faisceau d'homologie FB d'un faisceau différentiel B, le faisceau filtré d'homologie FB et le faisceau spectral d'homologie F, B d'un faisceau différentiel-filtré. Un complexe est un anneau différentiel à chaque élément duquel est associée une partie fermée de l'espace, son support; les supports sont assujettis à un système de conditions qui permettent de définir les transformés d'un complexe par l'inverse d'une application continue, l'intersection K O K' de deux complexes canoniques K et K' et l'intersection k Q B d'un complexe canonique K et d'un faisceau différentiel continu B; ce sont des complexes. K' O B désigne K O B muni d'une filtration définie par la donnée d'un entier / et d'une filtration de B. Un complexe, possédant une unité, est fin quand cette unité est somme d'éléments à supports arbitrairement petits. Une couverture est un complexe canonique dont la section par chaque point a pour anneau d'homologie l'anneau des entiers. Étant donnés un espace X, un entier et un faisceau différentiel-filtré-continu B défini sur X, soit X une couverture fine de X; nous prouvons que l'anneau spectral d'homologie de X'Ob, pour T^

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H ^ ( X ' O b ) , o u X < t, et H(X' Ob);

H .(X'OB) = H(XQF,B); liiti H (X'Q B) D GH (X'Q B) . J- Étant donnés une application continue ç de X dans Y, deux entiers l

H (çl^m Q x i Q b ), o u Z.

et H(ç^Y™ Qx' G B) , qui est H(X Q B) muni d'une cer­ taine filtration; ^(g^Y^O X' O B) = H(Y O ÇF^(X' G B) ;

lim H (Ç^Y^ G X' G B) D GHÜ^Y™ G X' Q B) . r-»-+oo Nous définissons enfin l'anneau spectral et l'anneau filtré d'homologie d'une application composée. Le Chapitre III obtient des invariants topologiques d'un espace X, d'une application continue g ou d'une application composée en choisissant B identique ou localement isomorphe à un anneau différentiel-filtré; les invariants d'un espace qu'on obtient en choisissant B identique à un anneau sont classiques, les invariants d'un espace qu'on obtient en choisissant B localement isomorphe a un anneau ont déjà été étudiés par Steenrod. The twenty-one published Exposés of the 1950-1951 Séminaire Henri Cartan "Cohomologie des groupes, suites spectrales, faisceaux" [28] included notes by Samuel Eilenberg (numbers 1, 2) and Cartan (numbers 3, 4) on homology of groups. Exposés were also given by J. P. Serre on "Applications algébriques de la cohomologie des groupes," (5,6,7). Eilenberg (8,9) and Serre (10) also spoke on "La suite spectrale." Cartan (11,12) and Serre (13) gave "Espaces avec groupes d'opérateurs," leaving thé subject of sheaves. Exposés numbered 14-20 were made by Cartan.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 73 The seminar on April 9, 1951 (number 14) utilized the work of Leray quoted above. The purpose [29, p. 14-01]

... de cet exposé et des suivants est de reprendre entièrement la théorie des "faisceaux et carapaces" qui a fait 1'object des exposés 12 à 17 du Séminaire 1948/49. Entre temps est paru le mémoire de LERAY (J. de Math., Pures et appl.t.29, 1950, p. 1-139) [149] Nous commencerons par une théorie axiomatique de la "cohomologie de îech" des espaces topologiques, à coefficients locaux les plus généraux, c'est-à-dire à coefficients dans un faisceau. Nous allons donc exposer tout d'abord les notions relatives aux faisceaux. N.B. - La terminologie s'écartera quelque peu de celle adoptée dans le Séminaire 1948/48. En particu­ lier, le sens du mot "faisceau" a été modifié, 1. - Définition d'un faisceau On va définir un faisceau sur un espace topologique X, sur lequel on ne fera d'abord aucune hypothèse restrictive. La définition qui suit est due, sous la forme "topologique" qui lui est donnée, à LAZARD: Définition: soit K un anneau commutatif à élément-unité (cas fréquents: anneau des entiers, ou corps.) Un faisceau de K-modules sur un espace topologique (régulier) X est un ensemble F, muni d'une application p (dite "projec­ tion") de F sur X et des 2 structures suivantes: 1) pour chaque point x e %, l'image réciproque p (x) = F est munie d'une structure de K-module; 2) F est muni d'une structure topologique (en général non séparée) satisfaisant aux deux conditions: (a) les lois de composition de F (non partout définies) définies par la structure de K-module des F^ sont conti­ nues; (3) la projection p est un homéomorphisme local (i.e.: tout élément de F possède un voisinage ouvert que p applique biunivoquement et bicontinûment sur un ouvert de #). Sections d'un faisceau: une section de F au-dessus d'un ouvert X C Z est une application continue s: X -»■ F qui, suivie de p, donne l'identité. L'image d'une telle application s (appelée aussi section) est un sous-ensemble ouvert de F, qui contient un élément de chaque module F^ tel que x e X. Tout élément de F appartient à une section au-dessus d'un voisinage de sa projection, en vertu de la condition (3). Pour chaque ouvert X C X, on notera r(F,X) l'ensemble ües sections dè F au-dessus de X. Cet

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ensemble n'est pas vide, car il contient la section s obtenue en associant, à chaque x e X, l'élément 0 e (1'ensemble des 0 de tous les F^ pour x e X est un ensemble ouvert de F, parce que, d'après la condition (a), l'opposé d'un élément de F est une fonction continue de cet élément.) L'ensemble r(F, X) est muni d'une structure de K-module d'une manière évidente. Si s est un élément de r(F,X), l'ensemble de joints X e X tels que s(x) / 0 est fermé; on l'appelle le support de la section s. Si X et Y sont deux ouverts de tels que X C Y on a un homomorphisme: r(F, Y) r(F, X), car la restriction à X d'une section au-dessus de Y est une section au-dessus de X. Pour X C Y C Z, 1'homomorphisme r(F, z) -*■ r(F, X) est composé de r(F, Z) -»• F (F, Y) et de F(F, Y) ^ F(F,X). Various operations on sheaves were defined [29, p. 14-04]: Homomorphisme de faisceaux: considérons deux faisceaux F et G sur le meme espace X (un cas plus général sera envisagé plus loin). Un homomorphisme de F dans G est une application continue ^ de F dans G, telle que, pour tout point x, la restriction de ^ à F soit un homomorphisme de F^ dans G^. L'ensemble des homomorphismes de F dans G est évidemment muni d'une structure de K-module. Si on a trois faisceaux F, G, L, un homomorphisme de F dans G, et un homomorphisme de G dans L, on définit un homomorphisme composé de F dans L. Un endomorphisme de F forment une K-algèbre (la multiplication étant la composition des endomorphismes). Si X est un ouvert de l'espace X, tout homomorphisme çp d'un faisceau F dans un faisceau G définit un homo­ morphisme des modules de sections: F (F, X) F (G, X). Il est défini comme suit: si x » s(x) est une section de F définie pour x e X, alors x

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 75 Puisqu'un homomorphisme de F dans G est une appli­ cation ouverte, l'image de F dans G est ouverte dans G; c'est un sous-faisceau G' de G, et, pour chaque point x, G^ est l'image de F^ ^ G^. Suite exacte de faisceaux et d'homomorphismes: on a la notion de suite exacte, puisqu'on a celle de noyau et d'image d'un homomorphisme de faisceaux. On consi­ déra notamment des suites exactes de la forme 0 F' -»■ F -»■ F" -+ G; pour une telle suite, F ' s'identifie à un sous-faisceau de F, et F" au faisceau-quotient de F par ce sous-faisceau. Étant donné une telle suite exacte, la suite des homomorphismes associés 0 r(F',x) -»■ r(F,x) -»■ r(F",x) est exacte, (trivial à partir des définitions). Mais, en général, 1 'homomorphisme r(F,X) r(F",X) n'est pas sur, bien que F -»■ F" soit sur. On reviendra plus loin sur cette question. Examples were given of exact sequences, homomorphismes. Operations on sheaves are defined [29, p. 14-07] Produit tensoriel de 2 faisceaux sur 2 espaces distincts. Soient X et J deux espaces, F un faisceau sur X, G un faisceau sur Y. Le produit tensoriel F a G va être un faisceau sur 1'espace-produit X & I. Pour le défi­ nir, supposons que F ait été défini par dés F^ relatifs à des ouverts X de %, et que G ait été défini par des Gy relatifs à des ouverts Y de J (et. numéro 2). Considérons, sur X x Y la famille des ouverts de la forme X 8 Y; attachons à X x Y le module x Y ~ F„ a G„. Ces modules avec des homomorphismes X i X X i évidents de ces modules les uns dans les autres, définissent un faisceau L sur X x Y. On a évidemment L, . = F a G , parce que le foncteur L ne change pas IXf Y) X y si on change les F^ et les G^ sans changer les faisceaux F et G qu'ils définissent. Le faisceau L est donc entièrement défini par la donnée des faisceaux F et G; c'est, par définition, le produit tensoriel F a G. Supposons données deux applications continues f : X -*■ X ' , et g: Y + Y', et soit h 1 ' application-product X X Y X' X Y'. Soient des faisceaux: F sur X, F' sur X', G sur Y, G' sur Y'; et soient des homomorphismes F' F et G' -»■ G compatibles avec les applications f et g respectivement. Alors l'homomorphisme F' a G' -» F a G défini par les homomorphismes F^^ ^ a G'^ ^ + G a G

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(produits tensoriels des homomorphismes F'F^ et ^ 9 (y) ^ est compatible avec l'application produit h. Cet homomorphisme du faisceau F ' a G' dans le faisceau F a G s'appelle le produit tensoriel des homomorphismes F' ->■ F et G' ^ G. Produit tensoriel de 2 faisceaux sur le même espace; si F et G sont deux faisceaux sur le meme espace Z, F S G est un faisceau sur Z a Z Plongeons Z dans Z x Z par 1'application diagonale ; le faisceau F a G induit un faisceau sur X, que nous noterons F O G, et appele- rons encore le produit tensoriel de F et G (sur l'espace Z). Il est clair que, pour tout point x e X, on a (P 0 G)^ = a G^. Soit donnée une application continue f de Z dans Z'; soient F et G deux faisceaux sur Z, F ' et G' deux faisceaux sur Z'; et soient des homomorphismes F' -»■ F et G' -»■ G compatibles avec l'application f. Alors 1 'homomorphisme F' 0 G' F 0 G défini par les homomor­ phismes a Gg^^j -»■ F^ a G^ (produits tensoriels des homomorphismes F^^^^ F^ et G^^^) G) est compatible avec l'application f. On l'appelle encore le produit tensoriel des homomorphismes F ' F et G' ^ G.

In Séminaire 15, April 16, 1951, "Faisceaux sur un espace topologique, II," first Cartan defined (and discussed) faisceaux fins [29, p. 15-01]J Définition; Un faisceau F de K-modules (éven­ tuellement, de K-modules gradués) est fin si, pour tout recouvrement localement fini de l'espace Z par des ouverts ü^, il existe des endomorphismes du faisceau F (cf. Exp. 14, numéro 3) tels que: 1° pour chaque i, 1'endomorphisme soit nul en dehors d'un fermé contenu dans U^; 2° la somme Z il ^ soit l'idehtité.

He then gave four examples : "Faisceau des fonctions à valeurs dans l'anneau," "Faisceau des cochaines d'Alexander-Spanier," "Faisceau des cochaines singulières,"

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and "Faisceau des formes différientielles sur une variété différentiable." He then defined (and discussed) "familles (j)" [29,

p. 15-03]: Soit (j) une famille de parties fermées, para- compactes , de l'espace X, satisfaisant aux conditions suivantes : ((j)l) Toute partie fermée d'un ensemble de ((> est dans (j) ; ( possède un voisinage fermé qui est dans F" ->■ 0; et soit G un faisceau fin.Alors, si l'un au moins des faisceaux F" et G est sans torsion, la suite 0 ^ r (F' 0 G) -» 9 r^(F 0 G) ->■ r.(F" 0 G) 0 est une suite exacte.

"faisceaux gradué avec cobord' [29, p. 15-06]: Une structure graduée, sur un faisceau F, est définie par la donnée de sous-faisceaux F^ (n parcourt 1'ensemble des entiers > 0 ou < 0), tels que, pour tout point x, le module F soit somme directe de ses sous- modules (F )^. Si, pour chaque n et chaque x, on considère le- projecteur (p^^^ de F^ sur (F^l^, la collection des (P^^^y quand n est fixé et que x parcourt l'espace, définit un endomorphisme p^ du faisceau F, qui l'applique sur F^, et est un projecteur (i.e.: est indempotent.) Un faisceau gradué avec cobord est un faisceau gradué F dans lequel on d'est en outre donné un endomorphisme 6 (du faisceau F) tel que 66 = 0 et 6p^ = p^ * ^6; ainsi 6 applique F^ dans ^ ^ (on dit que 6 est de degré + 1). Si F est un faisceau gradué avec cobord, les modules de sections r(F,X) sont des modules gradués avec cobord (noté encore 6); de même pour les modules r^(F). Pour définir un faisceau gradué avec cobord, on pourra, à chaque ouvert X (ou seulement aux ouverts

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d'une famille fondamentale) associer un module gradué avec cobord et, pour Y D X, se donner un endomor­ phisme f^y de Fy dans F^, qui soit compatible avec la structure graduée et avec le cobord et satisfasse aux conditions usuelles de transitivité; alors la limite inductive des F^ relatifs aux ouverts X contenant un point x, sera un module gradué avec cobord; et le faisceau F défini par ces F^ est un faisceau gradué avec cobord. On April 23, 1951, Cartan presented number 16, "Théorie axiomatique de la cohomologie." The exposé began [29, p. 16-01]: Introduction; Il s'agira ici de la cohomologie "de Cech"; plus exactement, dans le cas particulier d'un espace compact, la famille 9 étant la famille de tous les sous-espaces fermés, on retrouvera la cohomolo­ gie telle qu'elle a été définie par (ïech, au moins lorsque les coefficients forment un faisceau constant. Dans le cas général, la cohomologie qu'on va définir dépend de la famille $; elle dépend aussi des coeffi­ cients choisis; ceux-ci constituent, en général, un faisceau F (sans graduation ni cobord) sur l'espace considéré Z. Il s'agit donc de "coefficients locaux", non pas dans le sens (plus particulier) des coefficients locaux de Steenrod, mais tels que LERAY les a introduits dans ses Notes aux C, R. Acad. Sci. Paris, t. 222 (1946) p. 1366-1369 et 1419-1422 1145] 1144]; t. 223 (1946 p. 395-397 [141] et 412-415 [142] et dans son mémoire du J. de Math, pures et appl., t. 29, 1950, p. 1-139 [149]. Par contre, l'introduction systématique des familles $ est nouvelle: Leray s'était borné au cas où $ est la famille des compacts dans un espace localement compact. Six axioms were included, among them: "si F est fin (X,F) = 0 pour q > 0" [29, p. 16-02]. The existence and uniqueness of a 6ech cohomology theory was proved The concerns reported in Exposé 17 "Théorie de la cohomologie des espaces [29, p. 17-01]" from April 30, 1951 was with the exact sequence of cohomology relative to an open subspace. There is the following corollary [29, p. 17-02];

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Soit une suite exacte de faisceaux

0 K C o C. X ... C n ... telle que les soient 9 - injectifs et sans torsion (pour n > 0). Alors H^(X, F) « H*(r^(C OF)). ” 9 9 and consequently,

les C O F constituent use 9 - résolution du n faisceau F. (Ce résultat généralise celui obtenu (Expose 16) dans le cas où les sont fins). The definition: Le faisceau C = E C (faisceau gradué avec n>0 n cobord) s'appellera un faisceau 9 - fondamental s'il satisfait à l'hypothèse de l'énoncé précédent. is used in Théorème 2 [29, p. 17-06]: Un espace Z et une famille 9 étant donnés, ainsi qu'un entier n > 0, les conditions suivantes sont équivalentes : (a) pour tout faisceau 9 - fondamental C, le sous- faisceau des cocycles de C de dimension n est 9 - injectif; (b) il existe un faisceau 9 - fondamental de dimension < n; (c) lis modules de cohomologie (X, F) sont nuls pour tout faisceau F et tout entier q > n; (d) les modules de cohomologie ^ ^(X, F) sont nuls pour tout faisceau F. Exposé XVIII was entitled "Carapaces." The seminar was given May 21, 1951. Cartan began [29, p. 180]. 1. - Définition d'une carapace. L'espace topologique Z et l'anneau de base K étant donnés, une carapace est définie par la donnée d'un K-module A, et, pour chaque point x e Z, d'un homomor­ phisme

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(Car II) Si un élément a est tel que ^^(a) = 0 pour tout point x, alors a = 0. In the 1948-49 Séminaire Henri Cartan a "carapace" was a special faisceau. The exposé continued with [29, p. 18-02] 3. - Homomomorph i sme de carapaces (sur un même espace). Soient A et B deux carapaces (ou même deux pre- carapaces) sur l'espace J. On dit qu'un homomomorphisme h du K-modules A dans le K-module B est un homomorphisme de carapaces si, pour tout point x, h applique le noyau A' de (p dans le noyau B' de $ (on note çr et iji les X ^x X X X X homomorphismes qui définissent les structures de cara­ pace de A et B). Il s'ensuit que définit, par passage aux quotients, un homomorphisme h^ de A^ = A/A' dans B = B/x'. En termes de supports, la condition pour h est la suivante; a(h(a)) C o(a) pour tout a e A. En d'autres termes, 1'homomorphisme h diminue les supports. Cette condition est nécessaire et suffisante. It was completed with "Généralisation de.la notion d 'homo­

morphisme de carapaces" [29, p. 18-04]. On June 4, 1951 the seminar entitled "Théorème fondamentaux de la théorie des faisceaux" was given. In the exposé the theorems related local and global cohomologies. In theorem 4 Leray's original theorem and its discussion in XV of 1949 was enlarged. Before that, however, Cartan proved [29, p. 19-04] Théorème 3. - Soit, sur l'espace Z, un faisceau gradué F avec cobord de degré + 1, et soit donnée une "famille 9" d'ensembles fermés paracompacts. Supposons vérifiées les conditions suivantes; (1) H^ (Z, r), muni de l'opérateur cobord défini par le cobord de F, a une cohomologie nulle pour tout entier p > 1; (2) l'espace Z est de dimension finie, ou les degrés de F sont bornés intérieurement; alors il existe une suite spectrale, dont le terme E^ est

H? (X, h 'Ï(F)),

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et dont le terme est le module gradué associé au module H (r^ (F)) convenablement filtré.

A corollary followed which established the precise relations between the cohomology of a module A^, taken globally, and the local property of the cohomology sheaf H(F(A)). Consequently [29, p. 19-05], Cartan stated Théorème 4. - Soient, sur l'espace J, deux faisceaux gradués F et F', avec cobord de degré + 1 et soit f: F F' un homomorphisme de faisceaux, compatible avec graduation et cobord. Si 1'homomorphisme des faisceaux de cohomologie H(F) -*■ H(F'), défini par f, est un iso­ morphisme sur, et si les conditions (1) et (2) du théorème 3 sont satisfaites (par F et par F'), alors 1'homomorphisme (2) H(r^(F)) ^ H(r*(F')) est aussi un isomorphisme sur. In Exposé 20: "Théorie des Faisceaux: Application des théorèmes fondamentaux, étude de la structure multipli­ cative," Theorem 4 of Exposé 19 was applied in the seminar given June 11, 1951 [29, p. 20-01]: 1. - Cohomologie des espaces HLC [Homologically locally connected] Pour la définition (classique) des espaces HLC, voir Séminaire 1948-49, - exposés 13 et 14, Il y est prouvé que, dans un espace HLC, le faisceau des cochatnes singulières est un faisceau fondamental (il est inutile de supposer l'espace localement compact; cette hypothèse n'intervient pas dans la démonstration). Soit S ce faisceau. D'après 16, théorème 2, le faisceau S* peut être utilisé pour calculer la cohomologie de l'espace Z, quand Z est HLC; autrement dit: la cohomologie singu­ lière d'un espace HLC s'identifie canoniquement à la cohomologie de (fech de cet espace. On peut préciser comment se fait cette identification. Soit C le faisceau des cochaines d'Alexander-Spanier. Il existe un homomorphisme canonique de C dans S* et ceci pour tout espace Z (cf. 1948, Exp. 8, §4); il est compatible avec graduation et cobord. Pour tout faisceau de coefficients F, on a donc un homomorphisme de modules

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r*(c o' F) -V r^(s* o F), compatible avec graduation et cobord; d'où un homomor­ phisme des modules de cohomologie : H(r^(C o F) -» H(r^(S* o F)) qui conserve les degrés. Si maintenant X est un espace HLC, 1'homomorphisme des faisceaux de cohomologie H(C o F) -»■ H (S* o F) est un isomorphisme sur; alors le th. 4 de l'Exposé 19 permet de conclure que H(r^(C o F)) + H(r^(S* o F)), c'est-à-dire H^(Z, F) -» H(r^(S* o F)) est aussi un isomorphisme sur. Tel est 1'isomorphisme canonique de la cohomologie de Cech sur la cohomologie singulière d'un espace HLC. Differentiable varieties were discussed and Cartan's conclusions were: En bref: la cohomologie singulière s'identifie canoniquement à la cohomologie différentiable (et à la cohomologie de Cech, d'après le numéro 1 ci-dessus) [29, p. 20-02]. En bref: la 9 - homologie différentiable s'identifie canoniquement à la 9 - homologie singulière [29, p. 20-02]. En d'autres termes: si y désigne la classe fonda- mental de H (r (F)), 1'homomorphisme (1) (page 8) applique o 8 y sur l'élément de H^D + le (Tg(G o F) qui correspond précisément à a dans 1'isomorphisme canonique de nP(Z, G) sur H^ + ^ (T^(G o F)) [29, p. 20-10].

In the appendix the role of the Leray-Koszul spectral sequence was made explicit [29, p. 20-10]. Appendice: la suite spectrale d'un recouvrement r, localement fini, par des ensembles fermés. Pour simplifier, on se bornera aux deux cas sui­ vants : (a) l'espace X est paracompact, et la famille 9 est celle de tous les fermés; (b) l'espace X est localement compact, la famille 9 est celle des ensembles compacts, et les ensembles du recouvrement r sont compacts. Soit A la carapace basique des cochaines du recouvrement r, à valeurs dans l'anneau de base K, et à supports dans 9. Posons F(A) = F (faisceau gradué, a degrés > 0, avec cobord de degré + 1). On a classiquement H^(F) = 0 pour q / 0, et on a un

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isomorphisme canonique de (F) sur K. Le théorème 2 de 19 donne donc un isomorphisme canonique:

HP(r^(C 0 F) ) « h P(x , K) . Par contre, le théorème 1 de 19 n'est pas applicable en général; on a vu (Appendice de 19) qu'il l'est si les supports du nerf de r sont 9 - acycliques; la condi­ tion (C9) est remplie dans les cas (a) et (b) envisagés ici. La filtration considérée dans la démonstration du théorème 1 de 19 donne naissance â une suite spectrale, dont le terme est Z H^(Z, F ). Ici, H^(Z, F ) Pf g s'identifie au module des q-cochaînes du nerf de r(q-cochaînes quelconques dans le cas (a), q-cochaînes finies dans le cas (b)) qui, à chaque q-simplexe du nerf, associent un élément de H^(S., K), en désignant par le support de ce q-simplexe). En bref, on peut dire que H^(Z, F^) est le module des q-cochaînes du nerf de r, à valeurs dans la p-cohomologie des supports. L'opérateur différentiel de E^ est le cobord du nerf. Alors Eg peut s'appeler la cohomologie du nerf de r, à valeurs dans la cohomologie des supports; Eg est bi- gradué, comme d'habitude. Tel est le terme Eg d'une suite spectrale dont le terme E^ est le module gradué associé à H^(Z, K) convenablement filtré. On trouve ainsi, en gros, des relations entre le nerf du recouvrement, les propriétés de cohomologie de ses supports, et la cohomologie de l'espace. Ces relations ont été explicitées pour la première fois par LERAY. The final seminar of 1950-1951 was given by Serre on June 18. Entitled "La Suite spectrale attache à une application continue" the exposé began [29, p. 21-01]: Le but de cet exposé est de reprendre, dans le cadre des "9 - cohomologies", la théorie de 1'anneau spectral d'une application continue, développée par J. Leray dans des Notes aux C. R. Acad. Sci. Paris, ainsi que dans ses deux mémoires parus en 1950 au Journal de Math, pures et appl. L'outil essentiel va être le théorème fondamental de l'exposé 19, dont nous allons rappeler un cas particulier.

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1. - Rappel. Soit B un espace, 9 une famille de parties de B vérifiant les conditions données au paragraphe 7 de l'exposé 15, L un faisceau sur B vérifiant les condi­ tions suivantes : a) L est gradué (à degrés tous positifs), et possède un opérateur cobord de degre + 1. b) L est fin. Dans ces conditions, le théorème fondamental de l'exposé 19 s'applique et donne: Proposition 1. - Dans les conditions précédentes, ^ 3T 2 existe une suite spectrale (E ), où E = D cr oo P g ^ P H^(B, H^(L)) et où E est le module gradué associé a H(r.(D) convenablement filtré. 9 ------Remarque : La proposition précédente est valable sous des conditions sensiblement plus larges, comme on l'a dit dans l'exposé 19; celles que nous avons données nous suffiront par la suite. He proved [29, p. 21-04] Théorème 1. - Soit f: E -»■ B une application continue d'un espace E dans un espace B, 9 et 9 deux familles 9 bien adaptées de E et B. G est un faisceau de E , il existe une suite spectrale de coho- molgie (E^), où E? . _ _H?(B, H?(F_,G)), et où e " est — P g^p ® 9 D — le module-gradué associé à H^(E,G) convenablement filtré. Remarque: Nous n'avons démontré de théorème fondamental que sous des conditions assez restrictives; il y aurait intérêt à les affaiblir (tout particulière­ ment la condition (III)). Par example, le théorème est-il vrai lorsque E est un espace fibré localement trivial de base B, localement compact et paracompact, et que l'on prend pour familles 9 et 9 les familles de tous les fermés de E et de B? Cette question semble étroitement liée à celle de la cohomolgie des produits directs. Serre then applied the theory to "la théorie de la dimension (co-homologique)" [29, p. 21-05] and proved a proposition which he said was, the analogue of a well known theorem of the theory of dimension. He followed with applications to the "Théorème de Vietori^' [29, p. 21-05].

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In 1951 K. Stein published "Analytische Funktionen mehrer komplexer Veranderlichen zu vorgebenen Periodizitats- moduln und das zweite Cousihsche Problem" [199] in the Mathematische Annalen in which he introduced a new category of analytic complex varieties. According to Cartan [26, p. 160] "En fait, dès 1941 Stein avait explicité des condi­ tions de nature homologique pour la résolubilité du problème

de Cousin, sans faire appel à la théorie des espaces fibrés." The 1951-1952 Séminaire "Fonctions de analytiques de plusieurs variables" had twenty sessions for which exposés were made by Frangois Bruhat, Henri Cartan, Jean Cerf,

Pierre Dolbeault, Jean Frenkel, Michael Hervé, Malatian, Jean-Pierre Serre. Exposés 1-5 were on Kahler varieties (by

Cartan), thêta functions (Cartan and Dolbeault) and Abelian functions (by Jean Cerf). In Exposé 6. M. Hervé discussed the integral of A. Weil, particularly to remove restrictive hypothesis by substituting for the n-tuple integral a 2n-tuple integral extended over the whole space. This was the difficulty Cartan had found in 1934. In a paper "Problèmes globaux dans la théorie des fonctions analytiques de plusieurs variables complexes" presented in 1950 at the International Congress Cartan stated [26, p. 158] Les polyèdres analytiques ont été considérés explicitement pour la première fois par André Weil en 1935; l'intégrale de Weil qui généralise l'intégrale de Cauchy, exprime une fonction holomorphe dans un polyèdre analytique par une intégrale portant sur les valeurs de cette fonction sur les "arêtes" a n dimensions (réelles)

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 86 de polyèdre. Mais la construction du noyau de cette intégrale soulève, dans le cas général, des difficulties qui ne peuvent être surmontées que grâce à la théorie des idéaux de fonctions holomorphe, dont il sera parlé à la théorie des idéaux de fonctions holomorphe, dont il sera parle plus loin. In 7 and in 8 Cartan discussed his (and P. Thullen's) theory of convexity, domains of holomorphy, and domains of convergence as in their 1932 paper. In exposé 9 of the seminar given February 4, 1952 he continued with the theory

of convexity. In number 10 "Préliminaires à l'étude de l'anneau des fonctions analytique a l'origine," J. Frenkel discussed Rückertîs form of the Weierstrass preparation theorem which Oka mentioned in his footnote (as discussed above) and which he used in proving his fundamental theorem. 11 and 12 were about rings of power series and analytic varieties. In Exposé 13 H. Cartan introduced "La notion d'espace analyti­ que général et de fonction holomorphe sur un tel espace." In Exposé 15 Malatian presented the April 28, 1952, seminar on the study of the sheaf of relations between p holomorphic functions. This was [30, p. 15-1] Le premier d'une série d'exposés consacrés à la théorie globale des idéaux de fonctions analytiques, pour laquelle on renvoie surtout à: H. Cartan, Bull. Soc. Math, de France 78 (1950), p. 29-64. On adoptera le langage des faisceaux (voir Séminaire 1950-51, exposés 14 et suivants.) Dans le présent exposé, on utilize le langage des fonctions analytiques sur le corps complexe, mais les résultats sont valables pour le cas d'un corps valué complet non discret. 1. Le faisceau des fonctions analytiques. Soit E une variété analytique-complexe. A chaque point X E E associons l'anneau des fonctions (à valeurs scalaires) analytiques (ou holomorphes ce qui est synonyme) au point x. Pour tout ouvert ü C E.

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notons 0^ l'anneau des fonctions holomorphes dans U; si X E U, on a un homomorphisme canonique 0^ -> 0^, et la collection de ces homomorphismes identifie à la limite inductive des anneaux 0^ attachés aux ouverts U contenant x. Dans la réunion 0(E) des 0^ (x parcourant E), on dé­ finit une topologie: à chaque ouvert non vide ü de E, et à chaque f E 0^, on associe l'ensemble V(U,F) des images

de f dans les 0^ attaches aux divers points x e U. Par définition, les V(ü,F) sont des ensembles ouverts de l'espace topologique 0(E), et engendrent sa topologie. Il est classique que cette topologie est séparée. A chaque élément f e 0(E) associons le point x e E tel que f e 0^: ceci définit une application p de 0(E) sur E. Il est évident que p est un "homéomorphisme local": tout f e 0(E) possède un voisinage ouvert V tel que la restriction de p a V soit un homéomorphisme de V sur p(V). Ceci munit l'espace 0(E) d'une structure de variété analytique complexe. After the definition of analytic sheaves [30, p. 15-3] Faisceau des formes différentielles holomorphes: on a déjà défini les formes différentielles holomorphes sur une variété analytique-complexe E (cf. 1, pp. 12-13): ce sont les formes différentielles qui s'expriment avec les produits extérieurs des dz d'un système de coordonnées locales z^, les coefficients étant holomorphes. Soit F^ le module des formes différentielles holomorphes dans un ouvert U C E; F^ est un module sur l'anneau 0^. La limite inductive F^ des F^ relatifs aux ü contenant x est un module sur l'anneau O . Ceci définit bien un homomor­ phisme de faisceaux 0(E) oF F qui fait de F un faisceau analytique. Définition 2. Soient F et G deux faisceaux analyti­ ques sur E. Un homomorphisme de faisceaux F G s'appelle un homomorphisme analytique si la collection des homomorphismes

was followed by. [30, p. 15-4]

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 88 4. La notion de faisceau cohérent. Définition 3. On dit qu'un sous-faisceau analytique F de 0^(E) est cohérent au point x e E, s'il existe un voisinage ouvert ü de x et un système fini d'éléments u^ G 0^ jouissant de la propriété suivante: pour tout point y e ü, le sous-module de engendré par les u^ est précisément F^. On dit qu'un faisceau F est cohérent s'il est cohérent en tout point de E. Oka's theorem of 1950 was presented because it played a fundamental role in the sequence of exposés [30, p. 15-6]: Théorème. Pour toute suite finie (f^,...,fp) d'élé­ ments de O^E le faisceau des relations R (f_,...,f i p ) est cohérent. On va démontrer ce théorème par une double récur­ rence sur l'entier q et sur l'entier n (dimension de E). Pour des entiers n et q donnés, désignons par th (n,a) l'énoncé précédent. Il est évident que th(n,q) implique th(n,q') pour tout q' < q. D'autre part, th (0,q) est trivialement vrai. Pour établir le théorème, il suffira de prouver les deux assertions suivantes: (a) pour n > 0 et q > 1, th(n, q - 1) entraîne th (n,q); (b) Si, pour un n > 0, th (n - 1, q) est vrai pour tout q, alors th(n,1) est vrai. On May 5, 1952, J. Frenkel gave a seminar, number 16, "Faisceaux d'une sous-variété analytique" [30, p. 16-01]. Contrairement à l'exposé précédent (15), on supposera dans celui-ci, d'une manière essentielle, qu'il s'agit de fonctions analytiques sur le corps complexe, et de variétés ahalytiques-complexes. 1. Le faisceau d'une sous-variété analytique Soit E une variété analytique-complexe (au sens de 1), et soit V une sous-variété analytique de E (au sens de 12, n° 4). Pour chaque point x e E, soit V le germe de sous-variété défini par V.au point x (germe éventuelle­ ment vide). Soit F^ l'idéal du germe V^: c'est, dans l'anneau des fonctions holomorphes au point x, l'idéal formé des germes de fonctions qui s'annulent sur le germe V^. La collection des idéaux F^ satisfait visiblement à la condition (C) de l'exposé 15 (p.3), et définit donc un sous-faisceau analytique du faisceau 0(E). On l'appelle le faisceau de la sous-variété analytique V; on le notera F(V) .

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Le but de cet exposé est de démontrer le Théorème. Le faisceau F(V) d'une sous-variété, est un faisceau cohérent. (Pour la définition d'un faisceau cohérent voir 15, n° 4). In 17, Frenkel discussed "Un théorème sur les matrices holomorphes inversibles" from Cartan's 1940 paper on matrices and the connection with analytic faisceaux [30, 17-1]. Théorème 1. Soient, dans l'espace C^, deux polycy- lindres compacts A' = 6' X ... X 6', A" = s;' X ... X ô" 1 n 1 n tels que 6^ = pour tout i sauf un au plus. Supposons que l'intersection A' n A" (qui est un polycylindre compact) soit simplement connexe'. Soit M une matrice carrée (p lignes, et p colonnes), holomorphe et inversi­ ble dans A'"^A A",v c'est-à-dire une matrice dont les termes sont des fonctions holomorphes dans A' n A" (i.e., holomorphes dans un voisinage ouvert de A ' HA") et dont le déterminant est est / 0 en tout point de A' n A". Alors il existe une matrice M' holomorphe et inversible dans A', et une matrice M" holomorphe et inversible dans A", telles que M = M'*M" dans A'HA" (le signe • désigne la multiplication des matrices.) Ce théorème sera l'un des fondements de la théorie qu'on dévelopera dans les exposés suivants. Theorem 1 has an important application to the theory of analytic sheaves [30, p. 17-8]. Soit F un faisceau analytique sur une variété analytique complexe B. Pour tout compact (resp. ouvert) X de B, le module de cohomologie H^(X,F) n'est pas autre chose que le module des sections de F au-dessus de X; c'est un module sur l'anneau des fonctions holomorphes dans X. Soit Y C X; étant donnés des éléments u. e 0 0 X H (X,F), leurs images dans H (Y,F) engendrent un sous- module de H^(Y,F) pour la structure de module sur 0^. Pour abréger le langage, on dira simplement que c'est le sous-module de H®(Y,F) engendre par les üi E H°(X,F). Théorème 3. Soient A' et A" comme dans le théorème 1. Soit F un faisceau analytique dans la réunion A' U A" = A. Si des éléments en nombre fini

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U| E H°(A', F), v" e H°(A", F) engendrent le même sous-module de H^(A' n A", F), il existe des g, e H^(A, F) en nombre fini, qui engendrent dans H (A', F) le même sous-module que les u!, et 0 ^ ^ engendrent dans H (A'\ F) le même sous-module que les "j- In Exposé 18 "Faisceaux analytique sur les variétés de Stein" of the seminar given May 19, 1952, Cartan began [30, p. 18-1]; ... voici maintenant les résultats globaux de la théorie des idéaux de fonctions analytiques. Par rapport à ceux exposés dans le mémoire cité de H. Cartan (Bull. Soc. Math., 1950), ils sont plus complets et présentés autre­ ment: ces changements et améliorations sont dus, pour une bonne part, à J. P. Serre. 1. Généralisation de la notion de faisceau analyti­ que cohérent. Dans 15, on a seulement considéré des faisceaux sur une variété analytique-complexe E; désormais, on envisa­ gera aussi des faisceaux sur un sous-espace X de E (par example, sur un sous-espace compact X.) Un tel X porte une structure analytique-complexe; d'une façon précise, on a le faisceau 0(X) qui associe à chaque point x e X l'anneau 0 des fonctions holomorphes au point x (holomorphes dans la variété ambiante E). D'autre part, on ne se bornera pas à considérer des sous-faisceaux de 0^(X); la notion de faisceau analytique cohérent va être définie en toute généralité. Lorsqu'on remplace la considération de la variété E par celle d'un sous-espace X de E, la définition de faisceau analytique donnée dans 15 se transpose sans changement, ainsi que celle de sous-faisceau cohérent de 0^(X). La proposition 1 et la proposition 3 de 15 restent valables sans modification. Seule la proposition 2 (15, p.5) est à modifier: pour qu'on sous-faisceau analytique F de 0*^(X) soit cohérent, il faut et il suffit que change point x e X possède, dans la variété ambiante E, un voisinage ouvert U dans lequel on a un sous-module de 0^ qui engendre F^ en tout point y e U n X. Mais, même si X est compact, on ignore si un sous-module arbitraire M de 0^ engendre un faisceau

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cohérent sur X; pour qu'il en soit ainsi, il faut et il suffit que M contienne un sous module de type fini qui engendre, en chaque point y e X, le même sous-module de 0^ que M. (C'est un problème non résolu de savoir si cette condition est toujours vérifiée lorsque X est compact. General properties of coherent faisceaux were developed [30, p. 18-7]: Voici l'énoncé des deux théorèmes fondamentaux: Théorème A. Soit X une variété de Stein, ou un compact d'une variété de Stein identique à son enve­ loppe. Soit F un faisceau analytique cohérent sur X. Alors, pour tout point x e X, l'image, dans le 0„ - module F^, du module des sections H• 0 (X, F), engendre F pour sastructure de module sur O . X ------X Commentaire: il n'est pas question, bien entendu, de vouloir prolonger au-dessus de tout X n'importe quelle section de F donnée seulement au voisinage de X. Mais le théorème affirme que toute section de F, au-dessus d'un voisinage de x si petit soit-il, est combination linéaire (à coefficients holomorphes au point x) de sections prolongeables au-dessus de X tout entier. Théorème B. Soit X une variété de Stein, ou un compact d'une variété de Stein identique à son enve­ loppe. Soit un faisceau analytique cohérent sur X. Alors les modules de cohomologie H^(X, F) sont nuls pour tout entier q > 1. Séminaire 19 "Faisceaux analytiques sur les variétés de Stein: Démonstration des théorèmes fondamentaux," given on June 6, 1952, was written in October 1952. There Cartan demonstrated the difficult proof of his theorems A and B which were formulated in the previous exposé as well as in his paper of 1950. He utilized the theorem of Oka of 1936 [30, 19-10]: "...toute fonction holomorphe au voisinage de K est limite de polynômes par rapport aux fonctions de 0, la convergence étant uniforme dans un

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voisinage de K." The material of this important seminar is discussed below. Séminaire 20 "Applications de la théorie générale a

divers problèmes globaux" was given by J.-P. Serre June 16, 1952, and considerably rewritten for the exposé of November 1952. It was a "pioneer moment" according to Spencer. Applications were made and the theorems A and B "caractéri­ sent complètement les variétés de Stein" [30, p. 20-2]. This seminar is discussed in the next chapter. In Section II, X was a Stein variety. Cartan wrote

[30, 20-6] II s'ensuit que H^(X, C) est isomorphe au quotient de l'espace des formes différentielle holomorphe fermées de degré p par le sous-espace formé des cobords des formes de degré p - 1. C'est le "théorème de de Rham," mais démontré pour des formes holomorphes: le complexe des formes différentielles holomorphes sur X, muni du cobord de la différentiation extérieure, a même coho- mologie que X. 5. Relation avec le théorème de de Rham classique Soient le faisceau des germes de formas diffé­ rentielles de degré p (indéfiniment différentiables), Zq le sous-faisceau des formes fermées. Le faisceau est fin (partitions de l'unité) et on a donc H^(X, = 0 pour i > 0, ce qui permet de transcrire les raisonnements précédents en remplaçant partout 0^ par et par Z^. On obtient ainsi une démonstration du théorème de de Rham classique (différentiable) qui d'ailleurs est celle de A. Weil (traduite dans le langage des faisceaux.)... On tire de là: a) Toute forme différentielle fermée sur X est cohomologue à une forme holomorphe. b) Si une forme différentielle holomorphe est le cobord d'une forme différentiable, c'est le cobord d'une forme holomorphe.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 93 En particulier, il existe toujours une forme holo­ morphe et fermée de périodes (complexes) données.... Compte tenu des relations entre holomologie et cohomologie, ceci est équivalent à; Les groupes d'homologie Hp(X,Z) sont des groupes de torsion pour tout p > n. Si Hp (X, Z) est un groupe de type fini, cela équivaut à dire que les nombres de Betti B^ X sont nuls dès que p > n. On obtient ainsi une condition nécessaire purement topologique pour qu'une variété soit une variété de Stein (et a fortiori un domaine d'holomorphie). Il est facile de vérifier directement le résultat précédent pour divers types simples de variétés de Stein; pour les polycylindres c'est évident d'après la formule de Künneth; pour un groupe semi-simple algébri­ que de matrices X, cela résulte de la décomposition classique de X comme produit direct d'un sous-groupe compact K de dimension (réelle) n et de R^. Supposons maintenant que X soit une variété algébrique affine, soit V la variété algébrique projective contenant X, et soit W la section de V par l'hyperplan à l'infini. On a donc X = V - W; d'après la dualité de Poincaré appliquée à X, la relation "H^ (X, C) = 0 pour p > n" équivaut à" (X,C) = 0 pour p < n," où H* désigne la cohomologie à supports compacts de X; d'après la suite exacte de cohomologie qui relie V, W et V - W = X, cette dernière relation équivaut à son tour à:

h P(V,C) » H^(W,C) pour < n - 1;

(V,C) (W,C) est biunivôque, ce qui est un résultat classique de Lefschetz (d'ailleurs démontré avec Z au lieu de C). Section III "Espaces fibrés a groupe structural abélien." contained a topological digression. In [30, 20-10] IV, however Serre referred to the Deuxième problème de Cousin. Diviseurs. 10. Deuxième problème de Cousin C'est le suivant: étant donné un diviseur D, existe-t-il une fonction méromorphe f dont le divi­ seur est égal à D? Nous allons le reformuler en termes de faisceaux: soient G le faisceau des germes de fonctions méromorphes sur X (la loi de composition étant la

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multiplication), F le sous-faisceau de G formé des germes de fonctions holomorphes non nulles, D le faisceau quotient G/F. On a donc une suite exacte: 0-»-F-»‘G->D-»-0. (Il est clair que le faisceau D est le faisceau des germes de diviseurs sur X. Une section de D est donc un diviseur sur X). Écrivons la suite exacte de cohomologie: H°(X,G) + H°(X,D) + H°(X,F). And in conclusion [30, p. 20-14]; 13. Généralisations. Le faisceau des germes de sections d'un espace fibré peut se définir dans des cas plus généraux que celui du n° 11: si l'on a un espace fibré E de groupe structural le groupe GL (n, C) des matrices complexes inversibles (ou un sous-groupe de ce groupe) on peut lui associer un espace fibré E ' dont les fibres sont des espaces vec­ toriels complexes de dimension n. Le faisceau des germes de sections de E ' est encore un faisceau analy­ tique cohérent, auquel on peut donc appliquer les théorèmes A et B. En particulier, il possède toujours une section non triviale. Ce genre d'espace fibré a été considéré antérieure­ ment par Weil (Journal de Liouville, 1937), dans le cas ou la base X est la surface de Riemann d'une courbe algébrique. In 1953 at Colloque sur les fonctions de plusieurs variables, held in Brussels, Henri Cartan gave a survey of the results he had obtained about the cohomology of Stein varieties with coefficients in analytic sheaves. This was the same material as presented in the Séminaire Cartan 1951-1952 discussed above. In his review [196] of Cartan's survey, "Variétés analytiques complexes et cohomologie" [31], D.C. Spencer pointed out that "sheaves" is English for "faisceaux,"^ and concluded his

Spencer, in a private conversation, said that the mathematicians who gathered at Princeton for the seminars on algebraic geometry and topology held on the retirement of

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review stating that applications of Cartan*s theorems A and B are mentioned and that several unsolved problems are listed. Serre, also speaking at the same colloquium as Cartan on "Quelques problèmes globaux relatifs aux variétés de Stein/'discussed these applications as Spencer reported

[196]. First ...if X is a Stein variety, then H^(X,C) (the pth cohomology group of X with coefficients in the field C of complex numbers) is isomorphic to the d-cohomology of the holomorphic differential forms on X where d is exterior differentiation. The second application is to the second problem of Cousin. Let X be a paracompact complex-analytic variety of dimension n, and let D be a divisor on X. The second problem of Cousin is to find the conditions under which there exists a meromorphic function f on X with divisor (f) = D. 2 The third application is to show that, given c e H (X,Z), X a Stein variety, there exists a positive divisor D on X such that h(D) = c. Meanwhile in the United States, in 1949, (at the time of the first Séminaire Cartan), Hermann Weyl had invited a young (age 34) Japanese, Kunihiko Kodaira to Princeton to the Institute for Advanced Study. "Due to the isolation of the war years Kodaira had not been able to keep up with every­ thing that was going on in other countries and many of his results were developed more or less independently of the work of others in the same area" [127, p. vi]. Kodaira had

Lefschetz as department chairman decided, among themselves, to use "sheaf" for "faisceau,because "Stack was ambiguous. " Those present, besides Spencer, included Hodge and Kodaira.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 96 studied with S. lyanaga at the University of Tokyo, receiv­ ing his PhD there in 1949 by submitting a thesis on the subject of "Harmonic Fields in Riemannian manifolds" [118].^ He had been influenced by Weyl's book Die Idee der 2 Riemannschen Flache [229].

At the same time as the intensive investigation of questions related to the Cousin problems was being made by K. Oka, H. Behnke, H. Cartan, K. Stein and others, develop­ ments were taking place in the theory of harmonic differen­ tial forms. S. Bergman explained a type of hermitian metric on subdomains of complex n-space which became known as a "Kahler metric" after it was generalized by E. Kahler in "über eine bemerkenswerte Hermitschen metrik" [110] in 1933. In 1932 at Cambridge University, W. V. D. Hodge had begun to generalize the theory of harmonic functions on Riemann surfaces to harmonic differential forms on higher dimensional compact Riemannian manifolds and on complex manifolds which possess Kahler metrics, as he explained in The Theory and Applications of Harmonic Integrals of 1941. Although he did not know of this book until 1950, Kodaira developed and perfected Hodge's theory in the 1940s:. and generalized the theory to open manifolds and introduced the method of orthogonal projection in Hilbert space. 2 According to Dieudonné [55, p. 283] "... instead of following his predecessors in their constant appeal to 'intuition' for the definition and properties of Riemann surfaces, [Weyl] set out to give to their theory the same kind of axiomatic and rigorous treatment that Hilbert had given to Euclidean geometry. Using Hilbert's idea of defining neighborhoods by a system of axioms, and influenced by Brouwer's clever application of Poincaré's simplicial methods (which had just been published) he gave the first rigorous definition of a complex manifold of dimension 1 and a thorough treatment (without any appeal to intuition) of all questions regarding orientation, homology, and fundamental groups of these manifolds. Die Idee der Riemannschen Flache (1913) [229] immediately became a classic and inspired all later developments of the theory of differential and complex manifolds."

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Using Weyl's book as a guide, Kodaira said of the last

of his thesis [127, p. 588]; We shall consider the relations between various harmonic fields in a closed Riemannian manifold and show that there hold the reciprocity formulae and the generalized Riemann-Roch's theorem, just as in the classical case of Riemann surfaces. In the case where the dimension of the manifold is an even number we can moreover define a kind of analytic differential on the manifold, and give the complete analogy to the classical Riemann-Roch's theorem. This will be shown in another paper. In order to clarify the relation between our generalized Riemann-Roch's theorem and the classical one, we shall finally deduce the classical theorem from our stand point. The theme of the Riemann-Roch theorem also played a major role in Kodaira's later work, hence a summary of its devel­ opment follows. Georg Friedrich Bernhard Riemann, after studying in Gottingen with Gauss, went to Berlin to study with P. G. L. Dirichlet, C. G. J. Jacobi, J. Steiner and F. G. M. Eisenstein in 1847. At that time he was 21. There he formed ideas on the theory of functions of a complex vari­

able which led to most of his great discoveries. Back in Gottingen in 1857, Riemann published his memoir "Theorie der Abel'schen Functionen" [184]. This paper revolutionized the subject of abelian integrals by introducing new geometrical and topological considerations. He essentially created the topological study of orientable surfaces by applying the theory developed in his doctoral dissertation of 1851 "Griindlagen fiir eine allgemeine Theorie der Funktionen einer veranderlichen complexen Grosse" [183]. Riemann began "Theorie der Abel'schen Functioned'[184, p. 115];

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In der folgenden Abhandlung habe ich die Abel'schen Functionen nach einer Methode behandelt, deren Principien in meiner inauguraldissertation aufgestellt und in einer etwas veranderten Form in den drei vorhergehenden Aussatzen dargestellt worden sind. Zur Erleichterung der Uebersicht schicke ich eine kurze Inhaltsangabe vorauf. Die erste Abtheilung enthalt die Theorie eines Systems von gleichverzweigten algebraischen Functionen und ihren Integralen, soweit fur dieselbe nicht die Betrachtung von I-Reihen massgebend ist, und handelt im §.1-5 von der Bestimmung dieser Functionen durch ihre Verzweigungsart und ihre Dnstetigkeiten^ Der allgemeine Ausdruck einer Function w, die fiir m Punkte der Flache T, e^, ..., unendlich gross von der ersten Ordnung wird, ist nach dem Obigen

s = e^t^ + ggtg + ... + +

... + a w + const., P P worin t^ eine beliebige Function t(e^) und die Grossen a und 3 Constanten sind. Wenn von den m Punkten e eine Anzahl g in denselben Punkt n der FlSche T zusammen- fallen, so sind die g diesen Punkten zugehorigen Func­ tionen t zu erstzen durch eine Function t(n) und deren g ^ erste Derivirte nach ihrem Unstetigkeitswerthe. Die 2p Periodicitatsmoduln dieser Function s sind lineare homogene Functionen der p + m Grossen a und 6. Wenn m > p + 1, lassen sich also 2p von den Grossen a und 6 als lineare homogene Functionen der übrigen so bestimmen, dass die Periodicitatsmoduln sammtlich 0 werden. Die Function enthalt dann noch m - p + 1

^Georg Birkhoff pointed out that [11, p. 175] "Mathematicians of the 18th and 19th centuries made many profound and elaborate studies of integrals of algebraic functions, so-called Abelian integrals. In more detail, let f(x,y) be any polynomial, and let a(x) be the (usually many-valued) algebraic function defined by the equation f(x,y) = 0. Then since R(x, a(x)) is algebraic for any rational function R(x,y) = P (x,y)/Q(x,y) (P,Q polynomials), integrals of the form ^R(x,y)dx, f(x,y) = 0 f polynomial, are Abelian integrals."

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99 willkührliche Constanten, von denen sie eine lineare homogene Function ist, und kann als ein linearer Ausdruck von m - p Functionen betrachtet werden, deren jede nur fur p + 1 Werthe unendlich von der ersten Ordnung wird. Wenn m = p + 1 ist, so sind die Verhâltnisse der 2p + 1 Grôssen a und 3 bei jeder Lage der p + 1 Punkte e vôllig bestimmt. Es kônnen jedoch fur besondere Lagen dieser Punkte einige der Grossen 3 gleich 0 werden. Die Anzahl dieser Grôssen sei = m - y, so dass die Function nur fur y Punkte unendlich von der ersten Orduung wird. Diese y Punkte müssen dann eine solche Lage haben, dass von den 2p Bedingungsgleichungen zwischen den p + y übrigen Grôssen 3 und a p + 1 - y eine identische Folge der übrigen sind, und es kônnen daher nur 2y - p - 1 von ihnen beliebig gewâhlt werden. Ausserdem enthalt die Function noch 2 willkührliche Constanten. Es sei nun s so zu bestimmen, dass y môglichst klein wird. Wenn s ymal unendlich von der ersten Ordnung wird, so ist dies auch mit jeder rationalen Function ersten Grades vors der Fall; man kann daher für die Lôsung dieser Ausgabe einen der Punkte beliebig wâhlen. Die Lage der übrigen muss dann so bestimmt werden, dass p + 1 - y von den Bedingungsgleichungen zwischen den Grôssen a und 3 eine identische Folge der übriger sind; es muss also, wenn die Verzweigungswerthe der Flache T nicht besondern Bedingungsgleichungen genügen, p + 1 - y < y - 1 oder y > 1/2 p + 1 sein. Die Anzahl der in einer Function s, die nur für m Punkte der Flache T unendlich von der ersten Ordnung wird und übrigens stetig bleibt, enthaltenen willkühr- lichen Constanten ist in allen Fallen = 2m - p + 1. Then, in 1864, Riemann's student G. Roch proved one of the basic results of the entire subject of algebraic functions of one variable and hence his name became associated with Riemann's in his famous theorem. In "Uber die Anzahl der willkürlichen Constanten in algebraischen Functionen" [185] Roch began 1st s eine durch die Gleichung f(s, z) = 0 definirte algebraische Function von z, so kann nach Riemann (s. dessen Abhandlung über Abelsche Functionen, Band 54. dieses Journals) jede wie s verzweigte algebraische Function s' von z rational durch s und z ausgedrückt werden. Wird die Function s' in m Punkten der Flache T, welche die Verzweigungsart angiebt, unendlich erster Ordnung, so enthalt dieselbe nach §.5. der erwahnten Abhandlung m - p + 1 willkürliche Constanten. Schon die

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a. a. 0. untersuchte Bedingung der Existenz von Funct- iohen, die inweniger als p + 1 Punkten unendlich werden, zeigt, dass die Anzahl der wirklich vorhandenen Constanten eine grossere sein kann. Dies kann aber auch Statt finden, wenn m grosser als p ist. 1st z. B. s' der Quotient zweier Functionen

This result was later sharpened and reformulated by Brill and Noether (taking into account the work of Clobsch and Gordon) [16, p. 246]: Wird eine Function s' in m Punkten unendlich gross erster Ordnung and kônnen in diesen m Punkten g Fun­ ctionen — verschwinden, zwischen denen keine oH as linear Gleichung mit constanten Coefficienten besteht, so enthalt s' die Zahl m - p + 1 + q willkürlicher Constanten. The development of the Reimann-Roch theorem is dia­ grammed schematically according to significant contributors

as on the following page. In 1951 in the American Journal of Mathematics, Kodaira published "The theorem of Riemann-Roch on compact analytic surfaces" [120]. According to Weyl [231, p. 64] "Around 1950, Kodaira understood that the Riemann-Roch theorem for classical algebraic varieties of dimension 2

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1851

RIEMANN

RIEMANN - ROCH

1876 L(D)~L(A~D)“deg (D)+l—q VDEDEKIND WEBE^ ^ IDEAI.S f CLEBSH, GORDON> BRILL - M. NOETHER

1907 1900's 1920's CASTELNÜEVO IQUES SURFACES 2(|j)=q^"V-24z(w) -C ( ^ SCHMIDT ^ /SlVISORS (NOT IQEALS) 1949

0. GOLDMAN 1937 1930's EGAR, TODD 1951 1951

ZARISKI ■IflSZ KODAIRA

1952-1955

lODAIRA - SPENCER

1954 1954 1953 BXRZEBRUCH

1958

GROTHENDIECK

THE GENEOLOGY OF THE KIEMANN-ROCH THEOREM

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could be formulated as an equality between topological invariants of the variety instead of an inequality as with Zariski and the Italian geometers; but he still lacked the machinery to extend these results to higher dimensional cases." Kodaira had stated [120, p. 813] The main purpose of the present paper is to prove the theorem of Riemann-Roch on compact complex analytic manifolds of complex dimension 2 with Kahlerian metrics by means of the theory of harmonic integrals. Suppose such a manifold y as given and denote for an arbitrary divisor D on y which are multiples of—D byJF(D). The problem concerning the theorem of Riemann-Roch consists in expressing the dimension of the linear space f(D) in terms of the "characters" of D such as the topolo­ gical intersection-number of D with itself, the virtual genus of D, etc. In this paper, we shall mainly con­ sider the case that D is an arbitrary (reducible or irreducible) curve r. In Section 1, we shall first summarize the theory of harmonic currents due to G. de Rahm and several results of W. E. D. Hodge concerning analytic manifolds with Kahlerian metrics; then we shall prove a theorem concerning the existence of analytic differentials with given singularities. In Section 2, we shall prove a theorem due to A. Weil and, by means of that theorem, introduce the notion of the characteristic divisor class {6} on r which corresponds to the characteristic series of Italian geometers. In Section 3, we first introduce the linear space /(r,6) consisting of meromorphic functions on r associated with an arbitrarily fixed characteristic divisor 6e {6} and show that the singularities of any meromorphic function F e gT(r) can be represented by a meromorphic function f e /(r,6); secondly, using the existence theorem in Section 1, we prove that, for every f £ f(r,6), there exists on :y at least one additive meromorphic function which is a multiple of — r and has the singularities represented by f on I; thirdly, calculating the period of this additive meromorphic function, we find a neces­ sary and sufficient condition for f e f(r,6) in order that f should represent the singularities of a meromorphic function F £ jB(r) and deduce an expression of the difference dim f(r,6)--- dim f (r) in terms of the simple differentials of the first kind on y'. Our method applied here is a natural generalization of that of H. Weyl. The dimension of /(r,6) is given by the theorem of Riemann-Roch on reducible curves which will be proved in Section 4. Combining the result of Section 3 with that of Section 4, we shall prove, in the following Section 5, a formula expressing dim^(r)

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in terms of the topological intersection number of r with itself, the virtual genus of r and two non­ negative integers corresponding to the index of spe­ ciality appearing in the theorem of Riemann-Roch on algebraic curves. This formula may be called the theorem of Riemann-Roch on compact analytic surfaces with Kahlerian metrics, since it contains the theorem of Riemann-Roch for irreducible linear systems on algebraic surfaces. In the last part of this Section, we shall deduce a theorem of Castelnuovo concerning the superabundance of irreducible linear systems. In the following Section 6, we first prove a formula expressing the number of linearly independent double differentials on y which are multiples of r in terms of the virtual genus of r, the number of connected components of r and the number of simple Picard integrals of the first kind which are constant on each connected components of r, and then we deduce from that formula a theorem of F. Enriques concerning the deficiency of the series cut out on a generic curve of an arbitrary irreducible linear system by its adjoint system and a theorem of F. Severi concerning the superabundance of the adjoint systems of irreducible linear systems. Applying our results to algebraic surfaces, we shall prove, in the final Sec­ tion 7, several classical results of Italian geometers, e.g. a theorem of G. Castelnuovo concerning the deficiency of characteristic series. A strict algebraic proof of the theorem of Riemann-Roch for arbitrary divisors on algebraic surfaces was given recently by O. Goldman [78]. In the last part of this Section, we shall show that the theorem of Riemann-Roch for arbitrary divisors on algebraic surfaces can be readily deduced from our results in the same manner as in the proof of O. Goldman. In his Topological Methods in Algebraic Geometry [100],

F. Hirzebruch (whose contributions are discussed below) pointed out that by using the interpretation of divisors in

terms of line bundles and Hodge theory, Kodaira was able

to obtain for compact Kahler manifolds of complex dimension 2, a Riemann-Roch formula in which the missing terms from the formula found by the Italian geometers were

expressed by means of Chern classes.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 104 In 1952 he found a similar formula for Kahler manifolds of dimension 3. He began his paper "The theorem of Riemann-Roch for adjoint systems on 3-dimensional alge­ braic variétés" [121]; The main purpose of the present paper is to prove the theorem of Riemann-Roch for adjoint systems on 3-dimensional algebraic varieties by means of the theory of harmonic integrals. Let y be a non-singular alge­ braic variety of dimension 3 imbedded in a projective space, K be a canonical divisor on y and S be an arbitrary (possible reducible) surface on y with only ordinary singularities i.e. a double curve A on which there is a finite number of ordinary cuspodial points a,nd of triple points of S. The problem concerning the theorem of Riemann-Roch for the adjoint system |k + s| of S consists in expressing the dimension of |k + s| in terms of the virtual characters of K + S and other nu­ merical characteristics of S, e.g. the number of linearly independent simple or double differentials of the first kind on y vanishing on S, the number of connected components of S, etc. First, in Sections 1 to 7, we shall consider a more general case in which y is a 3-dimensional compact Kahlerian manifold and, using the theory of harmonic integrals, prove a formula expressing the number of linearly independent meromorphic triple differentials on y which are multiples of S in terms of the virtual arithmetic genus a(s) of S, the constant a (y) defined as a (y) = r^ - rg + r^, r^ being the number of linearly independent v-ple differentials of the first kind on y, and other numerical characteristics of S. In the following Section 8, we shall return to the case in which y is algebraic and show that the dimension of |K + S| is equal to a(S) + a(y) - 1 if the complete linear system (s| partially contains a hyperplane section E of y in the sense that ]S - E j contains a surface with ordinary singularities only. In section 9, we shall prove that the virtual arithmetic genus a(S) defined in terms of the arithmetic genera of irreducible components of S and of several numerical characteristics of the 3 2 singularities of S is represented as a(S) = (2S + 3KS + 2 K S + CS)/12 - 1 where C is the canonical curve on y and S 3 , KS 2 ,... denote the topological intersection numbers I(S,S,S), I(K,S,S),.... Then we shall define the virtual arithmetic genus a(D) of an arbitrary divisor D on y by setting a(D) = (2D^ + 3KD^ + K^D

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+ CD)/12 - 1. In Section 10, we shall prove the theorem of Riemann-Roch for the adjoint system |D| = |k + s| of S which says that dim |d | = - ir (D^) + a(D) - P_(y) + 3 - m + P - K + ô + n where ir (D^) = + 1/2 KD + 1, P^fy) = KC/12 - a(y) + 2, m is the number of connected components of S, 1 or k is the number of line­ arly independent double or simple differentials of the first kind on y which vanish on S, 6 is the deficiency of S with respect to simple differentials and n is the deficiency of the singularities of S. Then we shall deduce two postulation formulae and show that P^fy) is the arithmetic genus of y. In Section 11, we shall consider an arbitrary but fixed non-singular surface S on y and an arbitrary divisor D on y such that the complete linear system |S - D| contains a surface which is connected outside the surface S cut out by the com­ plete linear system |d ( in terms of simple differentials of the first kind on y. Then we shall deduce from it a sufficient condition for D in order that |d | cuts out on S a complete linear system. Using the results of Section 11, we shall prove in the following Section 12 a conjecture of Severi to the effect that the arithmetic genus P^(vi) is equal to a(y) = r^ - rg + r^. In Section 13, we shall first prove that the deficiency of the characteristic system of a sufficiently ample com­ plete linear system on y is equal to the number r^ of the linearly: independent .simple, differentials .of the first kind on y and then show that there exist exactly 2r^^ independent simple Picard integrals of the second kind on y. Finally, in the last Section 14, we shall deduce for an arbitrary irreducible non-singular surface S on y a formula expressing the dimension of the complete linear system |s|. The author wishes to express his grateful thanks to Professors A. Weil, 0. Zariski, S. Chern and D. C. Spencer for valuable suggestions given during the period of preparation of this paper. In particular, the proof of the adjunction formula is due to S. Chern. As soon as Serre and Cartan started using coherent faisceau in April 1952, Kodaira saw that this gave him one of his essential tools and in a few months he had (collaborating in part with D. C. Spencer, and independent of Serre as explained in the introduction to "On cohomology groups of compact analytic varieties with coefficients in

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 106 some analytic faisceaux") obtained far reaching results in that theory, using in particular theorems from the theory of elliptic partial differential equations. Encouragement and stimulation came from A. Weil. D. C. Spencer began a most fruitful collaboration with Kodaira which eventually produced 12 papers, four of them in 1953; "On arithmetic genera of algebraic varieties," May 5, 1953 [123], "Groups of complex line bundles over compact Kahler

varieties," May 29, 1953 [124], "Divisor class groups on algebraic varieties," May 29, 1953 [125], "On a theorem of Lefschetz and the Lemma of Enriques-

Severi-Zariski," October 8, 1953 [126]. The Riemann-Roch theorem was one of the early themes in these papers, as it was a powerful tool in the study of compact analytic surfaces (i.e^ compact complex manifolds of dimension 2). Spencer also realized the usefulness of faisceau for algebraic geometry, but the methods of Kodaira and Spencer differ from those of Serre (as discussed below). In his paper "Some results in the transcendental theory of algebraic varieties" [119] of 1949 Kodaira had written in the introduction Recently the theory of analytic sheaves, developed mainly by H. Cartan, has been successfully applied to algebraic geometry by J. P. Serre. This theory is inde­ pendent of potential theory. On the other hand it is possible to develop a theory of analytic sheaves on compact complex manifolds by a potential theoretic method. The present note is a survey of some results in this direction and their applications to algebraic geometry.

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H. Weyl wrote that Princeton decreed "sheaf" should be the translation of "faisceau". That occurred during 1953

when the Mathematical Review authors switched from using "fibre bundles" as Eilenberg had in 1946 to "faisceau" which was consistent between 1946 and 1953 to "sheaf" which began in 1954. (See footnote page 94.) F. Hirzebruch, then professor at the University of Erlangen, was awarded a scholarship to the Institute for Advanced Study for the years 1952-1955. In 1952 he had become interested in the Riemann-Roch theorem and he linked the Chern classes of vector bundles on algebraic varieties to earlier invariants introduced in algebraic geometry by

Egar and Todd. A little while later Serre was able to guess what the formulation of the general Riemann-Roch theorem would be using a theorem of finiteness of Kodaira's type. But a proof was still lacking. During the years Hirzebruch was at Princeton, he worked closely with Kodaira (using Kodaira's result that every Hodge variety is algebraic) and Spencer and was able to read their correspondence with Serre. He said his aim was to apply, alongside the theory of sheaves, the theory of characteristic classes and results of René Thom in his new theory of cobordism. In 1954 he was able to produce a sheaf theoretic treatment of the Riemann- Roch theorem. He knew then however that it was not simple and better proofs would come later (see the section on Grothendieck below) [100, p. vii].

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For Kodaira*s achievement in the theory of harmonic integrals and its applications to Kahlerian (and in particu­ lar algebraic) varieties he was co-recipient with Serre of the Fields medal at the International Congress of Mathe­ maticians, Amsterdam, 1954.^

The Fields Medal has been awarded to a number of the mathematicians named here— Kunihiko Kodaira (1954), Jean-Pierre Serre (1954), Paul Cohen (1970) , Alexander Grothendieck (1970)— and to others who made contributions to the development of sheaf theory— David Mumford (1974), Michael Francis Atiyah (1974) , and Heisuke Hironaka (1974). J. C. Fields, who set up a trust for the gold medals that constitute the award, said they should be made "in recogni­ tion of work already done and as an encouragement for further achievement on the part of the recipient." Georges de Rham (as chairman of the 1966 Fields medal committee) presented the awards to Atiyah, Cohen, and Grothendieck and pointed out that the committee confines its choice to candidates under the age of forty. Two medals were presented at the Congress held every fourth year since 1937 except during the war years. However, because of the vast development of mathematics since 1940, four were presented in 1966. Kodaira and Serre were presented their medals by Hermann Weyl, chairman of the 1954 committee, who spoke of their work in sheaf theory. In 1970 at the International Congress in Moscow, Henri Cartan spoke on Atiyah*s work, and Dieudonné gave the talk describing Grothendieck's work. However, Grothendieck was not present to receive the award. Grothendieck influenced Mumford, but both Mumford's and Hironaka*s principal teacher was Oscar Zariski.

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ABSTRACTION OF THE THEORY Oscar Zariski wrote in the preface to his Collected Papers [240, p. x] that when he was nearing forty he made a radical change in the direction of his research. The circumstances that brought about the change were personal and professional. As a student at the University of Kiev (1917-1921) he had become interested in algebra and number theory, but developed more of an interest in algebraic geometry when he became a student in Rome in 1921, at the age of 22. He remained there until 1927, working with his teacher Federigo Enriques^ and being influenced by Francesco Severi and Guido Castelnuevo. However ten years after he left Rome for Johns Hopkins he reoriented his research and began to introduce ideas from abstract algebra into alge­ braic geometry. From 1929 to 1937 he had often been in Princeton seeing Solomon Lefschetz at the Institute for

Federigo Enriques was by then 50 years old and his primary interest had become the philosophy and history of science. He was editor of a series of books entitled Per la Storia e la Filosophia delle Matematiche in which series Zariski published R Dedekind, Essenza e Significato dei Numeri. Continuatà e Numeri Irrazionali, Traduzione dal tedesco e note storico-oritiche di Oscar Zariski in 1926.

109

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Advanced Study and owed much of the change to his encourage­

m e n t . ^ In 1935 he published his Ergebnisse monograph Alge­ braic Surfaces [233]. In it he tried to present the underlying ideas that the Italian geometers had used and, discovering that their proofs were incomplete and imprecise, had "paid the price" of his "own personal loss of the geo­ metric paradise" in which he was so happily living before. Consequently, at age 37, he began studying modern algebra which had "come to life through the work of Emmy Noether and her pupil Bartel Leendert van der Waerden" [240, 2 p. vi]. Zariski said, "I had to begin somewhere and it was not by accident that I began with the problem of local uniformization and reduction of singularities. At that time there appeared the Ergebnisse monograph Idealtheorie [131] of Wolfgang Krull, emphasizing valuation theory" [240,

p. xi].

Lefschetz wrote in "Reminiscences of a mathematical immigrant" [139] that his appointment as a Research Professor which had no assigned duties in 1932 at Princeton, where he had been since 1923, and the establishment of the Institute for Advanced Study with mathematics as its strongest group where he came into contact with "a top authority in topology" — Alexander— caused his own mathematical work to progress. 2 Noether and van der Waerden's influence is of course observable in much of the literature pertinent to the creation of sheaf theory. For example her influence is vividly clear in Weyl's Gruppentheorie und Quantenmechanik, [230] and Weyl in turn was a major influence on Kodaira. P. S. Alexandroff addressed the Moscow Mathematical Society September 5, 1935 saying [3, p. 1]; "In 1924-25 the school of Emmy Noether made one of its most brilliant acquisitions; a graduating Amsterdam

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ill Zariski was the first person to use the general theory of valuations which Krull developed, no doubt to the surprise of Krull who had announced [240, p. xi] "the general concept of valuation (including nondiscrete valuations and valuations of rank > 1) was not likely to have applications in algebraic geometry." Contrariwise, Zariski saw that it could be "useful for the analysis of singularities and for the problem of reduction of singularities" [240, p. xi]. Zariski also used ideas of Marshall Stone, then professor at Harvard, who wrote in 1938 [201, p. 24] "A cardinal principle of modern mathematical research may be stated as a maxim: 'one must always topologize.'" As Saunders Mac Lane pointed out in "The Influence of M. H. Stone on the Origins of Category Theory" [158, p. 229] the "emphasis on the use of all relevant mathematical entities at hand is typical of Stone's influence on the development of mathematics. The decade 1929-1939 emphasized the study of a variety of mathematical structures, and so set the stage for many future developments." Mac Lane

student, B. L. van der Waerden, became her pupil. He was then 22 years old and one of the brightest young mathematical talents in Europe. Van der Waerden quickly mastered the theories of Emmy Noether, enlarged them with important new findings, and like no one else promoted her ideas. A course in the theory of ideals, given by van der Waerden in 1927 in Gôttingen, was enormously success­ ful. The ideas of Emmy Noether in the brilliant exposition of van der Waerden subdued mathematical opinion.... From 1927 the influence of the ideas of Emmy Noether on contemporary mathematics grew and along with it grew scientific praise for the author of those ideas."

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 112 summarized this decade by listing some of "the most influential books published in this period" [158, p. 228]: Alexandroff-Hopf, Topologie, 1935; Stefan Banach, Théorie des Opération Linéaires, 1932; Nicolas Bourbaki, Les Structures Fondamental de 1*Analyses, 1939; Bartel Eeen'deiSb van der Waerden, Moderne Algebra, 1930-1931; André Weil, Spaces with Uniform Structure and General Topology, 1938; Hermann Weyl, Gruppentheorie und Quantenmechanik, 1928. By 1942, Zariski, having examined the literature on algebraic surfaces, and having done the necessary prepara­ tory work stated [236, p. 402] "...we have to know a lot more about birational correspondences than we know at present before we can even attempt to carry out the resolution of singularities of higher dimensional varieties. A general theory of birational correspondences is a necessary prereq­ uisite for such an attempt." He simplified his 1939 proof that an algebraic surface over a ground field of character­ istic zero can be birationally transformed into one without singular points 1234] and in 1944 he proved a

desingularization theorem for [237] dimension 3. He had seen that some ideas which were algebraic could be better expressed in topological terms. For example, if V is the set of all valuations of K which vanish on k (K is an arbitrary of algebraic functions over a field k) Zariski defined on V a topology for which V became quasi-compact, not in general Hausdorff (i.e.^dimension 1). André Weil subsequently used the Zariski topology which is the structure wherein the intersection and finite

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unions of subvarieties on a variety V are also subvarieties; thus the subvarieties can be taken as the system of closed sets of a topology on V. While Weil was Professor for the Faculdad de Filosofia da Universidade de Sào Paulo, the Amer­

ican Mathematical Society published his book Foundations of

Algebraic Geometry [219]. There, in 1946, Weil freed algebraic varieties from projective space. He acknowledged the tremendous influence of Zariski, and commented that the reader [219, p. iv]" cannot easily imagine how much benefit I have derived... from personal contacts both with Zariski

and with Chevalley." In the introduction Weil wrote [219,

p. v] This book has arisen from the necessity of giving a firm basis to Severi's theory of correspondences on algebraic curves, especially in the case of characteristic p ^ 0, this being required for the solution of a long outstand­ ing problem, the proof of the Riemann hypothesis in function fields. ...Severi, van der Waerden and Zariski have made contributions to remedy the defects. The language and results in the book were applied to other theories, including that of abelian varieties, and according to Dieudonné, Weil [56, p. 251] "singlehandedly created the general theory of abelian varieties over fields of arbitrary characteristic." Weil remarked about Chapter VII [219,

p. iii] It is well known that classical algebraic geometry does not usually deal with varieties in affine spaces, but with so-called projective models; the main feature which distinguished the latter from the former is that they are, in a certain sense "complete" or in the topological case compact. Nevertheless, local properties of vari­ eties in projective spaces are almost always to be studied most conveniently on affine models of such varieties. There is now no reason why affine models, which can thus be pieced together so as to give a complete description of a given variety in a projective

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 1 4 space, may not be pieced together differently; and there are problems which cannot at present be handled other­ wise than by such a procedure. This leads to the following definition of abstract variety in Chapter VII:

Let now the V , for 1 < a < h, be h varieties; let, for each a, F a be a frontier on V a ; and let the T„ pa be coherent birational correspondences between the V^, such that the following condition is satisfied: (A) Whenever, for any a and g, is a point in V - F and P„ a point in V^ - F., such that (P^, P„) is u CX p p p CX p in Tg^, then P^ and P^ are regularly corresponding points of V and of V. by T. . ^ a 6 Ba When that is so, we say ththat the varieties V^, the n frontiers F , and the birational correspondences T ga define an abstract variety of which the V^ are called the representatives, and which we denote by [V^; F^; T_ ]; if, moreover, k is a common field of definition P OL for all the T„ga , hence also for all the V a , and if the F^ are all normally algebraic over k, then we say that k is a field of definition for the abstract variety [V^: F ; T. ], and that the latter is defined over k. The a ga abstract variety [V^: F^; T^^] being given, it always has fields of definition, e.g. any common field of defini­ tion for the Tg^ and for all the components of the F^; but there need not exist a smallest field of definition for it. However, if k is a field of definition for an abstract variety, then, by coroll. 2 of th. 1, Chap. IV, §1, and by prop. 12 of Chap. XV, §1, every field contain­ ing k is also a field of definition for that abstract variety. Next, Weil continued with a detailed treatment of divisors on a variety. In 1950 as professor at the University of Chicago Weil lectured on "Fibre Spaces in Analytic Geometry" [224]. He was able to extend to these abstract varieties the close

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relations he observed between the algebraic notion of divisors and the topological notion of fibre bundles. He observed that to a divisor D on a complex manifold was naturally attached a complex vector bundle of rank 1 (a line bundle). This idea of vector bundles had come from differ­ ential geometry, after Elie Cartan, the father of Henri Cartan, had invented moving frames in the 1930s.. ?•' Together then was formed a definition of vector bundles over differentiable manifolds. Weil then trans­ ferred vector bundles to abstract varieties using the Zariski topology.

As Dieudonné related in his biographical sketch of Eli Cartan [53, p. 96], Cartan's "guiding principle [in differential geometry] was a considerable extension of..."moving frames" of Darbou and Ribicour, to which he gave a tremendous flexibility and power, far beyond anything that had been done in classical differential geometry. In modern terms, the method consists in associating to a fiber bundle E the principal fiber bundle having the same base and having at each point of the base a fiber equal to the group that acts on the fiber of E at the same point. If E is the tangent bundle over the base, ...the corresponding group is the general linear group.... Cartan's ability to handle many other types of fibers and groups allows one to credit him with the first general idea of a fiber bundle, although he has never defined it explicitly. This concept has become one of the most important in all fields of modern mathematics, chiefly in global differen­ tial geometry and in algebraic and differential topology. ...Cartan's recognition as a first rate mathematician came to him only in his old age.... This was due partly to the fact that in France the main trend of mathematical research after 1900 was in the field of function theory, but chiefly to his extraordinary originality. It was only after 1930 that a younger generation started to explore the rich treasure of ideas and results that lay buried in his papers."

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 1 6 In 1953 Serre had the idea of transferring (in the same way Weil had done) the theory of sheaves to abstract

varieties using the Zariski topology, rather than the

topology Leray had used. As Zariski pointed out in his

report of the American Mathematical Society Second Summer

Institute which will be discussed below, "The Zariski

topology is much weaker than the usual Hausdorff topology,

but it has a perfect meaning in the abstract case [239, p. 120]." At the same time Serre observed that the concept of sheaf could be used to simplify Weil's definition of abstract variety, by using the idea of ringed space which had been invented by Cartan. As Cartan wrote in his "Analyse des travaux" [35, p. 11], C'est après 1950 gu'apparaît la nécessité de généraliser la notion de variété analytique complexe, pour y in­ clure des singularités d'un type particulier, comme on le fait en Géométrie algébrique. Par exemple, le quo­ tient d'une variété analytique complexe par un groupe proprement discontinu d 'automorphismes n'est pas une variété analytique en général (s'il y a des points fixes), mais c'est un espace analytique....Des 1951, BE H N K E et ST E I N tentaient d'introduire une notion d'es­ pace analytique en prenant comme modèles locaux des "revêtement ramifiés" d'ouverts de C^; mais leur définition était assez peu maniable. La première tentative date de mon Séminaire 1951-52 (Exposé XIII); j'ai repris cette définition des espaces analytiques dans mon Séminaire de 1953-54 en introduisant la notion générale d'espace annelé, qui a ensuite été popularisée par Serre, puis par GRAÜERT et GROTHENDIECK. En 1953-54, ma définition conduisait aux espaces analytiques normaux (c'est-à-dire tels que l'anneau associé à chaque point soit intégralement clos). C'est SERRE qui, le premier, attira l'attention sur l'utilité d'abandonner la condition restrictive de normalité.

The advantage of this kind of structure was that it could be collated along the open subsets and it is trivial to verify the conditions of compatability.

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Serre discovered that the sheaf theoretical methods which he had applied with Cartan in analytic geometry could be adapted to algebraic geometry over an arbitrary field.

The development of Serre's work in sheaves had the geneology shown in the following diagram. Serre wrote four papers developing these ideas, the first of which was available in mimeographed form at the Second Summer Institute. The only record of this is in the March 1956 Bulletin of the American Mathematical Society in a report written by Zariski. Jun-Ichi Igusa, who had come in 1953 from Japan as a re­ search associate at Harvard, Serge Lang, and Gérard Washitzer, were the most active members of the seminar, which also included Serre and Spencer. According to Zariski

[239, p. 117] The participants of the seminar had in their possession a short manuscript of Serre FAISCEAUX ALGÉBRIQUES written in the spring of 1954 in which he gave the defi­ nitions and stated (almost without proof) the basic results of the theory. It seemed fruitful to take this manuscript as the subject of the seminar and to try to reconstruct the proofs, this being the best way of acquainting ourselves with the new algebraic methods, ideas and results. The mimeographed paper which Serre wrote in the fall of 1953 "Faisceaux analytiques cohérents sur une variété algébrique" was not published, but Zariski included one sheaf theoret­ ical proof from it in his report. The proof used the m-fold differentials on V and was an adaption of an argument used by Weil in "Zur algebraischen Theorie der algebraischen Funktionen" of 1938 [218] in the one dimensional case [239, p. 138]:

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ENRIQUES

CASTELNUEVQ

SEVERI

E. ARTIN

VAN DER WAERDEN

KRULL

STONE

'LEFSCHETZ ZARISKI

WEIL

LERAY

KODAIBA SEREΠCARTAN,

:ARTAN - EILENBERC

DIEUDONNE lOLBEAULT CARTAN BUCHSBAUM GRAUERT

FRENKEL KAN

THE GENEOLOGY OF THREE IMPORTANT PUBLICATIONS 1. "Faisceaux algébriques cohérents", 2. "Sur quelques points d'algèbre homologiques", 3. Topologie Algébrique et Théorie des Faisceaux.

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We denote by of(D) the sheaf of germs of p-fold differ­ entials on V which are (locally) multiples of ——D. This is a coherent sheaf, since it is locally isomorphic with the (p)-fold direct product of the sheaf 0. Note that n^(D) = L(D). We write for of(0): this is the sheaf of germs of holomorphic differentials. First of all one proves that H^(V, is of dimen­ sion 1 over k. For m = 1 this is a consequence of the residue theorem for abelian differentials on an alge­ braic curve V ("there exists a differential with preassigned poles of order <1 and preassigned residues at these poles provided the sum of residues is zero"). The general case can be reduced to the case m = 1 by the inductive procedure described in §6. The fundamental exact sequence (4) of §6 to be used in this case is the one in which F is replaced by and C is replaced by C^, n large. A similar argument shows that H™(V, 0^(D)) is zero for any strictly positive divisor D. These two assertions can be regarded as generalizations of the "residue theorem" for abelian differentials. Now, if 0) is any element of (V, J2™(— D) ) (i.e., w is an m-fold differential which is globally a multiple of D), then the multiplication by w of each stalk of L(D) defines an element of Hom [L(D), fi™]. Therefore u also defines an element Jp(“) of Hom^ [h "^(V, L(D)), H^(V, fi’”)], and since H^(V, 0^) ^ k, Jq (“) is an element of the dual space of H^(V, L(D)). We thus have a mapping of (V, 0^(— D)) into the dual space of H^(V, L(D)). We shall denote this dual space by W(— D). Since (V, $2^(— D) ) is obviously isomorphic with H^(V, K — D), the duality relation (4) will follow if it is proved that is an isomorphism. If D is a multiple of D', the cokernel of the inclusion map i: L(D') -»■ L(D) is carried by a variety of dimension m - 1. Therefore i*:H^(V, L(D')) H^(V, L(D)) is an epimorphism, hence the dual of i* is a monomorphism as a mapping of W (— D) into W (—D ' ). We can consider the inductive limit W of W(-D) by these maps. It is easy to introduce in W a structure of a vector space over the function-field K of V such that the inductive limit J of is a K-linear mapping of the space of all global m-forms on V into W. Here,— and this is the key point of the proof— if J(w) is contained

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 120 in W(- D), then w is necessarily contained in H®(V) , D) ) . The proof is indirect and is as follows: If we assume that w is not contained in (V, D) ) , the reduced expression of ( w) - D is of the form where is non-negative and is strictly positive. Let and M 2 be cokernels of the inclusion maps L((u) - C^) -+ L(D) and L((w)) -»■ L((w) + , re­ spectively. Then we get the following commutative diagram of exact sequences:

+ " 1(V, M^) h "^(V, L((w) - C^)) -s- H^(V, L(D)) 0

" 1(V, Mg) h "^(V, L((w))) ->-0 + 0 We note that (u) : H^^V, L(w)) h "‘(V, fi"*) is an isomorphism. Moreover, since J(w) is contained in W(- D), by definition, = C^(w), i.e., the product

of H*xv, L((w) - C,± )) + H™XV, L((w))) and J, f oj; . (w) vanishes on the image of ^(V, M^) in L((w) - C^)). However the commutativity and the exact­ ness of the diagram show the opposite of this assertion, and this gives us the desired contradiction. As a simple consequence, we see that J is an isomor­ phism. On the other hand, W is of dimension one over K. In fact, let C be a general hyperplane section of V. Then, by an induction on m, we can show that ord (V)«n™/m! is the dominant term of dim H™(V, L(D - nC)) as a function of n, this function being equal to dim W(- D + nC). Since ord (V)»n™/m! is the highest term ofdim |nC| aspolynomial in n for n sufficiently large, we canreadily get a contradiction if W is not of dimension one over F(V). Since J is a monomorphism, it is now shown that J is an isomorphism. If we apply the key result which we have established earlier, we conclude that is an isomorphism. Thus (4) is proved. (4) h™(D) = h°(K - D) , where K is again a canonical divisor on V, i.e., the divisor of an m-fold differential on V. This equality follows readily from the so-called "lemma of Enriques- Severi-Zariski" proved by Zariski [238] which states

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that "if D is any divisor on V and is a general sec­ tion of V by a hypersurface of order n, then for n sufficientlylarge the linear system Tr^^ |D|, cut out on by the complete linear system |D|, is itself complete." An equivalent formulation is the following equality; (5) dim |d | = dim |D«C^|,

if n is large, since for large n we have dim |d | = dim Tr^ |D|. n Another result that is equivalent with the lemma of Enriques-Severi-Zariski is the following; (9) h^(D - C^) = 0,

if n is large. A seminar on complex manifolds was also held at the Second Summer Institute and the material there was also published in the Bulletin of the American Mathematical Society by Shing-shen Chern, then professor at the Univer­ sity of Chicago. Participants in that seminar, which will be discussed below, included Armand Borel, Kodaira, Spencer,

Wang and Weil among others. The paper of Serre discussed above, which he wrote in the fall of 1953, was followed by a presentation for the Seminare Bourbaki in March 1954 "Faisceau analytiques,"

[191]: Cet exposé fait suite à celui de Décembre 1952 auquel nous renvoyons pour tout ce qui concerne les faisceaux analytiques cohérents, les groupes de cohomologie, etc. Dans cet exposé il était principalement question de variétés de Stein (généralisation naturelle des domaines d'holomorphie); ici, au contraire, nous nous occuperons surtout de variétés algébriques projectives (donc compactes).

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 122 Serre stated several theorems. The first was a generaliza­ tion of results of Dolbeault, and utilized ideas of Grothendieck: THÉORÈME 1. - H*^(X, 0^) est isomorphe au q-ième groupe de cohomologie du complexe des sections des muni de l'opérateur cobord d. In section 2, "La Dualité analytique;" Les hypothèses étant celles du n° 1, soit A^'^ (resp. l'espace vectoriel des sections à supports quelconques (resp. compacts) du faisceau 0.^'^ (resp. K^'^). Si l'on munit A^'^ de la topologie de la conver­ gence compacte pour chaque dérivée, on obtient un espace de Fréchet (analogue a l'espace ç de Schwartz); on voit aisément que le dual de cet espace de Fréchet n'est autre que P/ où V* désigne l'espace fibre dual de V. En outre, le transposé de d" vis-à-vis de la dualité précédente, est égal à d" (au signe près). Par passage à la cohomologie, on obtient, compte-tenu du théorème 1; THÉORÈME 2. - Si les deux opérateurs A^' ^ ^ ^ A^'^ ^ A^'^ ^ ^ sont des homomorphismes, alors les espaces vectoriels H^(X, n^) et H* ^(X, ^) sont en dualité séparante. THÉORÈME 3. S i X est une variété compacte, les H^(X, F) sont des espaces vectoriels de dimension finie, quel que soit le faisceau analytique cohérent. (Ce démontre en utilisant deux recouvrements de Stein emboîtes, ainsi qu'un théorème de Schwartz sur les opérateurs complètement continus.) Cas particulier. - Si D est un diviseur de X, soit F^ le faisceau des germes de fonctions méromorphes > - D; on a = S^, où V est l'espace fibré (de dimen­ sion 1) associé à D; supposons qu'il existe au moins une forme différentielle méromorphe de degré n non identiquement nulle, et soit K son diviseur (diviseur canonique); on a alors

oÇ = Sy = et

^V* ~ ^ ® ~ ® D ~ - D ’

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 2 3 CORROLAIRE. - H^(X, F^) et H* '^(X, sont en dualité séparante. In Section 3, entitled "Faisceaux Analytiques Cohérents sur les Variétés Algébriques," Serre proved a sheaf is always algebraic in a compatible sense: THÉORÈME 4. - Soit F un faisceau analytique cohérent sur la variété algébrique X. Pour toute entier m assez grand, le faisceau F(m) = F S ^ F^^ vérifie les deux propriétés suivantes:

A) Pour tout X E X, H°(X, F(m)) engendre F(m) X (considéré comme 0^ - module) B) n9(X, F(m)) = 0 si q U. (Noter l'analogie avec les théorèmes A et B de la théorie des variétés de Stein). Pour démontrer le théorème 4, on se ramène tout d'abord au cas où X = P^(C) on procède alors par recur­ rence sur r, le théorème 3 jouant un rôle essentiel dans la démonstration (analogue à celle de [4]). As an application Serre discussed the lemma of Enriques- Severi, but did not include the proof which Zariski outlined above. Theorem 6 followed from theorem 4: THÉORÈME 6. - Tout faisceau cohérent d'idéaux est engendre par les polynômes qu'il contient. In the spring of 1954 Serre also wrote "Faisceaux algébriques," which was not published, but mentioned by

Zariski in his report. "Faisceaux algébriques cohérents" [192], was received for publication on the 8th of October, 1954, by the Annals of Mathematics and actually published in March 1955. There Serre developed the purely algebraic theory of sheaves and gave the first systematic application of cohomological algebra to abstract algebraic geometry. Serre began the introduction [192, p. 197]:

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On sait que les méthodes cohomologiques, et parti­ culièrement la théorie des faisceaux, jouent un rôle croissant, non seulement en théorie des fonctions de plusieurs variables complexes [cf, 191], mais aussi en géométrie algébrique classique (qu'il me suffise de citer les travaux récents de Kodaira-Spencer sur le théorème de Riemann-Roch) . The caractère algébrique de ces méthodes laissait penser qu'il était possible de les appliquer également à la géométrie algébrique abs­ traite; le but du présent mémoire est de montrer que tel est bien le cas. Chapter I was on the general theory of sheaves which has been in part translated as an introduction to this thesis. Serre pointed out that [192, p. 197] Au §3 sont définis les groupes de cohomologie d'un espace X à valeurs dans un faisceau F. Dans les applications ultérieures, X est une variété algébrique, munie de la topologie de Zariski, donc n'est pas un espace topologique séparé, et les méthodes utilisées par Leray [150] ou Cartan [31] (basées sur les "parti­ tions de l'unité," ou les faisceaux "fins") ne lui sont pas applicables; aussi avons-nous dû revenir au procédé de Cech, et définir les groupes de cohomologie H^(X, F) par passage à la limite sur des recouvrements ouverts de plus en plus fins. Une autre difficulté, liée à la non-séparation de X, se rencontre dans la "suite exacte de cohomologie" (cf. nos. 24 et 25): nous n'avons pu établir cette suite exacte que dans des cas particuliers, d'ailleurs suffisants pour les applications que nous avions en vue (cf. nos. 24 et 47). Chapter II included a definition of varieties analogous to that of Weil [192, p. 226]: 34. Définition de la structure de variété algé­ brique. DEFINITION. On appelle variété algébrique sur K (ou simplement variété algébrique) un ensemble X muni: 1° d'une topologie, 2° d'un sous-faisceau 0^ du faisceau F(X) des germes de fonctions sur X à valeurs dans K, ces données étant assujetties à vérifier les axiomes (VAj) et (VAjj) énoncés ci-dessous. La topologie de X sera appelée "topologie de Zariski" de X, et le faisceau sera appelé le faisceau des anneaux locaux de X.

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(VAj) - Il existe tin recouvrement ouvert fini B = (^6 l'espace X tel que chaque muni de la structure induite par celle de X, soit isomorphe à un sous-espace localement fermé d'un espace affine, muni du faisceau 0 défini au n° 31. ... ^i (VAjj) - La diagonale A de X x X est fermée dans X X X.

This is analogous to the separation condition of analytic

differentiable topologies. L 'axiome (VA^^) prend donc la forme suivante; (VAjj) - Pour tout couple (i,j), T^^ est fermé dans U. X U.. i 3 Sous cette forme, on reconnaît l'axiome (A) de Weil (cf. [224], p. 167), à ce près que Weil ne considère que des variétés irréductibles. The variety that Serre introduced here was slightly more general than that of Weil because it included reducible varieties (unions of finite numbers of Weil varieties.) The structure was defined by giving it [192, p. 224] 31. Sous-ensembles localement fermés de l'espace affine. Soit r un entier 2 0, et soit X = l'espace affine de dimension r sur le corps K. Nous munirons X de la topologie de Zariski; rappelons qu'un sous- ensemble de X est fermé pour cette topologie s'il est l'ensemble des zéros communs à une famille de polynômes e K[X^, ..., X^]. Puisque l'anneau des polynômes est noethérien, X vérifie la condition (A) du n° précédent; de plus, on montre facilement que X est un espace irré­ ductible. Si X = (x^, ..., x^) est un point de X, nous note­ rons l'anneau local de X, rappelons que c'est le sous-anneau du corps K(X^, ..., X^) formés des fractions rationnelles R qui peuvent être mises sous la forme; R = P/Q, où P et Q sont des polynômes, et Q(x) ^ 0, Une telle fraction rationnelle est dite régulière en x; en tout point x e X où Q(x) ^ 0, la fonction X ^ P(x)/Q(x) est une fonction continue à valeurs dans K (K étant muni de la topologie de Zariski) que l'on

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peut identifier avec R, le corps K étant infini. Les 0^, X e X, forment donc un sous-faisceau 0 du faisceau F(X) des germes de fonctions sur X à valeurs dans K (cf. n® 3); le faisceau O est un faisceau d'anneaux. Nous allons étendre ce qui précède aux sous-espaces localement fermés de X (nous dirons qu'un sous-ensemble d'un espace X est localement fermé dans X s'il est l'in­ tersection d'un sous-ensemble ouvert et d'un sous- ensemble fermé de X). Soit Y un tel sous-espace, et soit F(Y) le faisceau des germes de fonctions sur Y a valeurs dans K; si x est un point de Y, l'opération de restriction d'une fonction définit un homomorphisme canonique

= F ' X ' x * L'image de par est un sous-anneau de F(Y)^, que nous désignerons par 0 les 0 forment un sous- X / X X f y faisceau Oy de F CY), que nous appellerons le faisceau des anneaux locaux de Y. Une section de Oy sur un ouvert y de Y est donc, par définition, une application fiV K qui est égale, au voisinage de chaque point X e V, à la restriction à V d'une fonction rationnelle régulière en x; une telle fonction f sera dite régulière sur V; c'est une fonction continue lorsque l'on munit V de la topologie induite par celle de X, et K de la topologie de Zariski. L 'ensemble des fonctions régulières en tout point de V est un anneau, l'anneau rCV, Oy); observons également que, si / e r(V, Oy) et si /“(x) ^ 0 pour tout X e V, alors 1/f appartient aussi à r(V, Oy). ... On peut caractériser autrement le faisceau 0^: PROPOSITION 4. Soit U (resp. F) un sous-espace ouvert (resp. fermé) de X, et soit Y = U D F. Soit 1(F) l'idéal de K[X^, ..., X^] formé des polynômes nuls sur F. Si X est un point de Y, le noyau de la surjection e^:0^ ®x Y égal à l'idéal I(F)*.0^ de O x . COROLLAIRE. L'anneau 0^ y est isomorphe à l'anneau des fractions de K[X^, ..., X^]/I(F) relatif à l'idéal maximal défini par le point x. In paragraph 2 Serre proved the sheaf of local rings of the affine space is a coherent sheaf of rings, and then extended it to an arbitrary algebraic variety in

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Proposition 1; "Si V est une variété algébrique, le faisceau 0^ est un faisceau cohérent d'anneaux sur V" [192, p. 231]. He gave the basic definitions for [192, p. 231] 38. Faisceaux algébriques cohérents. Si V est une variété algébrique dont le faisceau des anneaux locaux est 0^, nous appellerons faisceau algébrique sur V tout faisceau de 0,^-modules, au sens du n® 6; si F et G sont deux faisceaux algébriques, nous dirons que ^o:F G est un homomorphisme algébrique (ou simplement un homo­ morphisme) si c'est un 0^-homomorphisme; rappelons que cela équivaut a dire que chacun des ^^:F^ G^ est 0 -linéaire et que transforme toute section locale X , V de F en une section locale de G. Si F est un faisceau algébrique sur V, les groupes de cohomologie H^(V, F) sont des modules sur r(V, O^), cf. n® 23; en particulier, ce sont des espaces vecto­ riels sur K. Un faisceau algébrique F sur V sera dit cohérent si un faisceau cohérent de 0 -modules, au sens du n® 12; vu la Proposition 7 du n® 15 et la Proposition 1 ci- dessus, un tel faisceau est caractérisé par le fait qu'il est localement isomorphe au conoyau d'un homomorphisme algébrique

Examples included [192, P« 231]j 1. 39. Faisceau d'idéaux défini par une sous-variété fermée. Soit W une sous-variété fermée d'une variété algébrique V. Pour tout x e V, soit (W) l'idéal de O formé des éléments f dont la restriction à W est X , V nulle au voisinage de x; soit I(W) le sous-faisceau de 0,^ formé par les I^(W). On a la proposition suivante, qui généralise le Lemme 2; PROPOSITION 2. Le faisceau l(W) est un faisceau algébrique cohérent.

2. Soit 0^ le faisceau des anneaux locaux de W, et soit 0^ le faisceau sur V obtenu en prolongeant 0^ par 0 en

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dehors de W (cf. n®5); ce faisceau est canoniquement isomorphe à 0^/1(W), autrement dit, on a une suite exacte; 0 + I(W) ->■ 0„ 0. V w Soit alors F un faisceau algébrique sur W, et soit F^ le faisceau obtenu en prolongeant F par 0 en dehors de W; on peut considérer F^ comme un faisceau de 0^-modules, donc aussi comme un faisceau de 0 -modules dont l'annu- lateur contient I(W). On a; PROPOSITION 3. Si F est un faisceau algébrique cohérent sur W, F^ est un faisceau algébrique cohérent sur V. Inversement, si G est un faisceau algébrique cohérent sur V dont 1'annulateur contient I(W), la restriction de G â W est un faisceau algébrique cohérent sur W. 3. 40. Faisceaux d'idéaux fractionnaires. Soit V une variété algébrique irréductible, et soit K(V) le faisceau constant des fonctions rationnelles sur V (cf. n® 36); K(V) est un faisceau algébrique, qui n'est pas cohérent si dim V > 0. Un sous-faisceau algébrique F de K(V) peut être appelé un "faisceau d'idéaux fraction­ naires," puisque chaque F est un idéal fractionnaire

PROPOSITION 4. Pour qu'un sous-faisceau algébrique F de K(V) soit cohérent, il faut et il suffit qu'il soit de type fini. 4. 41. Faisceau associé à un espace fibré à fibre vectorielle. Soit E un espace fibré algébrique, à fibre vectorielle de dimension r, et de base une variété algébrique V; par définition, la fibre type de E est l'espace vectoriel K^, et le groupe structural est le groupe linéaire GL(r, K) opérant sur a la façon usuelle (pour la définition d'un espace fibré algébri­ que, cf. [224]; voir aussi [193], n® 4 pour les espaces fibrés analytiques à fibres vectorielles), Si U est un ouvert de V, soit S(E)^ l'ensemble des sections de E régulières sur U; si V D U, on a un homomorphisme de restriction S(E)^ -»■ S(E)y; d'où un faisceau S(E), appelé le faisceau des germes de sections de E. Du fait que E est un espace fibré à fibre vectorielle, chaque S(E) est un r(U, 0^)-module, et il

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 2 9 s'ensuit que S(E) est un faisceau algébrique sur V. Si l'on identifie localement E à V x on voit que; PROPOSITION 5. Le faisceau S(E) est localement isomorphe à O^; en particulier, c'est un faisceau algébrique cohérent. Serre studied the theory of sheaves on affine varieties and showed that the cohomology is trivial in all dimensions > 0 [192, p. 238]: 46. Groupes de cohomologie d'une variété affine à valeurs dans un faisceau algébrique cohérent. Nous allons généraliser la Proposition 7; THÉORÈME 3. Soient X une variété affine, une famille finie de fonctions régulières sur X, ne s'annulant pas simultanément, et U le recouvrement ou­ vert de X formé par les X . = ü .. Si F est un faisceau ^ ^ a algébrique cohérent sur X, on a H^(H, F ) = 0 pour tout q > 0. COROLLAIRE 1. Si X est une variété affine, et F un faisceau algébrique cohérent sur X, on a H^(X, F) = 0 pour tout q > 0. According to Claude Chevalley "this seems to indicate that the natural domain of sheaf theory is the study of algebraico-geometric notions in which the completeness of the variety plays a crucial role" [43]. Serre showed [192, p. 230] THÉORÈME 5. Soient X une variété algébrique, et U = {U.un:recouvrement fini de X par des ouverts affines. Soit 0-^A-»-B-^C-»-0 une suite exacte de faisceaux sur X, le faisceau A étant algébrique cohé­ rent. L 'homomorphisme canonique H^ (II, C) -»■ H^ (11, C) (cf. n® 24) est bijectif pour tout q > 0. ... COROLLAIRE 1. Soit X une variété algébrique, et soit 0^A-»-B-»-C-»-0 une suite exacte de faisceaux sur X, le faisceau A étant algébrique cohérent. L'homomor­ phisme canonique H^CX, C) H^CX, C) est bijectif pour tout q > 0.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 3 0 COROLLAIRE 2. On a une suite exacte; HS(X,B) -s- H^(X,C) + yS + 1(X,A) + + ^(X,B) + ... He defined further [192, p. 240];

§4. Correspondance entre modules de type fini et fais­ ceaux algébriques cohérents 48. Faisceau associé à un module. Soient V une variété affine, O le faisceau des anneaux locaux de V; l'anneau A = r(V, 0) sera appelé l'anneau de coor­ données de V, c'est une algèbre sur K qui n'a pas d'autre élément nilpotent que 0. Si V est plongée comme sous-variété fermée dans un espace affine K^, on sait (cf. n® 44) que A s'identifie à l'algèbre quotient de K[X^, ..., X^] par l'idéal des polynômes nuls sur V; il s'ensuit que l'algèbre A est engendrée par un nombre fini d'éléments. Chapter III was concerned with "Faisceaux algébriques cohé­ rents sur les variétés projectives." Applying [192, p. 239]

THÉORÈME 4. Soient X une variété algébrique, F un faisceau algébrique cohérent sur X, et ü = un recouvrement fini de X par des ouverts affines. L'homomorphisme a(ll) ; H^(U), F) H^(X, F) est bijectif pour tout n > 0. and using the definition of a covering [192, p. 243],

Soit r un entier > 0, et soit Y = K^'*’ ^-{0}; le groupe multiplicatif K* des éléments ^ 0 de K opère sur Y par la formule ;

^(Pq/ ••»/ p^) — ^^^0' ***^ ^ p ^ ) . . Deux points y et y' seront dits équivalents s'il existe A e K* tel que y' = Ay; 1 ' espace ^otiént^ de Y par cette relation d 'équivalence sera noté P(K), où implement X; c'est l'espace projectif de dimension r sur K; la projection canonique de Y sur X sera notée ir. Soit I = {0, 1, ..., r}; pour tout i e I, nous désignerons par t. la i-ème fonction coordonnée sur r + 1 K , définie par la formule; Cp q , ..., p^) — ^i*

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Nous désignerons par V. le sous-ensemble ouvert de r + 1 ^ K formé des points où t^ est ^ 0, et par l'image de par ir; les. forment un recouvrement II de X. Si i e I et j e I, la fonction t^/t^ est régulière sur V^, et invariante par K*, donc définit une fonction sur que nous noterons encore t^/t^; pour fixé, les fonctions tj/t^, j ^ i, définissent une bijection K^.

Serre proved that each of the is isomorphic to K^. Hence [192, p. 244] PROPOSITION 2. Si P est un faisceau algébrique cohérent sur X = P^(K), 1 'homomorphisme a(U):H^(ll, F) -» H^(X, F) est bijectif pour tout n >0. Puisque U est formé de r + 1 ouverts, on a (cf. n® 20, corollaire à la Proposition 2 ;) COROLLAIRE; h ‘^(X, F) = 0 pour n > r. Ce dernier résultat peut être généralise de la façon suivante; PROPOSITION 3. Soit V une variété algébrique, iso­ morphe a une sous-variété localement fermée d'un espace projectif X. Soit F un faisceau algébrique cohérent sur V, et soit W une sous-variété de V telle que F soit nul en dehors de W. On a alors H^(V, F) = 0 pour n > dim W. .

In order to study the cohomology of X in an algebraic sheaf. Serre introduced [192, pp. 246, 248] §2. Modules gradués et faisceaux algébriques cohé­ rents sur l'espace projectif. 54. L'opération FCn). Soit F un faisceau algébrique sur X = P^CK]. Soit F^^ = F(U^) la restriction de F à U^Ccf. n® 51); n désignant un entier quelconque, soit Cn) 1'isomorphisme de F^(U^ O U^) sur tU^ n Uj) défini par la multiplication par la fonction t^/t^; cela a un sens, puisque tj/t^ est une fonction régulière sur H et à valeurs dans K*. On a (n)o 0 jj^(n) = 8^^(n) en tout point de H 6h peut donc appliquer la Proposition 4 du n® 4,

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et l'on obtient ainsi un faisceau algébrique, note F(n), défini par recollement des faisceaux F . = F (UL) au moyen des isomorphismes 6^^(n). 56. Modules gradués. Soit S = KEtg, ... , t^] l'algèbre des polynômes en t^, ... , t ; pour tout entier n > 0, soit S^ le sous-espace vectoriel de S forme par les polynômes homogènes de degré n; pour n < 0, on posera S^ = 0. L'algèbre S est somme directe des S^, n E Z, et l'on a S^'S^ C S^ ^ autrement dit s'est une algèbre graduée. Rappelons qu'un S-module M est dit gradué lorsqu'on s'est donné une décomposition de M comme somme directe: M = E n E z ,n les M n étant des sous-groupes de M tels que Sp'Mg C Mp ^ q, pour tout couple d'entiers (p, q). Un élément de M^ est dit homogène de degré n; un sous- module N de M est dit homogène s'il est somme direct des N n M^, auquel cas c'est un S-module gradué. Si M et M' sont deux S-modules gradués, un S-homomorphisme ç i M ->■ M' est dit homogène de degré s si f^(M^) C ^ ^ pour tout n E Z. Un S-homomorphisme homogène de degré 0 sera appelé simplement un homomorphisme. Si M est un S-module gradué, et n un entier, nous noterons M(n) le S-module gradué: M(n) = D p ^ z M(n)p' M(n)p = ^n + p. ... On a des isomorphismes canoniques: F CO)»F (n) (m)«F (n+m) De plus, F(n) est localement isomorphe à F , donc cohérent si F l'est; il en résulte également que toute suite exacte F -»■ F' -»■ F" de faisceaux algébriques donne nais­ sance à une suite exacte F(n) -»■ F' (n) -*■ F"(n) pour tout n E Z. On peut appliquer ce qui précède au faisceau F = 0, et l'on obtient ainsi les faisceaux 0(n), n e Z. Nous allons donner une autre description de ces faisceaux; si U est ouvert dans X, soit la partie de Ay = r'(TT ^(U), Oy) formée des fonctions homogènes de degré n (c'est-à-dire vérifiant l'identité f{\y) = /(y) pour X E K*, et y ETT ^(U)); les A^ sont des A^-modules, donc donnent naissance à un faisceau algébrique, que nous désignerons par 0'(n). Un élément de 0'(n)^ , x e X, peut donc être identifié à une fraction rationnelle P/Q,

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P et Q étant des polynômes homogènes tels que Q(x) / 0 et que deg P - deg Q = n. [192, p. 246] An important step was anticipated by Serre's remark [192, p. 254] "Nous verrons plus loin que l'on peut exprimer les H^(M) au moyen des Ext^." He established

§4. Relations avec les foncteurs Ext^ [192, p. 261] 68. Les foncteurs Ext^. Nous conservons les nota- s tions du n® 56. Si M et N sont deux S-modules gradués, nous désignerons par Hom^(M, N)^ le groupe des S-homomorphismes homogènes de degré n de M dans N, et par Hom (M, N) le groupe gradué E „Hom (M, N) ; c'est S II c Zi s II un S-module gradué; lorsque M est de type fini il coïn­ cide avec le S-module de tous les S-homomorphismes de M dans N. Lorsque M n'est pas un module de type fini, les Extg (M, N) définis ci-dessus peuvent différer des Extg(M, N) définis dans [36]; cela tient à ce que Hom^XM, N). n'a pas le même sens dans les deux cas. Cependant, toutes les démonstrations de [36] sont valables sans changement dans le cas envisagé ici; cela se voit, soit directe­ ment, soit en appliquant l'Appendice de [36]. 69. Interprétation des H^M) au moyen des Ext^. Soit M un S-module gradué, et soit K un entier > 0. Posons; [192, p. 262]

B^(M) ^ gHgtMfn)), ... Définissons d'abord la notion de module dual d'un S-module gradué. Soit M un S-module gradué, pour tout n e Z, est un espace vectoriel sur K, dont nous désignerons l'espace vectoriel dual par (M^)'. Posons; M* = D _M* , avecM* = (M - n) ' . n t z n n Nous allons munir M* d'une structure de S-module compatible avec sa graduation; pour tout P e S^, l'application m -»■ P*m est une application K-linéaire de dans M_^, donc définit par transposition une application K-linéaire de (M—n _)' = M* n dans —n—p (M ) ' = M* p ; ceci définitla structure deS-module de M*. On aurait également pu définir M* comme Hom (M, K), en désignant par K le S-module gradué S/(tg, ... , t^).

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Le S-module gradué M* est appelé le dual de M; on a M** = M si chacun des Mn est dedimension finie sur K, ce qui est le cas si M = r(F), F étant un faisceau algébrique cohérent sur X, ou bien si M est de type fini. Tout homomorphisme M ^ N définit par transpo­ sition un homomorphisme de N* dans M*. Si la suite M N -»■ P est exacte, la suite P* N* M* l'est aussi; autrement dit, M* est un foncteur contravariant, et exact, du module M. Lorsque I est un idéal homogène de S, le dual de S/I n'est autre que "l'inverse system" de I, au sens de Macaulay (cf.n® 25). Soient maintenant M un S-module gradué, et q un entier > 0. Nous avons défini au n® précédent le S-module gradué (M); le module dual de B^(M) sera noté T^(M). On a donc, par définition; T^(M) = D ^ + gT^fM)^, avec T^(M)^ = (H^(M(- n)))'.

Then Serre generalized Proposition 2 [192, p. 265]: Soit M un S-module gradué, pour q / r, les S-modules gradués T^ ^(M) et Ext^(M, n) sont isomorphes. De plus, on a une suite exacte. 0 Extg(M, n) T°(M) M* ^ Extg ^(M, Q.) -»■ 0.

where [192, p. 264] Nous désignerons par £2 le S-module gradué S(-r-l); c'est un module libre, admettant pour base un élément de degré r + 1. The passing from -n to n in the definition of T*^ (M) and the fact that the dual of a non-zero vector space is not non­ zero enabled Serre to prove [192, p. 268] 74. Nullité des groupes de cohomology H^(X, F(-n)) pour n -»■ + “ . THÉORÈME 1. Soit F un faisceau algébrique cohérent sur X, et soit q un entier > 0. Les deux conditions suivantes sont équivalentes; (a) H^(X, F(-n)) = 0 pour n assez grand. (b) Extg O^) = 0 pour tout X e X. X

and further [192, p. 269] extend

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75. Variétés sans singularités. Le résultat suivant joue un rôle essentiel dans l'extension au cas abstrait du "théorème de dualité" de [193]; THÉORÈME 3. Soit V une sous-variété sans singula­ rités de l'espace projectif P^fK); supposons que toutes les composantes irréductibles de V aient'la même dimen­ sion p. Soit F un faisceau algébrique cohérent sur V tel que, pour tout x e V, F^ soit un module libre sur O . On a alors H^ (V, F(-n)) = 0 pour n assez grand X/V et 0 < q < p. ... 76. Nous allons maintenant démontrer un résultat qui est en rapport étroit avec le "lemme d'Enriques- Severi", du a Zariski [238]; [192, p. 270] THÉORÈME 4. Soit V une sous-variété irréductible, normale, de dimension > 2, de l'espace project P^(K), Soit F un faisceau algébrique cohérent sur V tel que, pour tout x e V, F soit un module libre sur 0 . On 2 z X , v a alors H (V, F(-n)) = 0 pour n assez grand. PROPOSITION 5. Le genre arithmétique d'une variété projective V est égal à

_ _ 0 0 x(V, 0 ) =V (-1)9 dim h9(V, Ci ). V "q = 0 ^ ^ REMARQUES. (1) La Proposition précédente met en évidence le fait que le genre arithmétique est indé­ pendant du plongement de V dans un espace projectif, puisqu'il en est de même des h9(v, O^). (2) Le genre arithmétique virtuel (défini par Zariski dans [238] peut également être ramené à une caractéristique d'Euler-Poincaré. Nous reviendrons ultérieurement sur cette question, étroitement liée au théorème de Riemann-Roch. (3) Pour des raisons de commodité, nous avons adopté une définition du genre arithmétique légèrement différente de la définition classique (cf. [238]). Si toutes les composantes irréductibles de V ont la même dimension p, les deux définitions sont reliées par la formule suivante; x(V, 0^) = 1 + (-1)^ P^(V). [192, p. 276] This paper of Serre formed the subject of a seminar conducted at Harvard in 1954-1955 by Jun-Ichi Igusa and Oscar Zariski. The publication became extremely influential in the proliferation of papers on sheaf theory, as will be discussed below.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 3 6 Alexander Grothendieck had written several papers between 1950 and 1955 on topological vector spaces. As his teacher Jean Dieudonné said when Grothendieck won the Fields Medal in 1966, this work is utilized in functional analysis

[51]; ...notamment la theorie des espaces nucléaires, qui "explique" les phénomènes rencontrés dans la théorie des distributions. J'ai eu personnellement le privilège d'assister de près, à cette époque, à 1'éclosion du talent de cet extraordinaire "débutant" qui à 20 ans était déjà un maître; et, avec 10 ans de recul, je considère toujours que l'oeuvre de Grothendieck de cette période reste, avec celle de Banach, celle qui a le plus fortement marqué cette partie des mathématiques. In 1955 while at the University of Kansas for a seminar on homological algebra financed as a National Science Foundation Research Project on Geometry of Function Spaces, Grothendieck wrote "A General Theory of Fibre Spaces with Structure Sheaf" [83]. The purpose of this paper was to study [83, p. 2] the exact cohomology sequence associated with an exact sequence of sheaves e -»F-)-G-»H-)-2 ...[where] G is a sheaf of groups, F a subsheaf of groups, H = G/R, and according to various supplementary hypotheses on F (such as F normal, or F normal abelian, or F in the center) we get an exact cohomology sequence going from H^(X,F) (the group of sections of F) to H^(X,G) re- ~ 1 2 ~ “ spectively H (X,H), respectively to H (X,H), with more or less additional algebraic structures involved. The formalism thus developed is quite suggestive, and as it seems useful, in particular in dealing with the problem of classification of fibre bundles with a structure group G in which we consider a subgroup F, or the problem of comparing say the topology and analytic classification for a given analytic structure group G. The first four chapters had definitions of general fibre spaces, sheaves, fibre spaces with composition law

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 3 7 (including sheaves of groups) and fibre spaces with struc­ ture sheaf. Functors were always stressed and Grothendieck stated that they should have been stressed more. In Chapter 5 he defined the cohomology set H^(X,G) of X coefficients in the sheaf of groups G. Grothendieck stated that "one is led in a natural way to the notion of a struc­ ture sheaf G" by trying "to state in a general algebraic formalism the various notions of fibre space" [83, p. 2], general fibre space, fibre bundles, algebraic fibre spaces. And it was further suggested by interpreting the classes of fibre bundles on a space X, with abelian structure group G, as the elements of the first cohomology group of X with coefficients in the sheaf G of germs on continuous maps of X into G. Thus he wished to develop systematically the notion of fibre space with structure group G, where G is any sheaf of groups, and of the first cohomology set (X, G) of X with coefficients in G. Jean Frenkel, a student of Serre and Cartan, pub­ lished his thesis in 1957 Cohomologie non Abélienne et Espaces Fibres [70]. His inspiration came from the work of Cartan and Oka on the Cousin problems and Serre's 1952 conjecture on the validity of the "principle d'Oka" in the 1950 Séminaire Cartan Exposé 20. In the introduction Frenkel stated Le chapitre I est de caractère exclusivement technique; il est destiné à étendre, dans la mesure du possible, la notion de cohomologie d'un espace à valeurs dans un faisceau au cas où ce faisceau n'est pas abélien. On obtient ainsi une suite exacte à cinq, six ou sept

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termes selon les hypothèses faites sur les faisceaux; cette suite exacte est l'outil à peu près unique utilisé pour établir les résultats figurant dans ce travail. Elle est commodément utilisable grâce à la proposition 4.2 qui permet en quelque sorte de doter d'un élément neutre arbitraire le premier ensemble de cohomologie d'un espace. Cette technique, qui nous a été indis­ pensable pour établir en 1952 les résultats résumés dans [J. Frenkel, Sur une classe d'espaces fibrés analy­ tiques (C.R. Acad. Sc., & 236, 1953, p. 40-41] a été retrouvée indépendamment par M. P. Dedecker et M. A. Grothendieck [83]. Grothendieck acknowledged that Frenkel had gained these results sometime prior to his own, but only chose to publish them at this time. On February 4th and 11th, 1957, Grothendieck gave reports at the 9th annual Séminaire Cartan, speaking this

time on "Sur les faisceaux analytiques cohérents" [85]. He said the purpose of his exposé was to generalize Serre's theories, specifically the techniques in "Faisceaux algé­ briques cohérents" [192], "Géométrie algébrique et géométrie analytique" [194] and an article then in preparation

"Sur la cohomology des variétés algébriques," [190] which Serre published in 1956 in the Journal de Mathématiques Pures et Appliquées. He began with a review of Serre's FAC, pointing out that Pierre Cartier had also proved these theorems of Serre at the Séminaire Grothendieck in 1957. These seminars grew and grew into a considerable structure which is discussed below, in part prompted by Cartier's suggestion. Here Grothendieck chose to introduce schemes [85]: "Soit X un ensemble algébrique (sur un corps algé­ briquement clos K pour fixer les idées; mais les résultats

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de ce numéro et du suivant restent valubles pour les schémas, et même les schémas arithmétiques généraux...)." Grothendieck's article "Sur quelques points d'algè­ bre homologique" [87] was presented for publication in the Tohoku Mathematical Journal on March 1, 1957. Chapter I, II, IV, and part of III are essentially that of his Kansas article. His object was to develop a formal analogy between the theory of cohomology of a space with coeffi­ cients in a sheaf as was worked out in the Séminaire Cartan 1950-1951, 1951-1952 and the theory of derived functors of modules according to Cartan and Eilenberg's Homological Algebra [36] of 1956 (but which had been written in 1950-1953). In order to be sufficiently general Grothendieck

introduced categories in Chapter I [87, p. 120]. "L'intro­ duction des catégories additives préliminaires aux catégo­ ries abéliennes [exact categories in the sense of Buchsbaum] fournit un langage commode (par exemple pour traiter des foncteurs spectraux au Chapitre II)." An additive category is given by a collection of objects {A}, a collection of abelian groups {H(A,B)> for every pair of objects (A,B) and a law of composition H(A,B)x H(B,C) -*■ H(A,C). Axioms exist for associativity, identity, a zero object and direct sums. Abelian categories are additive categories in which the existence of kernel, image, cokernel and coimage are postulated for every map. Every object can be embedded in an injective object if each abelian category satisfies the following two axioms [87, p. 128, p. 129]:

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AB3) Pour toute famille (A^) i e I d'object de C, la somme directe des A^ existe. AB5) L'axiome AB3) est vérifié, et si (A^) i e I est une famille filtrante croissante de sous- trucs d'un A e C, B un sous-truc quelconque de A, on a (Z A.) n B = E (A. H B) . i ^ i 1 Chapter II gave the essential theory of derived functors of a given functor as in Cartan and Eilenberg, but using a 9-functor rather than "connected sequences of functors" [87, p. 139]: Chapitre II Algèbre Homologique dans les catégories abéliennes. 2.1. 9-foncteurs et 9*-foncteurs. Soient C une catégorie abélienne, C'une catégorie additive, a et b deux entiers (pouvant être égaux a + ~ ou - ») tels que a + 1 < b. Un a-foncteur covariant de C dans C , à degrés a < i < b, est un système T = (T^) de foncteurs covariants additifs de C dans C , (a < i < b), plus la donnée, pour tout i tel que a

(2.1.1.) T^(A') 4. T^(A) T^(A") 4- T^+lfA') est un complexe, i.e. le composé de deux morphismes con­ sécutifs dans cette suite est nul. Définition analogue pour un 9*-foncteur covariant, la seule différence étant que l'opérateur 9 diminue le degré d'une unité au lieu de 1'augmenter, Définition analogue pour les 9 foncteurs et 9*-foncteurs

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contravariants; les sont alors des foncteurs additifs contravariants et les opérateurs bords vont de T^(A') dans ^ ^(A") ou ^(A"). Si on change de signe les degrés i dans ou si on remplace la catégorie C par sa duale, les 9-foncteurs sont changés en 9*- foncteurs. Ainsi, on peut toujours se ramener à l'étude des 9-foncteurs covariants. Remarquons que si a = - », b = + », un 9-foncteur est une "corrected séquence of functors." Chapter II continued with a discussion of suite spectrales and foncteurs spectraux, again referring to Cartan-Eilenberg Chapters XV and XVII [87, p. 147]: Soient C, C ', C" trois catégories abéliennes, consi­ dérons deux foncteurs covariants F;C 4- c' et G : C 4- c". On suppose que tout object de C ou C' est isomorphe a un sous-truc d'un objet injectif, ce qui permet de considérer les foncteurs dérivés à droite de F,G,GF, on se propose d'établir des relations entre ces foncteurs dérivés. Soit A G C, soit C(A) le complexe associé à une résolution droite de A par des objets injectifs, considérons le complexe F(C(A)) dans C , il est déter­ miné à une homotopie près quand on fait varier C(A). Il s'ensuit aussitôt que les suites spectrales définies par G et ce complexe F(C(A)) ne dépendent que de A, ce sont donc des foncteurs spectraux cohomologiques sur C, ayant le même aboutissement, appelés foncteurs spectraux du foncteur composé GoF. Les formules données plus haut donnent immédiatement leurs termes initiaux;

lP^9(a) = (r P( (r ^g)F )) (A) = r Pg (r 9r (A) ) . 2 i De loin le cas le plus important pour l'obtention de suite spectrales non triviales est celui où F transforme objets injectifs en objets annulés par les r 9g , q > 1 (on appelle de tels objets G-acycliques), et où R^G = G (i.e. G est exact à gauche): alors Ig'9^ = 0 si q > 0, et se réduit a. R^(GF) si q = 0, d'où résulte que 1 'aboutissement commun des deux suites spectrales s'identifie au foncteur dérivé droit de GF. On obtient ainsi le THÉORÈME 2.4.1. Soient C,C',C" trois catégories abéliennes, on suppo'se que tout objet de C ou C est isomorphe à un sous-truc d'un objet injectif. Considé­ rons des foncteurs covariants F:C + C et G;C 4. c", on suppose que G est exact à gauche et que F transforme objets injectifs en objets G-acycliques (i.e. annulés

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par les R^G, q > 0). Alors il existe une foncteur spectral cohomologique sur C, à valeurs dans C", abou­ tissant au foncteur dérivé à droite R(GF) de GF, (convenablement filtré), et dont le terme initial est

(2.4.3) e |'^(A) = r Pg (r '^F (A) ) . REMARQUES 1. La deuxième hypothèse sur le couple (F,G) signifie que les foncteurs (R^G) F (q > 0) sont effaçables (cf. 2.2), ou encore que pour tout A e C, on peut trouver un monomorphisme de A dans un M tel que (R^G)(F(M)) = 0 pour q > 1; c'est sous cette forme que le plus souvent nous vérifierons cette hypothèse. 2. On vérifie aussitôt que pour calculer la deuxième suite spectrale d'un foncteur composé (i.e. celle dont il est question dans le théorème 2.4.1) il suffit de prendre une résolution C(A) de A par des qui sont F-acycliques (et non nécessairement injec- tifs) et de prendre la deuxième suite spectrale d'hyper- homologie du foncteur G par rapport au complexe FC(A). 3. Notons deux cas particuliers importants ou l'une des deux suites spectrales d 'hyperhomologie d'un foncteur F par rapport à un complexe K dégénère. Si K est une résolution d'un objet A de C, alors R ^ (K) = R^F(A), donc l'objet gradué (R^F(A)) est l'aboutissement d'une suite spectral de terme initial I^'^(K) = H^(R%(K)). Si les sont F-acycliques (i.e. r "*F(K^) = 0 pour m > 0) alors R^F(K) = H^(F(K)), donc l'objet gradué (H^(F(K)) est 1'aboutissement d'une suite spectrale de terme initial H^'^(K) = R^F(H^(K)). Combinant ces deux résultats, on trouve donc; si K est une résolution de A par des objets F-acycliques, on a R ^ (A) = (F (K) ) . In Chapter III, "Cohomologie a Coêfficients dans un faisceau," Grothendieck included "les suites spectrales classiques dé Leray." The theory developed did not need the restrictive hypotheses on the spaces, (applying to non-separated spaces) which was required in Serre's FAC and in the work of Pierre Cartier [87, p. 120]. Chapter IV [87, p. 120]

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. . .traite la question non classique des Ext de faisceaux de modules, on y trouvera en particulier une suite spectrale utile qui relie les Ext "globaux" et les Ext "locaux". La situation se corse au Chapitre V. où de plus un groupe G opère sur l'espace X, le faisceau d 'anneaux 0 donné sur X, et les faisceaux de modules sur 0 qu'on considère. On obtient en parti­ culier dans 5.2 un énoncé qui me semble être la forme définitive de la théorie cohomologique "dechiste" des espaces à groupe (non topologique) d'opérateurs, pou­ vant avoir des points fixés. Il s'exprime en introdui­ sant de nouveaux foncteurs H^(X; G, A) (implicites déjà dans bien des cas particuliers antérieurs); on trouve alors deux foncteurs spectraux, à termes initiaux remarqua­ bles, qui y aboutissent. While this paper was being written, Grothendieck's teacher, Roger Godement, was preparing an article on sheaves (which will be discussed below). Grothendieck referred to Godement for the commutative theory of cohomology [87, p. 120]:.

Des conversations avec R. Godement et H. Cartan ont été très précieuses pour la mise au point de la théorie, et en particulier l'introduction par Godement des faisceaux flasques et des faisceaux mous, qui se substituent avantageusement aux faisceaux fins dans bien des questions, s'est révélée extrêmement commode. Un exposé plus complet, auquel nous renverrons pour divers points de détail, sera donné dans un livre en préparation par R. Godement. Roger Godement's book Topologie Algébrique et Théorie des Faisceaux [77] is the watershed in thé creation of sheaf theory. By the time it was published in 1958 the history of mathematics gave clear evidence of the change in language, which was the external manifestation of the development. As Lucienne Félix stated in The Modern Aspect of Mathematics [69, p. 5] "Every epoch of great scientific achievement - particularly changes of ideas growing out of re-examination of fundamentals - is marked by a development of language."

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 4 4 Although the axiomatic theory of sheaves had been fully established in the Séminaire Cartan in 1950-1951 and more than 30 papers were written on the subject by 1953, no complete exposition was available. Courses and lectures were given at various universities following the American Mathematical Society Summer Institute in 1954. In 1954- 1955, Igusa and Zariski conducted a seminar at Harvard; Moore, Nickerson, and Spencer conducted a seminar at Princeton; Godement gave lectures at the University of Illinois and Grothendieck at the University of Kansas. In 1956 Dowker gave lectures on sheaf theory at the Tata Insti­ tute in Bombay, Koszul gave a course in Sao Paulo. Swan's lectures at Oxford in 1958 were published in book form in 1964. While Godement was a visiting professor at the Uni­ versity of Illinois in 1954-55 he wrote the first draft of Topologie Algébrique et Théorie des Faisceaux [77]. Called a classic by Cartan it was published in 1958. The need for exposition was clear as Godement stated in the preface [77, p. iv]: Alors que la technique des faisceaux envahit les branches les plus diverses des Mathématiques, une pareille situation ne pouvait être tolérée plus longtemps par les techniciens: c'est pourquoi un spécialiste de l'analyse fonctionnelle présente aujourd'hui un exposé complete, i.e. moins incomplet que les autres, de la théorie des faisceaux - with a vengeance. Godement built on the results of the papers already discussed, but included one new author's work, Daniel Marinus Kan [77, p. iv]:

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Comme de plus les travaux récents de Kan semblent prouver que ces complexes constituent le domaine naturel de validité d'une théorie complète de l'homotopie, on peut affirmer que la notion générale de complexe simpli- cial (due essentiellement à Eilenberg et Zilber) est appelée à jouer un rôle essentiel en topologie algébri­ que. Kan's book Adjoint Functors [112] was also published in 1958, although the recognition of adjoint functor had been around for some time (the book was in fact received for publication in 1956). Mac Lane, writing from an historical perspective of the situation, justifiably wondered why it all took so long [161, p. 140]: It is my own view that the climate of mathematical opinion in the decade 1946-1956 was not favorable to further conceptual development. Investigation of concepts as general as those of category theory were heartily discouraged perhaps because it was felt that the scheme provided by BOURBAKI's structures produced enough generality. It is to be noted that KAN, when developing adjoint functors, came at the time from a solitary position more or less outside active mathemati­ cal circles. It may even be that we should be on our guard lest the current very active mathematical life inhibit the development of ideas which fall outside the established directions of research. Kan, then at Columbia University, wrote on abstract homotopy in Proceedings of the National Academy of Sciences [111] and circulated other "papiers secrets" which Cartan used as a bibliography for his seminar "Sur la Théorie de Kan" [33] on December 12 and 17, 1956. Kan began Adjoint Functors 1112, p. 1]: 1. Introduction. In homology theory an important role is played by pairs of functors consisting of (i) a functor Hom in two variables, contravariant in the first variable and covariant in the second (for instance the functor which assigns to every two abelian groups A and B the group Hom (A, B) of homomorphisms fi A -»■ B) .

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(ii) a functor fi (tensor product) in two variables, covariant in both (for instance the functor which assigns to every two abelian groups A and B their tensor product A â B). These functors are not independent; there exists a natural equivalence of the form a : Hom (a, ) -)■ Hom ( ,Hom( , )) Such pairs of functors will be the subject of this paper. In the above formulation three functors Hom and only one tensor product are used. It appears however that there exists a kind of duality between the tensor product and the last functor Hom, while both functors Hom outside the parentheses play a secondary role. Let M be the category of sets. For each category y let H:y, y ^ M be the functor which assigns to every two objects A and B in y the set H(A,B) of the maps /:A B in Y. Let X and Z be categories and let S:X ^ Z and T:Z X be covariant functors. Then S is called a left adjoint of T and T a right adjoint of S if there exists a natural equivalence a : H(S(X) , Z) H(X, T(Z) ) . An important property of adjoint functors is that each determines the other up to a unique natural equivalence. The first chapter of Godement's book, "Algèbre Homologique" is set in the category of modules on a ring with unit (following Cartan-Eilenberg). Godement gave the definitions concerning categories, functors, additive and abelian categories. In a footnote [77, p. 15], however, he wrote "On trouvera a de sujet des renseignments beaucoup plus complet dans un article de D. Buchsbaum ("Exact Categories and Duality," Trans. Amer. Soc., (1955) pp 1-34) [18]." Chapter I of Godement's book is concerned with three important concepts: the theory of spectral sequences. Ext and Tor, and simplicial complexes (in section 3). The latter included [77, p. 4] "On trouvera dans ce paragraph 3

un exposé à peu près complet de la théorie des produits

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 4 7 (produit cartésien et cup-product), exposé dont la seule originalité est sans doute d'être imprimé." The theory of sheaves completed the rest of the book [77, p. Ii]: Après deux paragraphes de généralités sur les faisceaux d'ensembles et les faisceaux de modules, nous abordons au paragraph 3 le problème central de la théorie des faisceaux; celui du "prolongement" ou du "relèvement" des sections d'un faisceau. La notion essentielle de ce point de vue, à cause de sa simplicité et de son utilité, semble être celle de faisceau flasque: un faisceau F sur un espace X est flasque si toute section de F au-dessus d'un ouvert de X peut se prolonger à X tout entier. Tout faisceau F peut se plonger dans un faisceau flasque (par exemple, le faisceau des germes de sections non nécessairement continues de F); de plus, si l'on a une suite exacte

0 -»■ F° F^ ^- ... de faisceaux flasques de groupes abéliens, alors les sections de ces faisceaux au-dessus d'un ouvert quel­ conque forment encore une suite exacte. Dans les espaces paracompacts, il est important d'avoir aussi la notion plus faible de faisceau mou: un faisceau F est mou si toute section de F au-dessus d'un ensemble fermé se prolonge a l'espace ambiant. Cette notion semble devoir se substituer avantageusement à celle de faisceau fin, qui jouait un rôle essentiel dans la théorie antérieure, comme on le constatera expérimenta­ lement. Sur un espace paracompact, si l'on a une suite exacte de faisceaux mous de groupes abéliens, les sections de ces faisceaux au-dessus d'un fermé quelcon­ que forment encore une suite exacte. In Section 4 [77, p. III] La possibilité de définir, sans aucune hypothèse sur X, des groupes de cohomologie possédant ces propriétés, a été démontrée tout d'abord par A. Grothendieck en 1955, en utilisant le fait que tout faisceau se plonge dans un faisceau injectif au sens de l'algèbre homologique. Comme les recherches de Grothendieck sur ce sujet seront publiées prochainement, nous avons préféré utiliser, au lieu des faisceaux injectifs, les faisceaux flasques [qui évidemment sont spécialement adaptés à 1 ' étude du foncteur A -»■ r (A) ] ; il se trouve que l'on peut construire, de façon canonique, une résolution flasque 0 -»■ A C°(X; A) C^(X; A)

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de tout faisceau A, qui de plus est un foncteur "exact" par rapport à A; posant

C*(X;A) = r(C*(X;A)), H^(X,A) = H^(C*(X;A)), on obtient alors, de façon tout à fait élémentaire, les trois propriétés fondamentales des groupes de cohomologie. There was a demonstration using spectral sequences of the "célèbres 'théorèmes fondamentaux'" [77, p. III]: en particulier, toute résolution

A A ^ ... d'un faisceau A donne lieu à des homomorphismes cano­ niques

H^(r(L*)) ^ H*(X;A), lesquels som bijectifs si les sont flasques (ou mous, lorsque X est paracompact), ce qui montre bien entendu qu'il était en principe inutile, pour définir les groupes H^(X,A), de choisir une résolution flasque "canonique" de A. ... Le §5 étudie les relations entre les groupes K” (X;A) et les groupes H^(X;A) obtenus par la méthode de ?ech (laquelle, on le sait, ne donne pas de résultat satis­ faisant en dehors des espaces paracompacts, ou bien de catégories spéciales de faisceaux). Nous montrons d'abord que tout recouvrement U de X qui est soit ouvert, soit fermé et localement fini, définit une résolution C*iU;A) de tout faisceau A sur X; il en résulte des homomorphisme s canoniques H^(I/;A) H^(X;A) qui proviennent du reste d'une suite spectrale. A la limite, on trouve une suite spectrale reliant la cohomologie de Cech à la "bonne" cohomologie. On déduit de là qu'il y a i s omomorph i sme.

%^(X;A) = h "(X;A) pour tout faisceau si X est paracompact; et, si X n'est pas paracompact, il en est encore ainsi pour un faisceau donné s'il existe dans X "suffisamment" d'ensembles ouverts tels que, pour toute intersection fini ü de tels ouverts, on ait îf^(ü;A) = 0 pour n > I. Ce dernier résultat, dû à H. Cartan, permet de montrer que, pour les faisceaux algébriques cohérents étudiés par Serre, la cohomologie de îfech coïncide avec la "bonne" cohomologie. Nous avons pu d'autre part

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 4 9 d'autre part démontrer, sans hypothèse sur l'espace X, le résultat célèbre de Leray suivant lequel, étant donné un recouvrement M = (M.) de X tel que l'ont ait toujours H*(M. n ... nu. ;A) = 0 pour n > I, ""O ^p les homomorphismes

h ” (M;A) H*(X;A) sont bijectifs pourvu que M soit ouvert, ou bien fermés et localement fini. Nous n'avons malheureusement pas pu trouver une démonstration qui s'applique simulta­ nément à ces deux cas; dans le cas ouvert, il faut étudier le double complexe C*(M;C*(X;A)), et dans le cas fermé le double complexe C*(X;C*(M;A)), où C*(M;A) désigne la résolution de A définie par M. [77, p. IV] Finally in the section 6, Godement extended to the theory of sheaves the notions of cartesian product and cup-product

[77, p. V]: il existe, pour tout faisceau A sur un espace X, une résolution flasque 0 -»■ A ->■ F°(X;A) -4- F^(X;A) + ... qui est un foncteur exact en A, et dont la différen­ tielle résulte d'une structure semi-simpliciable (i.e. d'opérateurs de "face" et de "dégénérescence" analogues formellement à ceux de la théorie des complexes semi- simpliciaux d'Eilenberg-Mac Lane-Zilber). Si l'on utilise cette résolution pour définir les groupes de cohomologie, on constate que les formules "explicites" qui donnent le cup-produit dans la théorie simpliciale classique, disons la formule f U 9(Xg, ..., ~ •**' ®

g(Xp, ^p + g)' le donnent aussi en théorie des faisceaux. [Il en est bien entendu de même en cohomology de î^ech; malheureusement, la méthode de Cech n'est satisfaisante que sur les espaces paracompacts.] Cette remarque n'a pas seulement l'intérêt, après tout purement pédagogique, de montrer que la théorie multiplicative des faisceaux n'est qu'un cas particulier de la théorie général concernant les complexes

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"simpliciaux"; elle montre que toute notion reposant exclusivement sur l'existence d'une structure simpli- - ciale s'étend automatiquement à la théorie des faisceaux; en particulier, il est clair dès maintenant que les opérations de Steenrod peuvent se définir en théorie des faisceaux puisqu'elles supposent, tout au plus, une structure simpliciale et une application diagonale. In the last section Godement referred the reader to Grothendieck's article "Sur quelques points d'algèbre homo­ logique" [87] (which was at that time being prepared for publication in The Tohoku Mathematical Journal) for a more complete discussion of injective sheaves, derived functors, and the spectral sequence of Ext. [77, p. .V]: II semble raisonable de penser que la théorie des faisceaux est maintenant dans un état pratiquement final, attendu que d'une part elle a atteint le degré de généralité maximum que l'on puisse concevoir, et que d'autre part ses méthodes sont, à beaucoup d'égards, considérablement plus simples que celles des auteurs anciens; cela ne saurait cependant nous faire oublier le rôle qu'ont joué ceux-ci: il est clair que notre exposé ne comporte, la plupart du teirps, que des améliorations de détail par rapport à ceux de Leray et Cartan. Le progrès le plus important est probablement d'avoir pu construire une théorie raisonnable valable pour tout espace topologique; comme nous l'avons dit, on doit ce résultat a Grothendieck; mais il semble juste d'ajouter que, probablement, la question ne se serait pas même posée dans les travaux de Serre sur les variétés algébriques.

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APPLICATIONS

A. Sheaves and Ring Theory In his Topologie Algébrique et Théorie des Faisceaux Godement used the idea of sheaves in ring theory [77,

p. 12]: 2.1. - Faisceaux d'anneaux Nous avons, au chapitre I, §1, défini d'une façon générale la notion de préfaisceau à valeurs dans une catégorie; si les objects de cette catégorie sont des ensembles et si les homomorphi sme s de ces objects les uns dans les autres s'identifient à des applications, on pourra évidemment parler de faisceaux à valeurs dans la catégorie en question. Par exemple, sur un espace de base X, un faisceau de groupes (resp. de groupes abéliens, d'anneaux, d 'anneaux commutatifs avec élément unité, etc...) est un pré­ faisceau de base X, à valeurs dans la catégorie des groupes (resp. des groupes abéliens, etc...) et qui, en tant que préfaisceau d'ensembles, satisfait aux axiomes des faisceaux. Soit A un faisceau d'anneaux sur X; pour tout ouvert U, les ensembles A(ü) sont des anneaux, et pour ü D V l'application de restriction A(U)-»- A(V) est un homomor­ phisme d'anneaux. Il s'ensuit qu'à la limite les ensem­ bles ponctuels A(x) = lim. ind. A(U) U.3X sent canoniquement munis de structures d'anneaux. Du point de vue des espaces étalés on a donc dans A deux lois de composition (u, v) -»■ u + v et (u, v) uv, définies pour p(u) = pCv), induisant sur chaque fibre A(x) une structure d'anneau, et de plus continues. Si s et t sont deux sections de A au-dessus d'un ouvert U, il est clair que s + t et st sont les sections X -»■ s (x) + t (x) et X ->■ s (x) t (x).

151

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Exemple 2.I.I. - Si A est un anneau fixe, le faisceau simple de base X et de fibre A est canoniquement un faisceau d'anneaux, qu'on identifiera le plus souvent à l'anneau A lui-même. Exemple 2.1.2. - Si X est un espace topologique (resp. une variété différentiable, analytique complexe) le faisceau des germes de fonctions continues (resp. diffé- rentiables, holomorphes) sur X est un faisceau d'anneaux de façon évidente. Exemple 2.1.3. - Soit A un anneau commutatif avec l'élé­ ment unité: un idéal p ^ A de A est dit premier si l'anneau A/p est d'intégrité; rappelons que, si a est un idéal autre que A, l'intersection des idéaux premiers contenant à est formé des x tels que l'on ait € a pour un entier n au moins. Soit X = G(A; l'ensemble de tous les idéaux premiers de A ("spectre premier" de A); on peut définir une topolo­ gie sur X ("topologie de Zariski") en disant qu'une partie de X est fermée si elle est de la forme F(a), F(a) étant l'ensemble de tous les idéaux premiers contenant un idéal donné a; la vérification des axiomes des espaces topologiques est triviale en raison des formules r|F(a^) = F(y^aj); F (a) U F(b) = F(a fl b) .

Nous supposerons pour simplifier que A est un anneau d'intégrité; dans ce cas, l'intersection de deux ouverts non vides n'est jamais vide, car si l'on a F(a) U F(b) = X on voit que a D b est contenu dans tous les idéaux premiers de A, donc est nul, de sorte que l'un des idéaux a, b, au moins est lui-même nul. Nous allons maintenant définir un faisceau d'anneaux sur X comme suit. Soit K le corps des fractions de A; si p est un idéal premier de A, on note A^ l'ensemble des éléments de K qui sont de la forme x/y, avec x, y E A, y $ p; c'est un anneau ayant pour unique idéal maximal p.A^. Cela dit, pour tout ouvert non vide U de X posons

A(u) = n A ; peu P pour U D V nous définirons l'application de restriction A(U) ,-»■: A(V) comme étant l'application identique, ce qui a un sens puisque l'on a évidemment A(U) C A(V). Bien entendu on pose A(#) =0. Pour vérifier les axiomes des faisceaux prenons une famille (UL)^^^ d'ouverts non

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vides, de réunion U, et des s. E A(ü^) tels que, quels que soient i et j, les restrictions de s^ et s^ soient égales; comme n'est jamais vide, cela signifie que l'élément s^ de K est indépendant de i, d'où un élé­ ment s e A(U) = n A(U.) iel ^ et un seul qui "induit" s. dans chaque U^, ce qui prouve bien que les A(U) forment sur X un faisceau d'anneaux. On peut encore réaliser le faisceau précédent comme faisceau de germes de fonctions sur X. Soit un élé­ ment / de K; nous dirons que f est défini en un point p de X si f G A^; il est immédiat de vérifier que l'en­ semble D(/) des points où f est défini est ouvert dans X (considérer dans A l'idéal des q tels que q*/ E A); si / est défini en p, appelons valeur de f en p l'image de / dans le corps K(p) = Ap/p-Ap, qui n'est autre que le corps des fractions de l'anneau d'intégrité A/p; on associe ainsi à chaque / E K une fonction définie sur 1'ouvert D(f) et à valeurs dans les crops variables K(p); cela dit il est clair que A(U) n'est autre que l'anneau des / E K qui sont définis en tout point de ü. On notera que les f G K qui sont définis sur X ne sont autres que les éléments de A lui-même, autrement dit que A est l'intersection des anneaux locaux Ap (on peut même se borner aux idéaux p maximaux) comme on le voit immédiatement; comme de plus l'intersection de tous les idéaux premiers de A est nulle on voit que la correspondance entre un / E K et la fonction p -»■ f(p) sur X est biunivoque. Notons enfin qu'on peut déterminer facilement les anneaux ponctuels A(p); pour cela il faut calculer la limite inductive des A(U) lorsque U décrit l'ensemble filtrant décroissant des voisinages ouverts de p: évidemment, on trouve le sous-anneau de K réunion des divers A(U) pour U 9 p; cette réunion n'est autre que l'anneau local Ap; tout d'abord elle est évidemment contenue dans Ap,* reste à voir que tout f G appartient à un A(U) au moins - ce qui est évident si l'on prend ü =• D(f). Pour cette raison, on appelle A le faisceau des anneaux locaux de X.

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At the International Congress at Edinburgh in 1958, Grothendieck announced that he was undertaking an enormous generalization of all algebraic geometry including all previous developments starting from the category of all commutative rings (with unit). Grothendieck had realized

that the concepts in Serre's FAC could be generalized further, and he was further motivated to look for more generalizations because of the Weil conjectures, which are

discussed below. A whole series of talks were given at the Bourbaki seminars and in 1960 Grothendieck began his Sémi­ naire de Geometrie Algébrique which continued until 1972. In the first seminar he presented various new concepts and their applications, which material was then collected under the title Éléments de Géométrie Algébrique [91] with the collaboration of Jean Dieudonné. By 1967 EGA contained over 1800 pages. In his extensive review of Chapter I in the Bulletin of the American Mathematical Society, Serge Lang explained the reasons this treatise is different from previous ones, including the fact that there were no tools to deal with certain algebraic systems of varieties, such as Picard varieties [132, p. 240]; Most of algebraic geometry up to now has been concerned with varieties, say over arbitrary fields. It includes some results on algebraic families of varieties, but such results are few in number, and it has become increasingly clear in recent years that one was facing serious difficulties in dealing with such algebraic systems. ... In applications to number theory, it has been realized for some time that the reduction mod p of a variety defined over a number field was completely

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analogous to the situation of algebraic systems, a fiber being such a reduction. ... In order to deal efficiently with the above two points, it was necessary to incorporate from the start into the foundations the notion of a variety defined over a ring, not necessarily Noetherian, and having nilpotent elements (say to reduce mod p^, or to describe degenerate fibers in a system). This meant that a variety could not be regarded any more as a model of a "function field," and thus that it should be defined with a local description supplemented by a method for gluing local pieces together (sheaves being the natural tool here). Lang provided other reasons as well for a new approach. Grothendick dedicated EGA to Oscar Zariski and André Weil, for both men had a direct influence and contributed to the development of the point of view of schemes [91, p. 1]. Quant à l'influence de A. Weil, qu'il nous suffisse de dire que c'est la nécessité de développer l'outillage nécessaire pour formuler avec toute la définition de la "cohomologie de Weil" et pour aborder la démonstration de toutes les propriétés formelles nécessaires pour établir ses célèbres conjectures en Géométrie diophan- tinne, qui a été une des principles motivations de la rédaction du présent Traité au même titre que le désit de trouver le cadre natural des notions et méthodes usuelles en Géométrie algébrique, et de donner aux au­ teurs l'occasion de comprendre les dites notions et techniques. He recommended Serre's PAC because it [91, p. 2] "constituer une excellente préparation à celle de not Éléments" because it was an "exposé intermédiare entre le point de vue classique et le point de vue des schemas en Géométrie algébrique." In particular the theory of sheaves [91, p. 2] fournit le langage indispensable pour interpréter en termes

'géométriques' les notions essentielles de l'Algèbre commutative, et pour les ' globaliser' ." In Chapter 0 of

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this work Grothendieck gave the fundamentals of sheaf theory and the theory of ringed spaces, as defined by Cartan. In EGA I: "Le langage des schémas" Grothendieck

showed that for commutative rings the best frame for a functional representation theory is sheaf theory [91,

p. 201]: Dans tout ce paragraph, A désignera un anneau adique noethérien, I un idéal de définition de A. Soit X = Spec(A), X = Spf(A), qui s'identifie à la partie fermée V(I) de X (10.1.2). En outre, la définition (10.1.2) et la définition (10.8.4) montrent que le schéma formel affine X est identique au complété Xy^ du schéma affine X le long de la partie fermée X de son espace sous- jacent. A tout 0^-Module cohérent F correspond donc un 0^-Module de type fini Fy^., qui est d'ailleurs un faisceau de modules topologiques sur le faisceau d'anneaux topologiques 0^. Mais tout 0^-Module cohérent F est de la forme Sf, où M est un A-module de type fini (1.5.1); nous poserons (5)y^ = M^. En outre, si u : M -»■ N est un A-homomorphisme de A-modules de type fini, il lui correspond un homomorphisme ïï : M -»■ N, et par suite aussi un homomorphisme continu ùy^, : (M) y^, -» (N)y^,, que nous noterons u^. Il est immédiat que (vou)^ = v^ou^; on a ainsi défini un foncteur additif covariant M^ de la catégorie des A-modules de type fini dans celle des Oj^-Modules de type fini. Lorsque A est un anneau discret, on a M^ = M. Proposition (10.10.2). - (i) M^ est un foncteur exact en M; et il existe un isomorphisme canonique fonctoriel de A-modules r(Z, M^) ~ M. (ii) Si M et N sont deux A-modules de type fini, il existe des isomorphismes canoniques fonctoriels (M H ^N) ^ ~ M^ B Q

(Hom^(M, N))^ % Hony (M*, N^)

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(iii) L'application u ^ est un isomorphisme fonctoriel Hom^(M, N) ~ Hom^ (M*, N^)

However these results did not [105, p. 293] "enter the field of vision widely when the field of sectional represen­ tation began to blossom in the second half of the sixties as a field of interest which is independent of algebraic geom­ etry," according to Karl Heinrich Hofmann.^ The publication of a paper by Dauns and Hofmann and another by Pierce origi­ nated this era. A geneology of sheaves and ring theory is diagrammed schematically on the following page. John Dauns was a young (age 29) professor at Tulane in 1966 when he and

Hofmann pointed out that [105, p. 294] "the theorem of Grothendieck in which he establishes the correspondence between commutative rings and affine schemes" was "embedded into a formidable wealth of material contained in this monumental treatise about algebraic geometry," and was "lifted out of its encyclopedic environment into the more accessible form of a paperback by" I. G. Macdonald in 1968: Algebraic Geometry : Introduction to Schemes [154]. Macdonald's book was based on lectures he gave at the University of Sussex as an introduction to the language of schemes as defined in EGA for an audience of classical geometers (as he explained in the foreward). He concluded the introduction: "Any situation or theorem relating to affine varieties can be transcribed into one relating to their coordinate rings, and it has been recognized for a long time that in this way one gets more general statements, for generally the theorems of commutative algebra that arise are valid under much less restrictive hypotheses on the rings in question: often it is enough that they should be Noetherian. So, to obtain a satisfactorily general theory, one should start with a quite arbitrary commutative ring and construct something like an 'affine variety' from it, and then stick these objects together by means of structure sheaves to obtain generalized abstract varieties or preschemes."

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SEMINAIRE-CARTAN

DIXMIEP. GROTHENDIECK

GROT S DIEUDONNE GODEI-ENT KOSZUL

■— ______-X'

SWAN

D.VulîU S HOFMANN

R. S. PIERCE ) KOH TELEMAN

MACDONALD

HOFMANN

MULVEY

A GENEOLOGY OF SHEAVES AND RING THEORY

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Hofmann, also a young professor there (age 34) collaborated on "The representation of biregular rings by sheaves" [47]. In the introduction they wrote [47, p. 8] We will show that every biregular ring is isomorphic to the ring of global sections with compact supports in a sheaf of simple rings with identity over its maximal ideal space as base space and that every ring of global sections with compact supports in a sheaf of simple rings with identity over a locally compact, totally discon­ nected, Hausdorff base space is a biregular ring. They said that the methods used there were more or less stan­ dard in sheaf theory and they "assume familiarity with the elementary techniques of the theory of sheaves" as in Dowker's Bombay lectures [57], in Godement's Theorie des Faisceaux [77] and Swan's The Theory of Sheaves [202].^ Dowker's lectures of 1956 also follow closely the work of the 1950-1951 Seminare Cartan. The major difference was that Dowker was not as concise as the Seminare, and the greater detail and complete calculations in the notes written up by S. V. Adavi and N. Ramabhadran became very helpful in the spread of Cartan's ideas. Furthermore every new concept is carefully analysed with particular reference to "bad spaces" at specific instances. For example it is shown that fine and locally fine sheaves coincide on paracompact normal spaces and not for other spaces. Presheaves are defined as

In Chapter 10, Applications of Sheaf Theory to the Study of Rings, of A Radical Approach to Algebra [80] Mary Gray gave a particularly lucid explanation of the structure space of a ring and why the introduction of sheaves achieves a representation theory which gives more information than the structure theorems for general semisimple rings.

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opposed to sheaves. For applications to algebraic geometry or complex manifolds one had to look at the reports of Chern and Zariski (discussed above). Dauns and Hofmann wrote another memoir together pub­ lished in 1968 "Representation of rings by sections" [48]. As explained in the lengthy introduction to their monograph, they first laid the foundation of a.theory that generalized

the theories of sheaves and fibre bundles, in order to have the advantages of both and the disadvantages of neither. Consequently they introduced a "field of structures" [48, p. 2]: :

1.1 DEFINITION. Let tt:E -»■ B be a surjective continu­ ous map of topological spaces. The space B is called the base space, the space E the field space, and the subspaces TT~^(b), b e B, the stalks of ir. The set of all pairs (x,y) eE X E with ir(x) = n(y) will always be denoted by E V E. The diagonal {(x,x)|x s E} of E x E is called A(E). If V is an open subset of the base B, then a (contin­ uous) function a;V -»■ E is called a (continuous) local section if -rroa is the identity map of V. The domain and image of a local section a are denoted by dom a = V and ima. IfV=B> then a is called a global section. The set of all continuous local sections with domain V will be denoted with r(V,ir) or r (V) : the set of all continuous global sections we will abbreviate with r(n). In order to have a clear notation for distinguishing continuous sections from possibly discontinuous ones, S(ir) will denote the set of all sections. The restriction of any function a to a subset W C dom a will be denoted by o|w. The map it:E -»■ B is called a field of topological spaces if the following condition is satisfied; (1) E = U {a(V) |V open in B, a e r(V, tt)}.

Note that E V E = ( tt x ir) ^(AB) , that this set is an equivalence relation on E, and that it is a closed subspace of E x E provided that B is Hausdorff. Also observe that A(E) Cl E v E. The following examples will give a first illustration of the field concept. 1.2 EXAMPLE. Let B and K be topological spaces and ir:B X K B the projection onto the first factor. Then TT is a field of topological spaces. If C(B, K) denotes

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the set of all continuous functions from B into K, then the function s:r(ir) -»■ C(B, K) , which associates with a global section o of ir the continuous function sa:B ^ K defined by (sa)(b) = prgOCb), is a bijection, where pr 2 :B X K K is the projection onto the second factor. The field ir is called the constant field with base space B and stalk K. Algebraic and topological structures were added (under conti­ nuity conditions) and hence their new object did not have the disadvantage of necessarily discrete stalks (as do sheaves) or that of lack of variability of the stalks (as do fibre bundles). This concept was then applied to the representation of topological algebras and rings. However in Chapter 4 on weakly biregular rings they presented general methods of representation of rings as sections in a sheaf, and the discreteness provided by sheaves in a local ring over a zero-dimensional compact Hausdorff space. In conclusion [48, p. xi] "the. general'concept of a field and the.repre­

sentation by continuous sections may well be a frequently used tool in applications we cannot foresee." Dauns and Hofmann's characterization theorem for biregular rings [178, p. 2] "itself generalized results by Arens and Kaplansky about representation of certain biregular rings as function rings which in turn generalized the classical results of Stone about the representation of Boolean rings." R. S. Pierce, professor at the University of Hawaii, Honolulu, published "Modules over commutative regular rings," in the Memoirs of the American Mathematical Society in 1967 [178]. It was his intention to use the representation shown by Dauns and Hofmann [178, p. 1] "that bireguiar rings., can .

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be represented as the rings of global sections of a ringed space (X, K) where X is a totally disconnected, compact Hausdorff space, and K is a sheaf of simple rings over X" to study finitely generated modules over commutative, regular rings. His conclusion was that "the sheaf-theoretic repre­ sentation provided a method of constructing modules with almost any pathological property that can be imagined." Most of his results in the expository part of the paper [178, p. 1]" could be obtained by specialization from theorems in Grothendieck's work." Pierce pointed out that [178, p. 4] "The general idea of realizing algebraic objects by topolog­ ical and geometric structures goes back at least to Stone's topological representation of Boolean algebras...part one of this paper can be viewed as a direct generalization of Stone's theory." In his memoir. Pierce said "Of particular importance in this paper is the set of all central idempotents of

R. We denote this set by B(R). Thus B(R) = {e e Z(R)|e = e}." He continued [178, p. 8]; Denote by X(R) the set of all maximal ideals of B(R). A topology is imposed on this set in the usual way; if A C X(R) , the closure A“ of A is {M e X(R) | M 3 A A}. This hull-kernel topology makes X (R) a Boolean space, that is, a compact, totally disconnected Hausdorff space. ... The space X(R) with the hull-kernel topology will be called the decomposition space of R. Each prime ideal P of B(R) generates an ideal P of R which gives a stalk R/P of a sheaf. In Section 5, Pierce showed that given a sequence R -»■ (X(R) , K(R)) -»■ r(X(R), K(R)) the first and last terms are isomorphic "in an appropriate sense" [178, p. 19]. Part two of the memoir was [178, p. 53];

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Modules Over Commutative Regular Rings The second application of the representation theory developed in Part One is concerned with finitely gener­ ated R-modules, where R is a commutative, regular ring. Several topics are considered, including cyclic modules, direct sums of cyclic modules, the Grothendieck group of the category of finitely generated modules, projective and injective modules, and the torsion theory of modules. ...a concrete representation of the Grothendieck group of the category of finitely generated R-modules (R a commutative, regular ring) is obtained; it is shown that any two decompositions of a finitely generated R-module into direct sums of cyclic modules have a common refine­ ment; the problem of which commutative, regular rings admit finitely generated modules which are not direct sums of cyclic modules is solved; namely it is shown that R has this property if it satisfies the d.c.c.; and the self-injective, commutative regular rings are character­ ized as those commutative, regular rings for which every finitely generated, torsion free module is projective. . . . The representation theory of Part One makes it possible to translate most questions about modules over commutative, regular rings into problems concerning sheaves of K-modules over a space X, where (X, K) is a regular ringed space. The advantage of doing this comes from the abundant structure of regular ringed spaces and their sheaves of modules. Often, concepts and techniques which are perfectly natural in the context of sheaf theory seem very remote from the study of modules in an abstract setting. How to represent a (topological) ring of very general type by continuous sections in a canonical sheaf or field, was reported by Hofmann in a paper [105] published in the Bulletin of the American Mathematical Society in May 1972 (which developed out of an invited address of the Society in Athens, Georgia on November 20, 1970). It was in part a survey of the state of research in the area of sectional representation. Some of the material was based on earlier work done with Dauns and many aspects of the presentation were due to numerous discussions with Klaus Keimel, Kwangil Koh, and Silviu Teleman,who visited Tulane in the academic year

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1970-1971. This was during the Tulane Ring and Operator Year under a program made possible by the Ford Foundation. Visi­ tors from various institutions of the U.S.A. and abroad contributed series of lectures in which they covered recent advances in their own field of specialty. In 1970, Keimel published his thesis on the sheaf representation theory of lattice ordered groups and rings: Representation de groups et d'anneaux reticules par des sections dans des faisceaux [114]. He had published several other articles on this subject and at Tulane gave a direct treatment of the subject covered in his thesis, stating a general representation

theorem. In October 1970, Silviu Teleman of the Institutul de Mathemcatica, Bucharest, Rumania, came to Tulane University, where he was able to spend several months thanks to an exchange program between the National Academy of Sciences of the U.S.A. and the Academy of Sciences of the Socialist Republic of Rumania. At Tulane he gave lectures on harmonic algebras, a subject which originated in harmonic analysis. In 1937, M. H. Stone wrote his seminal paper "Applications of the theory of Boolean rings to general topology" [200]. This memoir, and others by I. M. Gelfand with M. A. Naimark, two Soviet mathematicians, helped to open a new field of func­ tional analysis, namely the functional representation of algebras. Norbert Wiener had published in 1932 "Tauberian theorems" [225], and in 1933 "The Fourier integral and certain of its applications" [226] . In 1947 G. E. Silov

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developed an abstract harmonic analysis in which the Tauberian theorem and the local theorem could be stated and proved. By 1956 Silov's theory had been extended to a class of non-commutative Banach algebras in a paper by A. B. Willcox "Some structure theorems for a class of Banach algebras" [227]. Willcox's theory was generalized in 1960 by Charles Rickart. Teleman wrote "Analyse harmonique dans les algêbres régulières" [203], which generalized "la théorie des algêbres de Banach régulières de A. B. Willcox" [203, p. 691]. He showed that the topology on the ring is not important and hence Wiener's local theorem is obtained in purely algebraic form. The two results discussed above, namely Grothendieck's that any ring (commutative with unit) is isomorphic to the full ring of global sections in a sheaf of local rings, and that of Dauns and Hofmann in which a representation of

sheaves was obtained for biregular rings (not necessarily commutative), suggested to Teleman to look for a functional representation by sheaves for the rings that in 1968 had been called regular, and later, in order to avoid confusion, harmonic. The theorem of representation by sheaves of regular (harmonic) rings had been obtained during the fall of 1967, but was only published in 1969. For its proof the generalization of the local theorem of Wiener played an important role.

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B. Sheaves and Categories In the decade 1929-1939 (which began with the depres­ sion and ended with the outbreak of the second World War) various structures in mathematics were emphasized. By the early 1950s secondary school teachers were urged to base their teaching on "structures." Thus, according to Saunders Mac Lane, the stage was set for the ideas of category

theory [158, p. 229]: Category theory asks of each type of mathematical struc­ ture "What are the morphisms?"... This emphasis on morphisms as such came in the 1940's and not in the . decade 1929-1939, when attention was focused rather on subobjects (monomorphisms) and quotients (epimorphisms). For example van der Waerden's Moderne Algebra, following the lead of Emmy Noether, studies homomorphi sms G H of groups, and of rings, but only such as map G onto H. The utility of considering the more general homomor- phisms G ->■ H of one group into another first became clear from the example of algebraic topology where one was forced to study continuous maps X -»■ Y of one topological space into another, and the corresponding homomorphisms on the homology groups. Indeed the notion of a functor as a morphism of categories is suggested by the decisive example of the homology functor H^ on the category of topological spaces to the category of abelian groups; it sends each space X to the corresponding nth (singular or Cech) homology group H^ (X) and each morphism f : X -»■ Y of spaces to the induced morphism H^(f); H^(X) -> H^ (Y) of homology groups.

In the article quoted above "The influence of M. H. Stone

on the origins of category theory" [158], Mac Lane illustrated the foreshadowing of "the general idea of a functor as a morphism of categories" by examples from Stone's

166

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work [158, p. 229]: "We may regard C [the ring of all continuous real-valued functions f: X -»■ R on the topolog­ ical space X] as a functor on the category of such spaces to the category of rings." The passage between topology and algebra was "fundamental" to Stone's investigations. Other categorical notions besides functor and morphism, were also foreshadowed in Stone's work, particularly natural isomorphism and equivalence of categories. Stone indicated that among the reasons he did not develop the language of categories himself was the disruption of the war [158, p. 240]-" I think that under other circumstances I would have inevitably been led to explore the significance of category theory as it then was for my own work." On the other hand wartime disruption in the 1940s provided Samuel Eilenberg and Saunders Mac Lane sufficient time to formulate their general observations about mathemati­ cal structure. Eilenberg had been a student of Kuratowski in Poland and had gained notable results on the topology of the plane. The 20 years 1918-1939 saw extraordinary progress in mathematics in Poland, also animated by a spirit of nationalism as in France. In 1938 he had first used cochains with coefficients in a homotopy group and continued this work in his paper "Cohomology and continuous mappings" [59] of April 26, 1939 which was submitted to the Annals of Mathematics two days before his arrival in the United States. Here he received [161, p. 135] "vital encourage­ ment" from Lefschetz (as had Zariski). After a 1941

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lecture Mac Lane delivered at the University of Michigan, Eilenberg remarked that some of the difficulties Mac Lane outlined had also come up recently in a paper by Steenrod. The two men then worked all night to solve the difficulties in order to prove that the p-adic solenoid was the dual of the p-adic integers. They [161, p. 136] "resolved to get to the bottom of this curious connection between algebra and topology." The fruitful collaboration lasted more than

14 years in which they produced more than 15 papers. Eilenberg and Mac Lane viewed their work as musical composi­ tions, and so the first paper "Group extensions and homology" [62] was Opus I. In it, according to Mac Lane [161, p. 136]:

We treated universal coefficient systems for Cech cohomology, so we had to handle limits of inverse systems of groups. To construct isomorphisms between such limits, we need transformations between two inverse systems; moreover we knew that the universal coefficient theorem for complexes was "natural" and we wanted to make this statement a real theorem and not a pious platitude. To do these things we had to discover the notion of a natural transformation. That in its turn forced us to look at functors, which in turn made us look at categories. These were very general notions indeed. A crucial step was then our willingness to write a paper on these generalities - Opus II, "General Theory of Natural Equivalences," published in the Trans­ actions of the AMS in 1945. In 1944, Mac Lane was on leave from Harvard to direct the Applied Mathematics Group of Columbia University and [161, p. 138] "was instructed to hire many fresh mathematical brains to help with the research side of the war effort." Samuel Eilenberg, professor at the University of Michigan was hired first. Mac Lane recalled [161, p. 138] "During

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the day we all worked hard at airborn fire control after which Sammy and I would go out to dinner followed by an evening of work on categories and/or cohomology." In the preliminary report. Opus Ila "Natural isomor­ phisms in group theory" [63} of October 1942, Eilenberg and Mac Lane proposed a precise definition of the "naturality" of the correspondences which they had observed in modern mathe­ matics, such as the isomorphism between two groups or between two complexes. In order to do this they explained what they mean by a "functor," by using properties of character groups. They regarded [63, p. 538] "the character group Ch(G) of G, a finite group, as a function of a variable group G, together with a prescription which assigns to any homomorphism y of G into a second group G ', y : G ->■ G', the induced homomorphisms

Ch(y) : Ch(G') -»■ Ch(G). The functions Ch(g) and Ch(y) jointly form what we shall call a "functor." In this report the authors also defined "natural equivalence" [63, p. 540]; Let T and S be two functors which are, say, both covariant in the variable G and contravariant in H. Suppose that for each pair of groups G and H we are given a homomorphism y(G, H) ; T(G, H) S(G, H) .

We say that t establishes a natural equivalence of the functor T to the functor S and that T is naturally equivalent to S whenever

(El) Each t (G, H) is a bicontinuous isomorphism of T(G, H) onto S(G, H); (E2) For each y : G^ + Gg and n : Hg (Gg, H^)T (y, n) = S (y, n)T (G^, H^)

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Mac Lane also pointed out that the idea of categories existed in the work of Stone long before it had a defini­ tion. For example Stone wrote in his well known 1937 article "Applications of the theory of Boolean rings to general topology" [200, p. 338]: The algebraic theory of Boolean rings is mathematically equivalent to the topological theory of Boolean spaces by virtue of the following relations; (1) every Boolean ring has a representative Boolean space; every Boolean space is the representative of some Boolean ring; and two Boolean rings are isomorphic if and only if their representatives are topologically equivalent; (2) the group of automorphisms of a Boolean ring is isomorphic to the topological group of an arbitrary representative of the ring; (3) the representatives of Boolean rings which are isomorphic to the various ideals in a Boolean ring A are characterized topologically as the open subsets of an arbitrary representative of A; in particular, G (a) is a representative of the ideal a in A; (4) the representatives of the homomorphs of a Boolean ring A are characterized topologically as the closed subsets of an arbitrary representative of A; in particu­ lar, G' (a) is a representative of the quotient ring A/a; (5) the representatives of Boolean rings with unit are characterized topologically by the property of bicom­ pactness . In Opus II, "General theory of natural equivalences [64]" published in 1945, Eilenberg and Mac Lane defined categories in order to provide a technique for clarifying concepts such as that of natural isomorphism. It has been said that categories are what one must define

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in order to define functors, and that functors are what one

must define in order to define natural transformations.^

The geneology of categories and sheaves is diagrammed sche­ matically on the following page. In order to deal with such situations as vector spaces and their linear transforma­ tions, groups and their homomorphisms, topological spaces and their continuous mappings, etc., they introduced [64, 234] a new concept; A category A. will consist of abstract elements of two types; the objects A (for example, vector spaces, groups) and the mappings a (for example, linear trans­ formations, homomorphisms). For some pairs of mappings in the category there is defined a product... Certain of these mappings act as identities with respect to this product, and there is a one-to-one correspondence between the objects of the category and these identi­ ties . In Chapter I, "Categories and functors" was the formal

definition [64, p. 237]; A category A = {A, a} is an aggregate of abstract elements A (for example, groups), called the objects of the category, and abstract elements a (for example, homomorphisms), called mappings of the category. Certain pairs of mappings a^, Og E A determine uniquely

a product mapping a = “2 “! ^ ' subject to the axioms Cl, C2, C3 below. Corresponding to each object A é A

Mac Lane once told Peter Freyd that there was an [71, p. 155] "intellectual ancestry for the words 'category' and 'functor' in Emanuel Kant's Critique of Pure Reason [113]." Kant intimated tliat in the deduction of categories ■Üiere appeared for the first time an endeavour to connect together into one organic whole the several elements entering into experience. According to the Kantian scholar Robert Adamson [1, 669] "the categories are restricted in their applicability to the schema, i.e. to the pure forms of conjunction of the manifold in time, and in the modes of combination of schemata and categories we have the foundation for the rational sciences of mathematics and physics."

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SAMUEL - MAC LAKE WEIL MURBMO.

ICM (WEIL) WEIL EH£NBERG & CARTAK £ BUCHSBAUK

;ro t h e n d i e (

GRAY GIRAUD

COBEK

GRAY . MITCH:

HAKIK

SCM (LAMERE)

THE (ZNEOL(X% OF CATEGORIES AND SHEAVES

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there is a unique mapping, denoted by e^ or by e(A), and subject to the axioms C4 and C5. The axioms are: Cl. The triple product defined if and only if is defined. When either is defined, the associative law

=3(*2*l) = (*3 *2 )"l holds. This triple product will be written as

02. The triple product 0 2 0 2 ^ 1 defined whenever both products and “2 “! defined. DEFINITION. A mapping e G A will be called an identity of A if and only if the existence of any prod­ uct ea or ge implies that ea = a and ge = g. 03. For each mapping a € 4 there is at least one identity e^ E 4 such that ae^ is defined, and at least one identity eg such that egOt is defined. 04. The mapping e^ corresponding to each object A is an identity. 05. Foreach identity e of . A there is a unique object A of 4 such that e^ = e.

In a paper published in 1948, "Groups, categories and duality" [155], Mac Lane called attention to the categories themselves. He introduced the idea of "bicategory" a term suggested by his colleague Professor Grace Rose [155,

p. 265]. A bicategory is a category with two distinguished classes of mappings, tlie "injections" and the "projec­ tions," subject to the following self dual axioms : (BC-1) Every identity is both an injection and a projection. (BC-2) The product of two injections (projections), when defined, is an injection (projection) (BC-3) Every mapping y can be represented uniquely as a product y = K 0 n, where ir is a projection, 6 anequivalence and K an injection. (BC-4) The product of two mappings within (upon), when defined, is a mapping within (upon) (BC-5) Two injections (projections) with identical domains and identical ranges are identical.

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He also observed that many statements about abelian groups were equivalent to statements about the category of abelian groups [155, p. 264]: A free abelian group F can be characterized in terms of homomorphisms of abelian groups by the following prop­ erty: for any homomorphism a: F -»■ A and any second homomorphism g: B ->■ A onto the image group A there exists a homomorphism y: F -»■ B with gz/ = a. An infinitely divisible abelian group D is one in which there exists for each d e D and each integer m a solution x of the equation mx = d. Any homomorphism of an abelian group A into D can be extended to any abelian group A containing A. This property characterizes the infinitely divisible abelian groups; as it may be stated in a form dual to the characteristic property of free groups: given a: A ^ D and in isomorphism g : A -> B of A into B, there exists a t/: B -»■ D with z/g = a. For an abelian group, free prod­ ucts reduce to direct products. If a factor group of an abelian group is a free group, it is a direct factor. Dually, if a subgroup of an abelian group is infinitely divisible, it is a direct factor. This duality for abelian groups appears in algebraic topology as a duality between homology and cohomology groups. This phenomena is especially striking in the axiomatic form of homology theory. Mac Lane pointed out that it was published and unpublished work of Eilenberg and Steenrod from which his ideas profited. In an invited address delivered before the Chicago meeting of the American Mathematical Society on November 27, 1948, Mac Lane spoke on "Duality for groups" [156]. He related [161, p. 136] "...this study arose from discussion with Sammy and my wish to do Eilenberg-Steenrod axiomatic homology theory with values not just abelian groups but objects of a more general abelian category." Mac Lane found conditions on a category such that many of the theorems true for the category of abelian groups held and he identified

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certain classes of statements that were true if and only if the dual statement was true. He called such categories "abelian." In giving some insight into the nature of the duality principle Mac Lane used illustrations of the dual nature of the frees and the divisibles in the category of abelian groups (the projectives and injectives, respec­ tively, in that category).^ He demonstrated that the additive structure of an abelian group is implied for other axioms, and proved that any map between extensions of the same objects was an isomorphism. In 1940 algebraic entities had been defined by the remnants of generators and rela­ tions. Mac Lane's definition of "product" as the solution of a universal mapping problem was revolutionary. He recognized later that it was so revolutionary that it was not immediately absorbed even "by the most category minded 2 people." He presented this [156, p. 508] DEFINITION. In a category with zero, a (simulta­ neous) free direct product of objects A and B is a diagram (18.1) A^AxB^B 1 2 consisting of an object A x B and four mapping r , r , A 1 , A,2 ;

^ A X B A, A X B -> B, ^A X B* “ “ " "A X B (18.2)

A 1 ;A->AxB, A^^_:B^“AxB,2 ^A X B" " ' " ^ "A X B

Illustrating the climate of the late 1940s ; in a footnote on page 486 he remarked that the dual of a free (nonabelian) group is a fascist group. 2 Statement at a conference in Dallas, Texas January 1973.

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with the following three properties (omitting the subscript A x B to simplify notation): (i) r V = r V = 0^, r V = 0^%, r V = (ii) For any pair of mappings a^; C + A, «g: C ^ B, there exists a unique y: C A x B with r^y = o,, 2 r y = Og; (iii) For any pair of mappings : A D, gg: B + D, there exists a unique 6: A x B D with ôA^ = g-, 2 ÔA^ = gg. Here property (ii) asserts that we have a direct product and (iii) that we have a free product, both in the sense of §3. DEFINITION. An abelian category (AC) is a category C satisfying the axiom Z (existence of zero) and the axioms AC-1. There exists an integral and a cointegral object in C. AC-2. There exists in C a free-and-direct product for any two objects of C. In 1951, Samuel Eilenberg and Norman Steenrod pub­ lished Foundations of Algebraic Topology [66]. The preface began [66, p. viii] "The principal contribution of this book is an axiomatic approach to the part of algebraic topology called homology theory," a function from topology to algebra [66, p. vii]. A homology theory assigns groups to topological spaces and homomorphisms to continuous maps of one space into another. To each array of spaces and maps is assigned an array of groups and homomorphisms. In this way, a homology theory is an algebraic image of topology. The domain of a homology theory is the topologist's field of study. Its range is the field of study of the alge­ braist. Topological problems are converted into algebraic problems. The apology was that an axiomatic treatment had been needed

for some years. Cartan and Leray had axiomatized the con­ cept of carapace (now called "grating ") on a space. In that paper they included the de Rham theorem which related the

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exterior differential manifold forms in a manifold to the cohomology groups of the manifold. The concepts of category and functor dominated the development of Foundations of Algebraic Topology. The authors pointed out that "success­ ful axiomatizations in the past have led invariably to new techniques of proof and a corresponding new language" [66, p. xi]. Here we find "exactness" [66, p. xi]" ... it is frequently the case that the image of the incoming homomorphism coincides with the kernel of the outgoing homomorphism. This property is called exactness. It asserts that the group at the vertex is determined, up to a group extension, by the two neighboring groups, the kernel of the incoming homo­ morphism, and the image of the outgoing homomorphism. Exact sequences of groups and homomorphisms occur throughout. Eilenberg and Steenrod [71, p. 157] ...recognized the importance of the choice (of this word) so that they used the word "blank" throughout most of the manuscript. After entertaining an unre­ corded number of possibilities they settled on "exact." It was initially suggested by history: the exact se­ quence in de Rham's theorem is about exact differen­ tials. It was chosen because it is descriptive, it is short, it translates easily, and it inflects well ("exactly," "exactness"). At the same time Steenrod and Eilenberg's Foundations of Algebraic Topology was being written, so was Cartan and Eilenberg's Homological Algebra [36], which, although pub­ lished in 1956, was written between 1950 and 1953. It

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became apparent that most propositions concerning finite diagrams of modules could be proved in a more general type of category and moreover that the number of such proposi­ tions could be halved through the use of duality. In 1955, Buchsbaum published a thesis which had been suggested by Eilenberg. "Exact Categories and Duality" {18], was a full fledged investigation refining the condi­ tions and giving convincing evidence that abelian categories allowed the full development of homological algebra as in Cartan and Eilenberg {36]. He used "exact" categories, at the same time (in unpublished work) Grothendieck was using "abelian." According to Freyd the word "abelian" has stuck {71, p. 156]," partly to honor Mac Lane who suggested the whole idea, partly because Grothendieck writes in French and 'abelian' seems to mean 'very nice structure' in French in Grothendieck's Tôhoku article. (There are two words Abelian and abelian)." This latter paper of Grothendieck, "Sur quelques points d'algèbre homolgique," [87] followed. On page 129 he introduced the important notions of AB 5 categories and on page 134 generators for a category (which Mac Lane had

touched on in 1950). According to Buchsbaum the generality that Grothendieck required to develop the necessary framework to state a formal analogy between the cohomology theory of a space with coefficients in a sheaf, and the derived functor

of functors of modules depended on categories. Abelian

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categories in the sense of Buchsbaum played THE fundamental

role in Grothendieck's paper "Sur quelques points' d'algèbre homologique." Grothendick also introduced the notion of additive category [87, p. 195]. Nous appelerons G-faisceau sur X = X(G) un faisceau (d'ensembles) A sur X, dans lequel G opère de façon compatible avec ses opérations sur X. Pour donner un sens a cette definition, on pourra par exemple consi­ dérer A comme espace étalé dans X (cf. 3.1); nous n'insisterons pas. On définit de même la notion de G-faisceau de groupes, d'anneaux, de G-faisceau abélien ( = G-faisceau de groupes abéliens) etc. De façon imagée, on peut dire que si X est muni de certaines structures et si un faisceau A sur X est défini en termes structuraux, alors A est un G-faisceau de façon naturelle si les opérations de G dans X sont des auto- morphismes. Ainsi, un faisceau constant peut toujours être considéré comme un G-faisceau ("G-faisceau trivial"), de même le faisceau des germes d'applications quelconques resp. continues de X dans un ensemble donné, le faisceau des germes de fonctions holomorphes si X est une variété holomorphe et si les opérations de G sont des automorphismes de la variété X, etc. On appelle G-homomorphisme d'un faisceau dans un autre un homomor­ phisme de faisceau qui permute aux opérations de G. Si les faisceaux envisagés sont des faisceaux de groupes par exemple, on sous-entend que 1 'homomorphisme respecte cette structure, comme d'habitude. Avec cette notion d 'homomorphisme, les G-faisceaux d'ensembles (resp. les G-faisceaux de groupes etc.) forment une catégorie (cf. 1.1), qui a les mêmes propriétés que la catégorie correspondante sans groupe d'opérateurs G, (qui en est d'ailleurs un cas particulier, correspondant au cas ou G est réduit à 1'élément neutre). En particulier, les G-faisceaux abéliens forment une catégorie additive, dont nous allons indiquer les propriétés. Plus généralement, soit 0 un G-faisceau d'anneaux avec unité, considérons les faisceaux A sur X qui sont à la fois des G-faisceaux abéliens et des 0-modules, les opérations de 0 sur A étant compatibles avec les opérations de 0 (i.e. 1'homomorphisme naturel de faisceaux d'ensembles 0 x A -»■ A étant un G-homomorphisme): un tel faisceau sera appelé un G-0- Module. Appelant G-O-homomorphisme Un homomorphisme de faisceaux qui est un 0-homomorphisme et un G-homomorphisme, on voit que la somme ou le composé de deux G-O-homomorphismes est un G-O-homomorphisme, donc les G-O-Modules forment une catégorie additive notéç ; si 0 est le faisceau constant des entiers (avec

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les opérations triviales de G) on obtient de nouveau la catégorie des G-faisceaux abéliens, notée . Si G opère trivialement sur X, la catégorie s'inter­ prète comme la catégorie des 0'-modules, ou 0' est un faisceau d'anneaux convenable, si par exemple G opère trivialement sur 0, ou aura 0' = 0 B (G), (ou Z(G) désigne l'algèbre de G par rapport à l'anneau Z des entiers, ou plutôt le faisceau constant qu'il définit). Les résultats de 3.1 restent valables à de petites modi­ fications près: PROPOSITION 5.1.1. Soit 0 un G-faisceau d'anneaux sur X. Alors la catégorie additive des G-0- Modules est une catégorie abélienne, satisfaisant les axiomes AB 5) et AB 3*) de 1.5, et admet un généra­ teur. " Eilenberg also influenced Kan's decisive paper of

1958, discussed above, on adjoint functors. According to Stone [200, p. 326] If one seeks to explain why the notion of an adjoint functor appeared in category theory as much as fifteen years after its first place, the concept had its origins and early development in analysis in the theory of differential equations. It was not a valid part of the mathematical experience of the algebraically-oriented pioneers in category theory. The other historical factor that has to be cited is the interruption of mathematical research imposed by World War II and the many professional readjustments that followed it. Although, as Mac Lane pointed out. Stone, and then von

Neumann had made a successful abstract codification of adjoint linear transformations, this was not the direct source of adjoint functor [157, p. 125]. "The proximate sources of this idea are the universal constructions of Samuel and Bourbaki (1948) and the basic paper of D. M. Kan (1958) defining adjoint functors." Samuel's paper "On uni­ versal mappings and free topological groupé' [187, p. 591] showed that the problem of "universal mappings" always has a

solution, providing that thé mappings satisfy certain axioms:

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Al. Every T-isomorphism is a T-mapping (where T is some structure, such as a group). A2. If f: E^ ->■ Eg and g: Eg + Eg are T-mappings, then the composite mapping g o f: E. + E_ is a T-mapping. A3. A necessary and sufficient condition for a one-to- one mapping f of E, onto E_ to be a T-isomorphism is — 1 ^ that F and F be T-mappings. 51 A subset of E composed of all the elements where a family of T-mappings takes the same value is T-closed. 52 Any intersection of T-closed sets is T-closed. 53 Cardinal (E') < certain function of cardinal (E'), a function which depends only on the structure T. Pi The projections (on the components) are T-mappings. P2 If the f^: E -»■ E^ are T-mappings, the product mapping f : E + n E^ (defined by f(x) = (f^(x))) is a T-mapping. [187, p. 593] Thus when given a set E with a structure S, and mappings ç»: E ->■ F of E into sets F with another structure T, the problem is to find a particular set F^ and a particular mapping (p^i E -»■ Fq such that every E -»■ F can be represented as a composite

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Without elements in the objects it was painfully difficult to prove even simple lemmas for abelian categories. Enough were proved however, so that mathematicians began to recognize a class of state­ ments, true for the category of abelian groups, which one would be confident were true for the abelian cate­ gories. A metatheorem was in order. It was provided, roughly simultaneously, by Lubkin [152], Heron [98] and the author. The proofs were entirely different.... The aim of this work is to serve as a basis for the theory of abelian categories. The "geodesic course," according to Freyd, of the book is toward the Mitchell embedding theorem. This remarkable theorem makes it possible to extend the results obtained for modules to the case of abelian categories and was published in the American Journal of Mathematics in 1964, after being presented in a thesis (under Buchsbaum [166 p. 635]): "Every small abelian category A admits a full, 15 exact, covariant imbedding into a category G for some ring R. " Mitchell was an instructor, working with Eilenberg at Columbia when he began his book Theory of Categories [166], published in 1965. In Chapter X he stated [166, p. 245] "Sheaves with values in sets, rings, and modules were defined by Godement in such a way as to open the road to a theory of sheaves with values in more general categories. This generalization has been carried out by J. Gray and R. Deheuvels independently." Deheuvels wrote "Homologie des ensembles ordonnés et des espaces topologiques" [49] in 1962, published in the Bulletin Société Mathématiques de France, and Gray passed out mimeographed notes at Columbia called Sheaves with Values in a Category in 1962.

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John Gray was also an instructor at Columbia at that time, and it was Freyd who stimulated him to write his paper, which was published in 1963 in Topology.^ Other mathematicians also wrote on the category of sheaves [97]. As Mac Lane noted [157, p. 124]; "The concepts of category theory required a long period of polishing, perfecting and adaption to put them in the most effective form for general use. For example, the notion of an abelian category required time, adjustment, and application to sheaf theory before it

became really workable." In 1965, Eilenberg and John C. Moore, a professor at Princeton, collaborated on a "beautiful and decisive" paper "Adjoint functors and triples" [65], published in the Illinois Journal of Mathematics. [65, p. 381] "A triple F = (F, n, u) in a category A consists of a functor F: A -> A and morphisms ri;l^-»-F, y;F 2 F satisfying some identi­ ties analogous to those satisfied in a monoid." Huber had pointed out in "Homology theory general categories [107]," "that whenever one has a pair of adjoint functors then the functor TS constitutes a triple in B and similarly ST yields a cotriple in A." The object of Eilenberg and Moore's

^Gray added a note at the end of his paper [81, p. 18]: "An earlier version of this paper appeared as NSF Report GI 9022 M. Without my being aware of it, much of this report was reproduced by Bourgin: Modern Algebraic Topol­ ogy [13], Chapter 17, under the notion of 'general sheaves.' Unfortunately in this version the exactness proofs are incomplete, and the adjointness proof for the functors on presheaves induced by a continuous map contains an incorrect statement."

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paper was [65, p. 381] "to show that this relation between adjointness and triples is in some sense reversible." Mac Lane wrote [161, p. 135] "this idea of a monad (the mis­ named triple) dominated category theory" for five or six years. Gray related that Godement really originated the idea of triples in his classic work on sheaves. Dieudonné urged [56] One should not minimize the impact of categories on many mathematical theories.... it provides (1) a general frame similar to the one given by set theory in earlier times; (2) very useful general notions such as representable functors and adjoint functors; (3) guiding principles such as: "Never define objects without defining morphisms between them." ...the most remarkable use of categories has been the very original concept of Grothendieck topologies (or 'sites') and sheaves defined on them ('topoi'), generalizing in an unsuspected and far-reaching manner the classical topological notions. This generalization of topological spaces had come about when Grothendieck and his students were working out the abstract machinery necessary to attack the Weil conjectures. Weil had formulated these conjectures as extensions to algebraic varieties of arbitrary dimension of his work on the Zeta function of algebraic curves over finite fields. In "Numbers of solutions of equations in finite fields [22]," presented for publication to the American Mathematical Society on October 2, 1948 (the date of the first Séminaire Cartan), Weil gives a brief history of equations of the type n n^ n A X ° + A, X- + ... + A x_ = b. o o 1 r r Gauss, Jacobi, and Lebesgue all made contributions, and then much later Hardy and Littlewood, and later Davenport and Hasse (1935). Then in 1948 Hua and Vandiver, and others

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brought them up for discussion again. Weil proceeded to give a complete exposition of the topic in a new mode of presentation and concluded with [221, p. 498] "some conjec­ tures concerning the numbers of solutions of equations over finite fields and their relation to topological properties over variétés defined by the corresponding equations over the field of complex numbers" [221, p. 507]: Let V be a variety without singular points, of dimension n, defined over a finite field k with q ele­ ments. Let be the number of rational points on V over the extension k.^ of k of degree v. Then we have

^ = 3 0 1 ° 9 2 <0 )' 1 where Z(U) is a rational function in Ü, satisfying a functional equation W glX/ZyXg(U) with X equal to the Euler-Poincaré characteristic of V (intersection-number of the diagonal with itself on the product V X V). Furthermore, we have: Pn(U)P-(U) ... P,n _ 1(U) Z(U) = ^ zn ± , PQ(U)Pg(U) ... Pgn(U)

with P q (U) = 1 - Uy Pg^^U) = 1 - q ^ , and, for 1 < h < 2n - 1:

Pj^(u) = n (1 - “hi^) i=l where the a, . are algebraic integers of absolute value gk/:. Finally, let us call the degrees of the poly­ nomials P^(U) the Betti numbers of the variety V; the Euler-Poincaré characteristic % is then expressed by the usual formula % =^^h (-l)^Bj^. The evidence at hand seems to suggest that, if V is a variety without

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 8 6 singular points, defined over a field K of algebraic numbers, the Betti numbers of the varieties derived from V by reduction modulo a prime ideal p in K, are equal to the Betti numbers of V (considered as a vari­ ety over complex numbers) in the sense of combinatorial topology, for all except at most a finite number of prime ideals p. At the International Congress in Amsterdam in 1954, Weil also proposed these conjectures in his talk "Abstract versus Classical Algebraic Geometry" [223], pointing out the need for invariants analogous to topological ones, of varieties over fields of characteristic p. In 1958 Grothendieck spoke at the next International Congress on "Cohomology Theory of Abstract Algebraic Varieties" [88] giving an outline of the main topics of cohomological investigation then, four years after they had been introduced by Serre in FAC. The initial aim was to find the "Weil cohomology," which, it was soon realized, must be different than Serre's approach. As Grothendieck related: [88, p. 104] "Such an approach was recently suggested to me by the connections between sheaf-theoretic cohomology and cohomology of Galois groups on the one hand, and the classifi­

cation of unramified coverings of a variety on the other." He was making reference to a talk by Serre in Mexico, in August 1956. The second topic for cohomological methods in the cohomology theory of algebraic coherent sheaves [88, p. 104] : (a) General finiteness and asymptotic behaviour theorems. (b) Duality theorems, including a cohomological theory of residues. (c) Riemann-Roch theorem, including the theory of Chern classes for algebraic coherent sheaves. (d) Some special results, concerning mainly abelian varieties.

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The third topic consists in the application of the cohomological methods to local algebra. The remainder of his talk centered around the second topic and he concluded

[88; p. 117]: As a quite general fact, it is believed that a better insight in any part of even the most classical Algebraic Geometry will be obtained by trying to re-state all known facts and problems in the context of general schemata. Grothendieck had defined schema [88, p. 107]: We call pre-schema a topological space X with a sheaf of rings 0^ on X, called its structure sheaf, such that every point of X has an open neighborhood isomorphic to some Spec (A). If X and Y are two pre­ schemas, a morphism f from X into Y is a continuous map f: X ^ Y, together with a corresponding homomorphism f*z Oy Ojj for the structure sheaves, submitted to the one condition: if x e X, y = /(x), then the inverse image by f*z 0„ „ O., ^ of the maximal ideal in 0 X/V A/X A/X is the maximal ideal in 0„ . If X and Y are the prime % f y spectra of rings A and B, then it can be shown that the morphisms from X into Y correspond exactly to the ring homomorphisms of B into A, as they should. As was explained before, if we consider morphisms fz X S with a fixed pre-schema S, S plays the part of a ground-field. Besides, if S = Spec (a), then X is a pre-schema over S if and only if the sheaf of rings is a sheaf of A-algebras. In the category of all pre­ schemata, over a given S, there exists a product (which corresponds to the tensor product of algebras over a commutative ring A). Using this, one can define objects like pre-schemata of groups, etc., over a fixed ground- pre-schema S. One can also use products in order to introduce a mild separation condition on pre-schemata, suggested by the usual condition in the definition of algebraic varieties, so that one gets what can be called a schema. Again the conclusion [88, p. 117]; This work is now begun, and will be carried on in a treatise on Algebraic Geometry which, it is hoped, will be written in the following years by J. Dieudonné and myself, and which is expected to give a systematic account of all the questions touched upon in this talk.

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Many other people worked on the Weil conjectures, in­ cluding Serre, Monsky, Washnitzer, and Weil of course. In results published in 1960, Dwork proved the rationality of the Zeta function by methods not involving cohomology. But Weil had pointed out the need for a "good" cohomology theory to gain the required results. Grothendieck was the first person to define a "good" cohomology theory in algebraic ge­ ometry in 1962 according to Saul Lubkin [153]. Lubkin, at that time a graduate student at Harvard, said that [153, p. 444], "It had been well known for quite a while before that Grothendieck had a 'secret' definition of cohomology." In that same spring Lubkin presented his own cohomology in a seminar published as "On a conjecture of André Weil" [153]. It contains proofs of Weil's conjectures on the factorization and functional equation of the Zeta function of a complete non-singular variety X liftable to characteristic zero and defined over a finite field of characteristic p > 0. He also proved Weil's conjecture that the group of numerical equivalence classes of cycles on X is finely generated. Mimeographed notes of Grothendieck's seminar at Harvard were written up by his student Michael Artin [5]. The first topology to be defined on a scheme was the Zariski topology, discussed above. Serre took the Zariski topology to define a cohomology theory of coherent sheaves on a variety. He then proved that for a projective variety over the field of complex numbers, the theory was the same as the analytic theory. In 1958, Grothendieck found a

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general version of sheaf theory which enabled him to define the notion of etale cohomology of schemes. This broadened Serre*s work, and provided an approach to etale cohomology. In 1955 Y. Kawada and J. Tate had published a paper using etale coverings for cohomology theory of a variety. Grothendieck's definition of a Grothendieck topology: EXAMPLE (0, 0). Let X be a topological space and let T be the category whose objects are open sets of X and with Hom (U, V) consisting of the inclusion map if U C V, empty otherwise. Then a presheaf F on X with values in a category C is a contravariant functor F: T^ C. A sheaf F is a presheaf satisfying the following axiom: For U e T, {U^} an open covering of U the sequence F(U) n F(U, ) Î n F(U, nu.) 1 1 i,i 1 ] is exact with the canonical maps. (We assume C has products. For the definition of exactness, see (Sem. Bourb. #195)). Note that in T (in fact in the category of topological spaces) U, n u .~ U. x U.. Therefore, J- J ^ u ] DEFINITION (0, 1). A Grothendieck topology T consists of a category Cat T and a set Gov T of families {Uj— J of maps in Cat T called coverings (where in each covering the range U of the maps 0^ is fixed) satisfying (1) If 0 is an isomomorphism then {0} e Cov T. (2) If ^ Ü} e Cov T and (V^j -»■ U^} e Cov T for each f then the family {V^j ^ U} obtained by composition is in Cov T. (3) If {Uj^ -> U} e Cov T and V ^ U e Cat T is arbitrary then U.x V exists and {U.x V -»■ V} e Cov T. We will abuse language and call T a topology. [5, p. 7]. Artin's notes show, how categorical sheaf theory can become. As Tennison pointed out [207, p. 155] "It was in working with the abstract machinery to attack the Weil conjectures

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that Grothendieck and his students were led to this gener­ alization of topological spaces, over which one can still do sheaf theory." Séminaire de Géométrie Algébrique (SGA) had been held in various places since 1960 and these semi­ nars built up the material for EGA. Then in Bois Marie in 1963, 1964, Artin, J. L. Verdier, Grothendieck, P. Deligne, B. Saint Donat, and N. Bourbaki collaborated on Cohomologie des Schemas [6]. A category together with a Grothendieck topology on it is called a site. To every site can be assigned a certain full subcategory of the category of pre­ sheaves, called the category of sheaves, analogous to the definition of sheaves on topological spaces. Because the Grothendieck topologies on a category form a lattice there is a "finest" topology such that a given class of presheaves are sheaves. The canonical Grothendieck topology is the finest for which the representable presheaves are sheaves. In 1962/1963 Giraud proved [73]: From any site we many construct a new one by considering the category of sheaves for the site with its canonical Grothendieck topology. The category of sheaves of the latter site is equivalent to the category of sheaves of the former. Consequently the name topos was introduced in SGA 4 "Theorie des topos et cohomologie seale des schemas" as categories of set valued sheaves. In the initial seminar it was the "cohomologie étale des schemas" that was important, but there occurred a change in perspective and a consequent change in the title [6, vi]. The two theorems on the change of base are technically the central result of

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the seminar. Verdier and Grothendick presented the defini­ tion of topos due to Giraud : Definition 1.1. On appelle U-topos, ou simplement topos si aucune confusion n'est à craindre, une catégorie E telle qu'il existe un site C e U tel que E soit équiva­ lente à la catégorie C~ des U-faisceaux d'ensembles sur C. 1.1.1. Soit E un U-topos. Nous considérons toujours E comme muni de sa topologie canonique (II 2.5), qui en fait donc un site, et même, en vertu de 1.1.2 d) ci- dessous, un U-site (II 3.0.2). Sauf mention expresse du contraire, nous ne considérons pas d'autre topologie sur E que celle qu'on vient d'expliciter.

Some of the motivations were given in the preface to the second edition: Notre principe directeur a été de développer un langage et des notations qui soient ceux qui servent déjà effectivement dans les divers applications, de sort à ne pas perdre effectivement dans les diverses applications, de sorte à ne pas perdre contact avec le contenu "géométrique" (ou "topologique") des divers foncteurs qu'on est amené à considérer entre sites. Pour ceci, les notions de topos et de morphism de topos semblent être le fil conducteur indispensable, et il convient de leur donner la place central, la notion de site devenant une notion technique auxiliaire. Cela nous a amené en particulier à étoffer considérablement la rédaction de 1'exposé IV consacré à ces notions, et à reprendre cornétement l'exposé VI (consacré aux sites et topos fibrés) dans cet esprit. Grothendieck and those around him had long maintained that in sheaf theory it is the topos itself - i.e., the whole category of sheaves-that is important and not the site, or small category, from which it is derived. He himself, however, had never consequently developed this point of view. Interested in the no-man's- land between category theory and logic, William Lawvere wrote a thesis under..

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Eilenberg's direction at Columbia in 1963: Functorial Semantics of Algebraic Theories [133]. He later wrote about that period: • . .The same year in which I finished my doctoral dissertation five distinct developments in geometry and logic became known, the subsequent unification of which has, I believe, forced upon us the serious consideration of a new concept of set. These were the following: Non Standard Analysis (A. Robinson) Independence Proofs in Set Theory (P. J. Cohen) Semantics for intuitionistic predicate calculus (Kripke) Elementary Axioms for the Category of Abstract Sets (Lawvere) The general theory of topoi (Giraud) In a partial summary of his thesis published in 1963 [133], Lawvere thanked Eilenberg, Mac Lane and Freyd for

their inspiration and encouragement, in particular Eilenberg's suggestion of [133, p. 869] "functorizing the study of general algebraic systems." Lawvere did this by using Freyd's refinement of Kan's adjoint functors. He began by defining an algebraic theory as [133, p. 1] "a small category A whose objects are the natural numbers 0,1,2,... and in which each object n is the categorical direct product of the object 1 with itself n times." He then showed that algebraic theories and the mappings between them form a category. Furthermore [133], Each algebraic theory A determines a large category (A) S whose class of objects is just the equational class (variety) of all algebras of type A, and whose maps are all (into) homomorphi sms between these. An algebra of type A can be viewed as a product preserving functor A S from A to the category of sets; a homomorphism of algebras is then just a natural transformation between such functors. If A is the algebraic theory whose only n-ary operations are projections (i.e., A is equivalent to the dual of the category of finite sets),

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then the category of algebras S (Al is just the category S of sets. Every map /: A ^ B of algebraic theories deter­ mines in an obvious way a functor -»■ which (f) preserves underlying sets, i.e., for which = S where U^i S S, U^i ^ S are the underlying set functors (notice the order in which we write composition). We call any functor of the an algebraic functor, and we call any category of the form S (A) an algebraic category. Any algebraic theory A is equivalent to the dual of the full category of finitely generated free algebras in the associated algebraic category. These ideas "marked a turning point in category theory" [158,

p. 137] according to Mac Lane. The basic idea is that substitution should be represented by composition of arrows. Shortly afterward Lawvere published "An elementary theory of the category of sets" [134]. We adjoin eight first-order axioms to the usual first-order theory of an abstract Eilenberg-Mac Lane category to obtain an elementary theory with the follow­ ing properties; (a) There is essentially only one category which satisfies these eight axioms together with the additional (nonelementary) axiom of completeness, namely the category S of sets and mappings. Thus our theory distinguishes S structurally from other complete categories, such as those of topological spaces, groups, rings, partially ordered sets, etc. (b) The theory provides a foundation for number theory, analysis, and much of algebra and topology even though no relation G with the traditional properties can be defined.... AXIOM 1 (Finite Roots). There is a terminal object 1 and an initial object 0. Every pair A^, A 2 of objects Pk has a product with projections A^ x ^ and a sum ^k with injections A^ -»■ A^^ + A^ satisfying the usual uni­ versal mapping properties. Every pair A 3 B of mappings has an equalizer K ->■ A and a coequalizer B -»■ Q satisfying the usual universal mapping properties.... AXIOM 2 (Exponentiation). For any pair A, B of objects there is an object B and an "evaluation" mapping A X B^ $ B with the property that given any object X and

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^ h A any mapping A x X B, there is a unique X -»■ B such that (A X h) e = f. AXIOM 3. There is an object N together with mappings 1 N ^ N such that given any object X together with mappings 1 X ^ x, there is a unique mapping N 5 X such that X q = zx and xt = sx. £ AXIOM 4 (1 is a generator). If mappings A ^ B are different, then there is an element x G A such that xf xg. AXIOM 5 (Axiom of Choice). If the domain of f has elements, then there is g such that f g f = f. AXIOM 6. If A is not an initial object, then A has elements. AXIOM 7. An element of a sum is a member of one of the injections. AXIOM 8. There exists an object with more than one element. Unfortunately with these original axioms, characteristic functions [134, p. 1506] were hard to handle, but as Lawvere stated any topos satisfied (i) finite completeness and finite cocompleteness (ii) Cartesian closedness (iii) the existence of a subobject classifier. Therefore any category satisfying these axioms was called an elementary topos, and topoi in the old sense are called Grothendieck topoi. Freyd considered [134] "the development of elementary topoi to be the most important event in the history of categorical algebra." Lawvere went a step further in his program of doing algebraic logic by means of catego­ ries when he introduced the idea that quantifiers were "adjoints to substitution." In the year 1969-1970 Myles Tierney, an associate professor at Rutgers University, had a senior fellowship to Dalhousie University, where he could collaborate with

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Lawvere. Tierney had also taken his PhD at Columbia (in 1965). During this time they worked on establishing the independence of the Continuum Hypothesis from other axioms of the category of sets, and showed that the axioms proposed above were [136, p. 1] "adequate for all the usual exactness properties of toposes as well as for the construction of sheaf categories and the proof that they are again toposes." They also simplified the original axioms. During the year Tierney and Lawvere collaborated, some of their work was concerned with [136, p. 1] "establishing the independence of the Continuum Hypothesis from other axioms of the category of sets." These results were published in 1972 by Tierney [211]. The paper used several results about topoi— i.e. categories of sheaves, and in conclusion Tierney stated [211, p. 41] "it seems that the topological interpretation of intuitionism can be thought of simply as mathematics done in sheaves (T) where T is a topological space. Many independence results in intuitionistic algebra and analysis should be provable by topos methods, though only the surface has been scratched to date." Interest had arisen in the Continuum Problem with Paul Cohen's proof in 1963 as a consequence of which he was awarded the Fields medal in 1966. In presenting the award Alonzo Church stated that the results he accomplished would be greatly improved, but it was the initial breakthrough that was significant. It was this proof that was formulated in topos-theoretic language, partly

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because an essential step in Cohen's proof turns out to be the topological process of turning a presheaf into a sheaf.

In 1970 at the International Congress of Mathematicians Lawvere gave a talk entitled "Quantifiers and Sheaves" [134] describing his and Tierney's axiomatic theory of sheaves. As Mac Lane said [160, p. 126] The basic discovery of Lawvere resides in the observation that the characteristic functions required for the elementary theory of the category of sets could be efficiently described by this axiom [each of our catego­ ries E has a subobject classifier] and that it also applied to the categories of sheaves. This was the connection with the theory of sheaves which had been developed by the Grothendieck School in France. In a course held at Varenna in September 1971, Tierney gave an introduction to this axiomatic theory of sheaves. He pointed out that his and Lawvere's first goal was to develop Grothendieck's point of view that it is the whole category of sheaves that is important, but that Grothendieck had never done this. It was later that they thought about topos as a [212] "kind of set theory useful for dealing with many kinds

of 'sets' other than just 'abstract' sets." In January 1971 seventy mathematicians participated in a conference on connections between Category Theory and Algebraic Geometry & Intuitionistic Logic held at Halifax, Nova Scotia, Canada, under the sponsorship of Dalhousie University. There were formal talks and many lively discus­ sions and informal talks. These were still political times as Lawvere's political activities relating to the war in Vietnam created tensions at his university. "The conference sharpened the development whereby two previously unrelated

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trends in modern mathematics are now each applying concepts and methods developed by the other" [136, p. 1]. Initial progress was made in [136, p. 1] the development on the basis of elementary (first-order) axioms of a theory of "toposes" just good enough to be applicable not only to sheaf theory, algebraic spaces, global spectrum, etc. as originally envisaged by Grothen­ dieck, Giraud, Verdier and Hakim but also to Kripke semantics, abstract proof theory, and the Cohen-Scott- Solovay method for obtaining independence results in set theory. A topos is any category which is cartesian closed and has a subobject-representor, and thus "summarizes in Objective categorical form the essence of higher-order logic" [131, p. 1]. It provides a [136, p. 3] "useful generalization of set theory to the consideration of sets which internally develop." Lawvere continued. The first class of toposes to be studied as categories was the class of E of the form E = all 6-valued sheaves on some topological space. In such an example our axioms are verified in terms of the section functor r as follows

r(U, Y^) = Hom (X|U, Y|U) for all sheaves X and Y and all open sets U, and r(U, 0) = Set of all open subsets of U.

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APPLICATIONS

C. Sheaves and topology

Leray invented a faisceau while working in the area of algebraic topology. Faisceaux were then developed to clarify and solve a problem in several complex variables. In conclu­ sion I would like to make a brief reference to sheaf theory and solving problems in topology illustrating how the theory

enriches the branch in which it originally started. Swan began the introduction to his book [202, p. 1] by asking the question "What are they good for?" and concluding that "Sheaves are very useful in proving theorems." He illustrated his answer with the examples of the Vietoris mapping theorem [202, p. 133] using the spectral sequence of a map, and two duality theorems— Poincaré-Lefschetz and Alexander-Lefschetz [202, p. 138]. He also proved that singular and Cech cohomology agree for paracompact HLC spaces [202, p. 101]. In 1960 A. Borel and J. Moore published "Homology theory for locally compact spaces" [12]. According to W. S. Massey (who had introduced Swan to sheaves at Princeton in 1955) [164], "They seemed to be mainly interested in defining a (fech-type homology theory (for arbitrarily locally Hausdorff spaces) with coefficients in a sheaf, so that they could

198

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state and prove the most general version possible of the Poincare duality theorem for generalized manifolds." Glen Bredon had given a course in the theory of sheaves at Berkeley in 1964, and that series of lectures was published in 1967— Sheaf Theory [14]. In this textbook is a full discussion as to the importance of Borel-Moore homology. Sheaf theory has applications in the branch of mathe­ matics that was one of its sources and is useful in simpli­ fying previously proven problems, as diagrammed schematically on the following page. The solution of the Cousin problem meant that sheaf theory met the final test of every new mathematical theory: its success in answering questions that the theory was not designed to answer. Furthermore, although the Cousin problem was important, sheaf theory has shown itself to be useful in unexpected ways in ring theory and category theory, and in proving theorems in topology.

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SEVERAL COMPLEX ALGEBRAIC VARIABLES TOPOLOGY

COUS

faisceau

ALGEBRAIC GEOMETRY

SHEAF THEORY

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IMPLICATIONS FOR MATHEMATICS EDUCATION

One of the purposes of mathematics education is to help the students studying mathematics to find an answer to the question "What is mathematics?" The majority of the mathe­ matics students are exposed to throughout their high school and undergraduate education is mathematics that is completed and refined until it seems to be a manipulation of symbols. Very few students are exposed to the construction (or uncov­ ering) of mathematics. While students are learning mathematics in high school, they often ask how the branches they study— geometry, calculus, linear algebra— relate to one another. They wonder what significance the subject matter they are learning has to do with the bulk of mathematics. In order to answer these questions a study of the history of mathematics is essential, in part because the various branches of mathematics are historically determined. Looking at the chronological development of the subject we can see that the vast majority of mathematics has been created since 1900. Consequently, what is mathematics now is cer­ tainly not what it was, for example, in the year 1500. Is it the essential similarities which determine what makes up mathematics, or is it the dynamic change itself which is its

201

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distinguishing characteristic? As mathematics is taught, students see it as the former, whereas a study of the history of the subject would indicate it probably is the latter. The educational implications of the creation of sheaf theory are: 1) an establishment of an example of what is meant by the history of mathematics, beyond a chronology of a definition of a theory. Moreover, sheaf theory will demon­ strate how a theory is determined, and that it is in part the recognition of theories that make up the history of mathe­ matics; 2) a practical approach to the teaching of the history of mathematics in school, so that a case study of this type can provide teachers with a perspective to determine whether they are teaching mathematics as invention or as discovery;

3) the case study can help to determine what place the history of mathematics has in school mathematics because, as Piaget has said, if we know how human kind discovered mathematics we will know better how a child learns it and that it is not detached from the human condition. 1. The creation of sheaf theory is an example of significant growth of knowledge in mathematics. However, sheaf theory, as it is presented in text books, does not reveal that its creation was of particular importance. It is only in looking at how it came about and why it was needed that we can see the level of its importance. It satisfies Hilbert's criterion: success in answering questions, success in answering questions which are preexistent, and success in answering questions that the theory was not designed to

answer.

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"The final test of every new mathematical theory is its success in answering preexistent questions that the theory was not designed to answer." [99, p. 384]. In reading such a history students studying sheaf theory can see the dynamics of the subject and can come to a better understanding of the mathematics itself. Furthermore, as it exhibits significant growth in a discipline, it is even more important to under­ stand it in terms of our cultural heritage as well as to make students aware of how our culture affects the growth of mathematics. As a case study of Hilbert's criterion, it shows

how different branches of mathematics affect one another. Students using sheaves in their own mathematical research would have a better understanding as to the background of the subject and hence the significance of their own work. As has been indicated by Cartan, he needed Weil's integral, but he didn't have i .. Consequently, a student's outlook will be broadened by seeing the way the mathematics they are doing fits into their own culture, as well as how the speciality fits into mathematics as a whole . . . how it arose in the first place and where it should go . . . and just as Gronwall found the gap in Cousin's proof, they will be helped in detecting the gaps where new concepts are needed. It is possible that even more significantly they will see the broad areas where new structures provide unification and consolida­

tion of seemingly diverse concepts . . . for example how Grothendieck undertook to rewrite all of mathematics by using sheaves and the whole idea of FAC. They will also see what

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is as important, that theories don't simply appear to mathe­ maticians, but are worked out laboriously by means of letters to other people, seminars and visits back and forth to various institutions. This will show them how important these insti­ tutions really are in the growth of mathematics, and that in general it is a very human activity, not only in the develop­ ment of the terminology but in the progress of it as a body of knowledge. Hilbert's criterion enables us to see that significant

growth is only going to take place when an important problem has been solved. Studying the history of the subject is the only way we can determine what is truly an important problem, i.e., by questioning whether it came before the theory which was used to solve it. Students doing research at whatever

level should understand how and why the introduction of a new idea such as sheaves leads to the clarification and solution of a problem thirty-seven years after it was established as a problem. By so doing they will understand how certain ideas seem to be around at a time . . . how Oka and Cartan independently solved a problem in the same year although they were separated by half the world. Thus in conclusion, we see that the creation of sheaf theory is important for research students and for teachers. 2. How can this case study or one of this type help provide a practical approach to the teaching of mathematics and the history of mathematics? As has been established here, a branch of mathematics is an area of mathematical research

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historically determined with its own distinctive items, problems, methods, techniques and formal theories. It was the combination of the resources of several complex variables and algebraic topology that brought about the establishment of sheaf theory as a theory. In particular we see that the establishment of the terminology and the change in language determines that we have something worth testing as a theory. Now when a teacher is imparting knowledge he or she must emphasize the language. The research done in this study shows this to be true for mathematics and shows also that it is not simply the manipulation of symbols. Furthermore, in having some understanding of the history of mathematics by either doing such a case study or studying others that have been made, the teacher should be aware of his or her own attitudes about whether mathematics is discovered or invented. The view taken here is that it is invented, but the other viewpoint is only a subjective distinction. However, in the teaching of mathematics the training and attitudes of the teacher on this question are worthwhile to investigate because

of the subsequent effects on styles of teaching as well as the consequent effect on students' attitudes toward mathe­ matics. Ignoring this question may be the cause of much of the debate in current research in mathematics education. The attitude of Platonic mathematicians is that mathe­ matics exists and is waiting to be found by those clever enough to discover (or uncover) it in the wilderness. On the other hand, humanist mathematicians hold that mathematical

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structures are invented so that their work requires quite a different kind of creativity— choosing just the right set of axioms on which to build a structure. These two viewpoints are important for mathematics teachers and for others doing research in mathematics education— and it is only through some consideration of the dynamics of the history of mathe­ matics that mathematics teachers will become aware of these differences. It is not so much that answering the question one way or the other is the important issue, but that the teacher's attitudes toward the question be clarified because, if mathematics is all discovery then it is truly worthwhile to develop the necessary skills and follow them by a rapid survey of previous mathematics until a new frontier is reached— a wilderness on which a young mathematician can set his foot. In contrast, if mathematics is invented it is creativity that matters, and a student's technique must be developed in that area. Polya has said that a productive mathematician is free to prefer any aspect of mathematics and he should prefer the one that is most profitable for his work. However on the high school teaching level we do not have quite the same choice.

3. Unless we live with a mathematician we learn mathe­ matics totally detached from the human condition. However in using Hilbert's criteria to establish that sheaf theory is in fact a theory, we have to look at the human element as well. If a problem existed before the theory then it may be consid­ ered a sufficiently important problem to satisfy the criteria.

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The only way it could be an important problem is that some mathematician couldn't solve it. If he couldn't solve it, then it is important that others try to do so, probably over a long period of time, and so it takes on importance because mathematicians— human beings— have deemed it important. Now what all mathematicians do is solve problems, whether those originated by others or those they have themselves originated. Significant growth takes place in mathematics when the problem solved is deemed important. A teacher must experience mathematics as a growing theory in order to be able to elicit an appreciation of mathematics from the students. Perusing this dissertation would allow a student (particularly of mathematics education) to select those original works of greatest interest to him or her. It is in studying these original works with no intention to mastery of the material that a teacher of mathematics can experience the flavor of the subject. After all, the study of mathematics is an intellectual pursuit and hence our wishes to come into some contact with its greatness which is clearly not in textbooks. It is in following the gradual development of an idea, such as faisceau, as it penetrated different branches of mathematics, that a prospective teacher gains an appreciation of the chief ideas that moved about. It seems that the most efficient personal approach to the discipline is along historically dynamic lines. In teaching structured mathematics the teacher is apt to put an undue emphasis on proofs. However looking at the history of mathematics.

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particularly here in sheaf theory the papers of Oka and Cartan, we can see that the main contribution of proofs to mathematical progress in the development of sheaf theory is that the proofs actually caused conceptual change to occur. They are agents of change not the completion of an idea. Furthermore, we can see how one theory feeds off another, as in the development of the Cousin problem out of the work of Weierstrass, Mittag-Leff1er and Poincaré. In conclusion, the educational implications of my dissertation are that in educating a student in mathematics a teacher can use a case study such as this to determine what it is they are really teaching: Is it a fixed body of knowledge or is it dynamic? Is it discovered or invented? Furthermore, a teacher can determine what is meant by growth in mathematics through a determination of what is really a theory and how a theory takes place. The teacher will then see the enormous human interchange and effect culture itself has on the growth of mathematics.

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