Asia Pacific Mathematics Newsletter 1 GrothendieckGrothendieck and and Algebraic Algebraic Geometry Geometry

LucLuc Illusie Illusie and and Michel Michel Raynaud Raynaud

Grothendieck’s thesis and subsequent publica- it was Grothendieck who showed all its wealth tions in the early 1950’s dealt with functional and flexibility. Starting with a category , to each C analysis. This was remarkable work, which is object X of one can associate a contravariant func- C attracting new attention today.a Still, his most tor on with values in the category of sets, h : C X important contributions are in algebraic geometry, Sets, sending the object T to Hom (T, X). By a C→ C a field which occupied him entirely from the late classical lemma of Yoneda, the functor X h is → X 1950’s on, in particular during the whole time he fully faithful. To preserve the geometric language, was a professor at the Institut des Hautes Etudes´ Grothendieck called hX(T) the set of points of X Scientifiques (IHES)´ (1959–1970). with values in T. Thus, an object X is known when Algebraic geometry studies objects defined by we know its points with values in every object T. polynomial equationsb and interprets them in a Grothendieck applied this to algebraic geometry. geometric language. A major problem faced by This was revolutionary as, until then, only field algebraic geometers was to define a good frame- valued points had been considered. work and develop local to global techniques. As an example, suppose we have a system of In the early 1950’s complex analytic geometry equations showed the way with the use of sheaf theory. f1(x1, ..., xn) = = fN(x1, ..., xn) = 0, (1) Thus, a complex analytic space is a ringed space, ··· with its underlying space and sheaf of holomor- where the fi’s are polynomials with coefficients in Z. Let be the opposite category of the category phic functions (Oka, H Cartan). Coherent sheaves A of modules over this sheaf of rings play an impor- of rings, and let F be the (contravariant) functor tant role.c In 1954 Serre transposed this viewpoint sending a ring A to the set F(A) of solutions (x ), x A, of (1). This functor is nothing but the to algebraic geometry for varieties defined over i i ∈ functor h for X the object of corresponding to an algebraically closed field. He employed the X A Zariski topology, a topology with few open sub- the quotient of Z[x1, ..., xn] by the ideal generated sets, whose definition is entirely algebraic (with by the fi’s: the functor F is represented by X, an no topology on the base field), but which is well affine scheme. Points of X with values in C are adapted, for example, to the description of a pro- points of a complex algebraic variety — that one jective space as a union of affine spaces, and gives can possibly study by analytic methods — while rise to a cohomology theory which enabled him, points with values in Z, Q, or in a finite field for example, to compare certain algebraic and are solutions of a diophantine problem. Thus the analytic invariants of complex projective varieties. functor F relates arithmetic and geometry. Inspired by this, Grothendieck introduced If the fi’s have coefficients in a ring B instead of Z, the analogous functor F on the category schemes as ringed spaces obtained by gluing (for A the Zariski topology) spectra of general com- opposite to that of B-algebras, sending a B-algebra mutative rings. Furthermore, he described these A to the set F(A) of solutions of (1) with values objects from a functorial viewpoint. The language in A, is similarly represented by the spectrum of categories already existed, having appeared in X of a B-algebra (quotient of B[x1, ..., xn] by the the framework of homological algebra, following ideal generated by the fi’s), a scheme over the the publication of Cartan–Eilenberg’s book (Ho- spectrum of B. In this way a relative viewpoint mological Algebra, Princeton Univ. Press, 1956). But appears, for which the language of schemes is perfectly suited. The essential tool is base change, aSee [G Pisier, Grothendieck’s Theorem, Past and Present, Bull. Amer. Math. Soc. (N.S.) 49(2) (2012) 237–323]. a generalisation of the notion of extension of bHowever, we seldom saw Grothendieck write an explicit scalars: given a scheme X over S and a base equation on the blackboard; he did it only for basic, crucial change morphism S S, we get a new scheme cases. → c Cf. Cartan’s famous theorems A and B. X over S, namely, the fiber product of X and S

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over S. In particular, X defines a family of schemes covering, satisfying a small number of properties,

Xs parameterised by the points s of S. The above similar to those satisfied by open coverings in functor F then becomes the functor sending a topological spaces. The conceptual jump is that S-scheme T to the set of S-morphisms from T to the arrows U U are not necessarily inclusions.f i → X. A number of useful properties of X/S (such Grothendieck developed the corresponding no- as smoothness or properness) can nicely be read tions of sheaf and cohomology. The basic example g on the functor hX. The two above mentioned is the ´etaletopology. A seminar run by M Artin properties are stable under base change, as is at Harvard in the spring of 1962 started its sys- flatness, a property that plays a central role in tematic study. Given a scheme X, the category to algebraic geometry, as Grothendieck showed. In be considered is that of etale´ maps U X, and → 1968, thanks to M Artin’s approximation theorem, covering families are families (U U) such that i → it became possible to characterise functors that U is the union of the images of the Uis. The defi- are representable by algebraic spaces (objects very nition of an etale´ morphism of schemes is purely close to schemes) by a list of properties of the algebraic, but one should keep in mind that if X is functor, each of them often being relatively easy a complex algebraic variety, a morphism Y X → to check. But already in 1960, using only the is etale´ if and only if the morphism Yan Xan → notion of flatness, Grothendieck had constructed, between the associated analytic spaces is a local in a very natural way, Hilbert and Picard schemes isomorphism. A finite Galois extension is another as representing certain functors, at once super- typical example of an etale´ morphism. seding — by far — all that had been written on For torsion coefficients, such as Z/nZ, one the subject before. obtains a good cohomology theory Hi(X, Z/nZ), Nilpotent elementsd in the local rings of at least for n prime to the residue characteristics schemes appear naturally (for example in fiber of the local rings of X. Taking integers n of the products), and they play a key role in questions form r for a fixed prime number , and pass- of infinitesimal deformations. Using them system- ing to the limit, one obtains cohomologies with r atically, Grothendieck constructed a very general values in Z = lim Z/ Z, and its fraction field differential calculus on schemes, encompassing Q. If X is a complex←−− algebraic variety, one has arithmetic and geometry. comparison isomorphisms (due to M Artin) be- In 1949 Weil formulated his conjectures on tween the etale´ cohomology groups Hi(X, Z/rZ) varieties over finite fields. They suggested that and the Betti cohomology groups Hi(Xan, Z/rZ),h it would be desirable to have at one’s disposal thus providing a purely algebraic interpretation of a cohomology with discrete coefficients satisfying the latter. Now, if X is an algebraic variety over an analogue of the Lefschetz fixed point formula. an arbitrary field k (but of characteristic ) (a In classical algebraic topology, cohomology with k-scheme of finite type in Grothendieck’s lan-

discrete coefficients, such as Z, is reached by cut- guage), k an algebraic closure of k, and Xk de- ting a complicated object into elementary pieces, duced from X by extension of scalars, the groups i such as simplices, and studying how they overlap. H (Xk, Q) are finite dimensional Q-vector spaces, In algebraic geometry, the Zariski topology is too and they are equipped with a continuous action of coarse to allow such a process. To bypass this the Galois group Gal(k/k). It is especially through obstacle, Grothendieck created a conceptual rev- these representations that algebraic geometry in-

olution in topology by presenting new notions of terests arithmeticians. When k is a finite field Fq, in gluing (a general theory of descent,e conceived al- which case Gal(k/k) is generated by the Frobenius ready in 1959), giving rise to new spaces: sites and substitution a aq, the Weil conjectures, which → topoi, defined by what we now call Grothendieck are now proven, give a lot of information about topologies. A Grothendieck topology on a category these representations. Etale´ cohomology enabled is the datum of a particular class of morphisms Grothendieck to prove the first three of these

and families of morphisms (Ui U)i I, called → ∈ fMore precisely, monomorphisms, in categorical language. dAn element x of a ring is called nilpotent if there exists an gThe choice of the word ´etale is due to Grothendieck. integer n 1 such that xn = 0. hBut not between Hi(X, Z) and Hi(Xan, Z): by passing to e ≥ i The word “descent” had been introduced by Weil in the case the limit one gets an isomorphism between H (X, Z) and i an of Galois extensions. H (X , Z) Z. ⊗

2 January 2015, Volume 5 No 1 Asia Pacific Mathematics Newsletter3 conjectures in 1966.i The last and most difficult Galois representations, are still largely conjectural. one (the Riemann hypothesis for varieties over finite In this field, the progress made since 1970 owes fields) was established by Deligne in 1973. much to the theory of automorphic forms (the When Grothendieck and his collaborators Langlands programme), a field that Grothendieck (Artin, Verdier) began to study etale´ cohomology, never considered. the case of curves and constant coefficients was In the mid 1960’s Grothendieck dreamed of known: the interesting group is H1, which is es- a universal cohomology for algebraic varieties, sentially controlled by the Jacobian of the curve. It without particular coefficients, having realisa- was a different story in higher dimension, already tions, by appropriate functors, in the cohomolo- for a surface, and a priori it was unclear how to gies mentioned above: the theory of motives. He attack, for example, the question of the finiteness gave a construction, from algebraic varieties and of these cohomology groups (for a variety over algebraic correspondences between them, relying an algebraically closed field). But Grothendieck on a number of conjectures that he called stan- showed that an apparently much more difficult dard. Except for one of them,l they are still open. problem, namely a relative variant of the question, Nevertheless, the dream was a fruitful source of for a morphism f : X Y, could be solved simply, inspiration, as can be seen from Deligne’s theory → by d´evissage and reduction to the case of a family of absolute Hodge cycles, and the construction by of curves.j This method, which had already made Voevodsky of a triangulated category of mixed Grothendieck famous with his proof, in 1957, of motives. This construction enabled him to prove the Grothendieck–Riemann–Roch formula (although a conjecture of Bloch–Kato on Milnor K-groups, the devissage,´ in this case, was of a different and paved the way to the proof, by Brown, of the nature), suggested a new way of thinking, and Deligne–Hoffman conjecture on values of multi- inspired generations of geometers. zeta functions. In 1967 Grothendieck defined and studied a The above is far from giving a full ac- more sophisticated, second type of topology, the count of Grothendieck’s contributions to alge- crystalline topology, whose corresponding coho- braic geometry. We did not discuss Riemann– mology theory generalises de Rham cohomology, Roch and K-theory groups, stacks and gerbes,m enabling one to analyse differential properties of group schemes (SGA 3), derived categories and varieties over fields of characteristic p > 0 or the formalism of six operations,n the tannakian p-adic fields. The foundations were written up by viewpoint, unifying Galois groups and Poincare´ Berthelot in his thesis. Work of Serre, Tate, and groups, or anabelian geometry, which he developed Grothendieck on p-divisible groups, and problems in the late 1970’s. concerning their relations with Dieudonne´ theory All major advances in arithmetic geometry and crystalline cohomology launched a whole during the past forty years (proof of the Rie- new line of research, which remains very active mann hypothesis over finite fields (Deligne), of today. Comparison theorems (solving conjectures the Mordell conjecture (Faltings), of the Shimura– made by Fontainek) establish bridges between Taniyama–Weil conjecture (Taylor–Wiles), works etale´ cohomology with values in Qp of varieties of Drinfeld, L Lafforgue, Ngo)ˆ rely on the founda- over p-adic fields (with the Galois action) on tions constructed by Grothendieck in the 1960’s. the one hand, and their de Rham cohomology He was a visionary and a builder. He thought (with certain extra structures) on the other hand, that mathematics, properly understood, should thus providing a good understanding of these arise from “natural” constructions. He gave many p-adic representations. However, over global examples where obstacles disappeared, as if fields, such as number fields, the expected prop- l erties of etale´ cohomology, hence of the associated The hard Lefschetz conjecture, proved by Deligne in 1974. mThese objects had been introduced by Grothendieck to provide an adequate framework for non abelian coho- iThe first one (rationality of the zeta function) had already been mology, developed by J Giraud (Cohomologie non ab´elienne, proved by Dwork in 1960, by methods of p-adic analysis. Die Grundlehren der mathematischen Wissenschaften 179, jAt least for the similar problem concerning cohomology with Springer-Verlag, 1971). Endowed with suitable algebraic struc- proper supports: the case of cohomology with arbitrary supports tures (Deligne–Mumford, Artin), stacks have become efficient was treated only later by Deligne using other devissages.´ tools in a lot of problems in geometry and representation k The so-called Ccris, Cst, and CdR conjectures, first proved in theory. full generality by Tsuji in 1997, and to which many authors nCurrently used today in the theory of linear partial differen- contributed. tial equations.

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Alexander Grothendieck at the IHES´ in 1960’s during his famous Seminar of Algebraic Geometry. [Photo courtesy IHES]´

by magic, because of his introduction of the right SGA: S´eminaire de G´eom´etrieAlg´ebriquedu Bois- concept at the right place. If during the last Marie, SGA 1, 3, 4, 5, 6, 7, Lecture Notes in decades of his life he chose to live in extreme Math. 151, 152, 153, 224, 225, 269, 270, 288, 305, isolation, we must remember that, on the contrary, 340, 589, Springer-Verlag; SGA 2, North Holland, between 1957 and 1970, he devoted enormous 1968. energy to explaining and popularising, quite suc- cessfully, his point of view. Acknowledgements

**************** We thank Jean-Benoˆıt Bost, Pierre Deligne, and Grothendieck’s three major works in algebraic Jean-Pierre Serre for their remarks on a prelimi- geometry are: nary French version of this text. The first author thanks Robin Hartshorne, Nicholas Katz, William ´ ´ EGA: El´ements de g´eom´etrie alg´ebrique,redig´ es´ Messing, and Arthur Ogus for their help with its avec la collaboration de J. Dieudonne,´ Pub. Math. translation into English. IHES´ 4, 8, 11, 17, 20, 24, 28 et 32. FGA: Fondements de la g´eom´etriealg´ebrique, Ex- traits du Seminaire´ Bourbaki, 1957–1962, Paris, Secretariat´ mathematique,´ 1962.

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Luc Illusie Université de Paris-Sud, France [email protected]

Luc Illusie, born 1940, is an honorary professor of mathematics at the Université de Paris-Sud. He was a student of Grothendieck in the 1960’s. His main contributions are on the theory of the cotangent complex and deforma- tions, crystalline cohomology and the de Rham-Witt complex, Hodge theory, and logarithmic geometry.

Michel Raynaud Université de Paris-Sud, France [email protected]

Michel Raynaud, born 1938, is an hononary professor of mathematics at the Université de Paris-Sud. He was a student of Grothendieck in the 1960’s. His main poles of interest are group schemes, Néron models and rigid geometry. He proved Abhyankar’s conjecture on the fundamental group of the affine line on an algebraically closed field of positive characteristic.

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