ATELIER INTERNATIONAL SUR

LA THORIE ALGBRIQUE et ANALYTIQUE DES RSIDUS

ET SES APPLICATIONS

PARIS IHP du au MAI

La thorie des rsidus est un domaine qui connat une rcente et intressante activit

Plusieurs group es de travail se sont p enchs sur ce thme durant cette anne Cet atelier est

donc lo ccasion dune prsentation confrontation et synthse des ides sur le sujet aussi bien

dun p oint de vue thorique que pratique

Quelques p oints dintrt

Unication des direntes appro ches algbriques sur les rsidus

Connexion entre les appro ches analytiques et algbriques

Complexes rsiduels et leur constructions

Thormes de BrianonSkoda

Calculs eectifs de sommes de rsidus

Applications des rsidus la thorie de llimination aux formules de reprsentations

aux rsolutions de systmes p olynomiaux

Orateurs invits

Reinhold HBL Craig HUNEKE Ernst KUNZ Joseph LIPMAN

Organisateurs

Marc CHARDIN Centre de Mathmatiques

Ecole p olytechnique F Palaiseau France

Mohamed ELKADI Dept de Mathmatiques

Univ de Nice SophiaAntipolis Parc Valrose Nice France

Bernard MOURRAIN INRIA Pro jet SAFIR

route des Lucioles F Valbonne France

Les organisateurs ont t aids dans leur travail par les participants de deux sminaires

informels sur ce sujet lun Nice Formes Formules et lautre Paris et aussi par

David EISENBUD et JeanPierre JOUANOLOU

Cet atelier a reu le soutien du Lab oratoire GAGE cole p olytechnique Palaiseau et

du pro jet SAFIR Nice SophiaAntipolis travers le programme HCM SAC

INTERNATIONAL WORKSHOP ON

ALGEBRAIC AND ANALYTIC THEORY OF RESIDUES

AND ITS APPLICATIONS

PARIS IHP MAY

Residues theory is a eld which has known recent and interesting activities Several

working groups have fo cus on this theme during the last years The workshop is the o ccasion of

a presentation confrontation and synthesis of ideas on this sub jects as well from a theoretical

p oint of view as for its applications

Some p oints of interest

Unication of the dierent algebraic approaches on residues

Residual complexes and their construction

Connection b etween the analytic and algebraic approaches

BrianonSkoda theorems

Eective computation of sums of residues

Application of Residue Theory to elimination theory representation formulas and p oly

nomial systems solving

Invited sp eakers

Reinhold HBL Craig HUNEKE Ernst KUNZ Joseph LIPMAN

Organizers

Marc CHARDIN Centre de Mathmatiques

Ecole p olytechnique F Palaiseau France

Mohamed ELKADI Dept de Mathmatiques

Univ de Nice SophiaAntipolis Parc Valrose Nice France

Bernard MOURRAIN INRIA Pro jet SAFIR

route des Lucioles F Valbonne France

The organizers were and are help ed in their task by the participants of two informal

seminars on this sub ject one in Nice and one in Paris and by and Jean

Pierre JOUANOLOU

This workshop is supp orted by the Lab oratoire GAGE Ecole p olytechnique Palaiseau

the pro ject SAFIR Nice SophiaAntipolis via the HCM programm SAC

PROGRAMME PROGRAM

JEUDI THURSDAY

MATIN MORNING

h h Amnon YEKUTIELI

Adeles and the De RhamResidue Complex

h h Uwe STORCH

Resultants for homogeneous regular sequences

APRESMIDI AFTERNOON

h h Carlos BERENSTEIN

On the residue formula of Jacobi and its applications

h h Alain YGER

Integral representation formulas and multidimensional residues

h h Alicia DICKENSTEIN

Residues in Toric Varieties

h h MarieFrancoise ROY

Quadratic forms and Bezoutians

VENDREDI FRIDAY

MATIN MORNING

h h Craig HUNEKE

Tight closure and theorems of BrianconSkoda type

h h Joseph LIPMAN

Formal Duality Fundamental Class and the Residue Theorem

APRESMIDI AFTERNOON

h h IChiau HUANG

An explicit construction of residual complexes

h h Leovigildo ALONSOTARRIO Formal Completion and Duality

h h Mikael PASSARE

Courants rsiduels et faisceaux dualisants

h h Teresa KRICK

Elimination by arithmetic circuits The duality to ol

SAMEDI SATURDAY

MATIN MORNING

h h Ernst KUNZ

Generalization of a theorem of Chasles

h h Reinhold HBL

Generalization of a theorem of Waring

APRESMIDI AFTERNOON

h h Ab dellah AL AMRANI

Fibrs pro jectifs tordus et classes de Chern

h h Salomon OFMAN

Application de lalgorithme de GelfandLerayShilov aux courantsrsidus

h h Djilali BOUDIAF Interprtation des rsidus comp oss laide des courants

Interprtation des rsidus comp oss laide des courants

rsiduels

Djilali Boudiaf

Rsum

Soient X une varit analytique complexe de n et F fY Y g p n

p

e

une famille ordonne dhypersurfaces complexes de X de runion Y et dintersection Y telle

que chaque sousfamille F fY Y g i p soit en p osition dintersection complte

i i

e

cestdire dim F n i o F Y Y On dsigne par E Y resp D

C i i i

X X

D le complexe des faisceaux des germes des formes semimromorphes sur X p les

Y

e

sur Y resp des courants sur X supp orts sur Y et par X D lensemble des sections

Y

X

globales de D supp ort dans Y Suivant HerreraLieb erman p our le cas p on

X

montre que le diagramme de cohomologie suivant est commutatif

e

I Y

q q

e e

H X E Y H X nY C

X

p p

r es R

F F

y y

q p

q p

H X C H X D

Y

X

Y

Y

e

o I Y est lisomorphisme de Grothendieck est un isomorphisme comp os de

Y

lisomorphisme de Poly

q p

H X D H X D H Y C

Y nq p Y X nq p

X

q p

X C et de H Y C H

nq p

Y

inverse de lisomorphisme de dualit de Poincar X obtenu par cappro duit par la classe

p

fondamentale de X est lhomomorphisme rsiduel dni dans partir de lop rateur R

F

p q p q p

e

est lhomomorphisme rsidu et enn r es Y D E de ColeHerrera R

F Y X F

compos dni par Poly partir de travaux de Sorani Cest la gnralisation de

lhomomorphisme de Leray et Norguet en eet si les hypersurfaces Y sont lisses et

j

p

e

en position gnrale alors lhomomorphisme r es asso cie toute classe de cohomologie de

F

p

e

e

X n Y la classe de cohomologie prsidu compos de LerayNorguet note r es Si est

F

p

e

une forme semimromorphe p les simples sur Y alors la forme prsidu compos r es

F

est dnie et on a la relation

p p

p

I Y r es i R

F F

o le premier membre est le courant rsiduel asso ci et I Y est le courant dintgration

de Lelong sur Y

Rfrences

Boudiaf D Sur les courants rsiduels Thse Universit Paris VI

Coleff N et Herrera M Les courants rsiduels asso cis une forme mromorphe

Lecture Notes in Math Springer Verlag

Coleff N Herrera M et Lieberman D Algebraic cycles as residues of meromorphic

forms Math Ann

Grothendieck A On the de Rham cohomology of algebraic varieties Inst Hautes

tudes Sci Publ Math

Herrera M et Lieberman D Residues and principal values on complex spaces Math

Ann

Leray J Le calcul direntiel et intgral sur une varit analytique complexe Problme

de Cauchy I I I Bull So c Math France

Norguet F Introduction la thorie cohomologique des rsidus Sm P Lelong Anal

yse Anne Lecture Notes in Math SpringerVerlag

Poly JB Formules des rsidus et intersection des chanes sousanalytiques Thses

Universit de Poitiers

Poly JB Morphismes de MayerVietoris et rsidus comp oss paratre

Sorani G Sui residui delle forme diferenziali di una variet analitica complessa Rend

Mat Appl V Ser

Authors Address

Djilali Boudiaf Universit de Poitiers Mathmatiques Avenue du

Recteur Pineau POITIERS

Residues in Toric Varieties

Eduardo Cattani David Cox and Alicia Dickenstein

Toric residues provide a to ol for the study of certain homogeneous ideals of the homogeneous

co ordinate of a toric variety such as those app earing in the description of the Ho dge

structure of their hypersurfaces BC They were introduced in C where some of their

prop erties were describ ed in the sp ecial case when all of the divisors involved were linearly

equivalent The main results of the present work are an extension of the Isomorphism

Theorem of C to the case of nonequivalent ample divisors a global transformation law for

toric residues and a theorem expressing the toric residue as a sum of lo cal Grothendieck

residues

Assume that X is a complete simplicial toric variety of dimension n S denotes the

homogeneous co ordinate ring of X which is the p olynomial ring S Cx x Here

nr

the variables x corresp ond to the generators of the dimensional cones in the fan which

i

determines X and hence to torusinvariant irreducible divisors D of X S is graded by

i

P

d

nr nr

i

declaring that the monomial x has degree d D in the Chow group A X

i i n

i i i

C

P

nr

We let degx A X denote the anticanonical class on X Then given

i n

i

for i n we dene their critical degree to b e homogeneous p olynomials F S

i

i

P

n

A X Each H S determines a meromorphic nform H

i n F

i

H

on X BC

F F

n

If in addition the F dont vanish simultaneously on X then relative to the op en cover

i

n n

U fx X F x g of X this gives a ech class H H X The toric

i i F

X

residue

Res S hF F i C

F n

is given by the formula

Res H Tr H

F F

n n

where Tr H X C is the trace map

X

Our rst main result is the following Global Transformation Law

for i n Suppose and G S Theorem Let F S

i i

i i

n

X

G A F

j ij i

i

where A is homogeneous of degree and assume the G dont vanish simultaneously

ij j i i

on X Let be the critical degree for F F Then for each H S H detA is of

n ij

the critical degree for G G and

n

Res H Res H detA

F G ij

The pro of uses a ech co chain argument One application of this transformation law is that in

certain cases we can describ e explicit elements of S with nonzero residue For this purp ose

assume X is complete and its fan contains a ndimensional simplicial cone Then denote

the variables of the co ordinate ring as x x z z where x x corresp ond to

n r n

the dimensional cones of Also supp ose that are Qample classes which means

n

that some multiple is Cartier and ample In this situation each F can b e written in the

j

form

n

X

A x F A z z

ij i j j r

i

Then the n n determin ant detA is in S and has the following imp ortant

ij

prop erty

Theorem Assume X is complete and is simplicial and ndimensional Suppose

that F S for i n where is Qample and the F dont vanish simultaneously

i i i

i

on X Then

Res

F

We also prove the following Residue Isomorphism Theorem

Theorem Let X b e complete and simplicial and assume that F S for i n

i

i

where is ample and the F dont vanish simultaneously on X Then

i i

i The toric residue map Res S hF F i C is an isomorphism

F n

ii For each variable x i n r we have x S hF F i

i i n

In the case when all the are equal to a xed ample divisor this theorem follows from

i

the fact that F F are a regular sequence in the CohenMacaulay ring S S

n k k

C In the general case the pro of relies on the use of the Cayley trick and results

of Batyrev and Cox BC concerning the cohomology of pro jective hypersurfaces in toric

varieties to show that

dim S hF F i

n

when X is simplicial and the divisors F are ample with empty intersection Then

i

the rst and main part of the Residue Isomorphism Theorem follows immediately from

Theorem and the second part is a consequence of the rst using Theorem and Cramers

Rule

As a corollary of Theorems and we get a simple algorithm for computing toric residues

in terms of normal forms

We also show that for simplicial toric varieties the toric residue may b e computed as a

sum of lo cal Grothendieck residues The toric setting is not essential here and in fact it is

convenient to work with the more general notion of a V manifold or orbifold see Sa The

pro of of the following lo calglobal theorem is based on the theory of residual currents CH

Theorem Let X be a complete simplicial toric variety of dimension n and let

F F be homogeneous which dont vanish simultaneously on X If H S

n

where is the critical degree and D fx X F x i k g is nite then the toric

i

k

residue is given by

X

H

k

Res Res H

k x F

F F

n

xD

^

k

Note that the niteness condition holds automatically whenever the divisor fF g

k

is Qample Under appropriate conditions Theorem gives a framework for the study of

sums of lo cal residues b oth in the ane and toric cases as a global residue dened in

a suitable toric compactication It is p ossible for example to interpret in this light the

results of CDS which corresp ond to the case when the toric variety under consideration is

a weighted pro jective space

Finally we show that in the equal degree case the toric residue equals a single lo cal

residue at the origin of the ane cone of X This generalizes the observation in PS that

n n

toric residues on P can b e written as a residue at the origin in C

References

BC V Batyrev and D Cox On the Hodge structure of projective hypersurfaces in toric

varieties Duke J Math

CH N Cole and M Herrera Les Courants Residuels Asso cis une Forme Meromorphe

Lecture Notes in Math SpringerVerlag Berlin Heidelb erg New York

C D Cox The homogeneous coordinate ring of a toric variety J Algebraic Geom

C D Cox Toric residues Ark Mat to app ear alggeom

CDS E Cattani A Dickenstein and B Sturmfels Computing multidimensional residues

Pro c MEGA Birkhuser to app ear

PS C Peters and J Steenbrink Innitesimal variation of Hodge structure and the generic

Torelli theorem for projective hypersurfaces in Classication of Algebraic and Analytic

Manifolds K Ueno editor Progress in Math Birkhuser Boston Basel Berlin

Sa I Satake On a generalization of the notion of manifolds Pro c Nat Acad Sci USA

Authors Address

E Cattani Department of U of Massachusetts

Amherst MA USA cattanimathumassedu

D Cox Department of Mathematics Amherst College Amherst MA

USA daccsamherstedu

A Dickenstein Departamento de Matemtica FCE y N Universidad

de Buenos Aires Ciudad Universitaria Pabelln I Buenos Aires

Argentina alidickmatedmubaar

Generalization of a Theorem of Waring

Reinhold Hbl

In this talk we will present results obtained jointly with E Kunz

th

In the century many b eautiful theorems have b een proved ab out the intersection of

two plane curves Many of these results can b e obtained by rather explicit calculations using

the equations of the curves In some cases however a pro of can also b e given using residues

of dierential forms and the residue theorem on curves as already introduced by Serre and

these pro ofs very often generalize to curves in higherdimesional spaces As an example of

this metho d we will present a generalization of a theorem of Waring cf Co p

n

Let K b e a eld of characteristic Recall that for a zerodimensional subscheme S A

K

i i

of degree N with supp ort jS j fP P g P a a the centroid S of S is

r i

n

dened to b e the vector sum

r

X

i i

S a a l g O

SP

n i

N

i

n n

in A For a reduced and irreducible curve A an asymptote to is dened to b e

K K

a tangent A to an analytic branch R of the pro jective closure at a p oint P at

P i i

innity The asymptote cycle A of is dened to b e the formal sum over all asymptotes of

n

counted with multiplicities For an arbitrary curve C A with reduced and irreducible

K

comp onents C C the asymptote cycle AC of C is given by

t

t

X

l g O AC AC

CC i

i

i

In this situation we have the following higherdimensional analogue of Warings theorem

n

Theorem Let C A be a curve such that none of its asymptotes is contained in the

K

n

be a hyperplane having no common points with C at hyperplane at innitiy and let H A

K

innity Then

C H AC H

This theorem is proved by expressing centroids in terms of residues and by showing that

we get the same expression if we replace C by its asymptote cycle

As an immediate consequence of Warings theorem we conclude the following result which

in the case of plane curves already was known to Newton

Theorem For a curve C as above let H run through a family of parallel hyperplanes

which are not asymptotic to C Then al l centroids C H are on a line

n

Such a line is called a diameter of C A closed p oint M A is called a center of C

K

if it lies on all diameters Warings theorem also allows a classication of those curves that

have centers in the sense dened ab ove

References

Co Coolidge J L A treatise on algebraic plane curves Dover Publications New York

HK Hbl R and E Kunz On the intersection of algebraic curves and hypersurfaces

Preprint

H Hbl R Residues of regular and meromorphic dierential forms Math Ann

KW Kunz E and R Waldi Asymptotes and centers of ane algebraic curves Preprint

Se Serre JP Groupes Algbriques et Corps de Classes Hermann Paris

Authors Address

Reinhold Hbl Universitat Regensburg NWF I Mathematik D

Regensburg Germany HUEBLvaxrzuniregensburgdde

Tight closure and theorems of BrianonSkoda type

Craig Huneke

A theorem of Brianon and Skoda later generalized by Lipman and Sathaye and still later

by Lipman and Teissier states that if R is a lo cal ring which is a rational singularity of

d

dimension d then the integral closure of the dth p ower of an arbitrary ideal I denoted I

is contained in the ideal I Recall the denition of the integral closure

Denition Let I b e an ideal of a No etherian ring R An element x is in the integral closure

k k i

of I I if x satises an equation of the form x a x a where a I

k i

A more general version of the theorem of Lipman and Teissier states

Theorem Let R m be a ddimensional local ring which is a rational singularity Let

I be any ideal of R Then for any w

w

dw

I I

A tight closure version was given by Ho chster and Huneke in The denition of tight

closure in p ositive characteristic is

Denition Let I b e an ideal of R and x R An element x is said to b e in the tight

closure of I if there exists an element c not in any minimal prime of R such that for all large

e q q q

q p cx I where I is the ideal generated by the q th p owers of all elements of I We

denote the tight closure of I by I

A denition can also b e given for all No etherian rings containing a eld by using reduction

to characteristic p

The theorem of Ho chster and Huneke states

Theorem Let R be a ddimensional Noetherian ring containing a eld Let I be any

ideal generated by n elements For al l w

w

nw

I I

In particular for al l ideals I of R

w

dw

I I

The test ideal of the ring R is the intersection of I I where the intersection runs over all

ideals I of R We denote the test ideal by R If R is excellent reduced and has an isolated

singularity the test ideal is either the whole ring or is mprimary where m is the maximal

ideal of R In general if R is regular c has a p ower which is a test element This leads to the

c

following theorem for an excellent reduced lo cal ring R m of dimensional d with isolated

t

singularity if m is contained in the test ideal then for every ideal I the integral closure of

dt

I is contained in I

t

Of particular interest then is the p ower t of the maximal ideal m such that m R

If R is graded over an algebraically closed eld of characteristic a conjecture is that one

can take t a where a is the ainvariant of R namely the highest degree of a nonzero

element in the top lo cal cohomology of R This conjecture has b een veried by Karen Smith

and myself for graded rings which are complete intersections with isolated singularity It

turns out that this conjecture is equivalent in the Gorenstein case to another conjecture which

I will discuss which is in turn a generalization of the Ko daira vanishing theorem

The talk will fo cus on the relationship b etween the BrianonSkoda type theorems the

test ideal and theory of tight closure and the Ko daira vanishing theorem

Reference

IM Ab erbach and C Huneke An improved BrianonSkoda theorem with applications to

the CohenMacaulayness of Rees rings Math Ann

IM Ab erbach and C Huneke A theorem of BrianonSkoda type for regular local rings

containing a eld to app ear Pro c Amer Math So c

IM Ab erbach C Huneke and NV Trung Reduction numbers BrianonSkoda theorem

and the depth of Rees rings to app ear Comp ositio Math

J Brianon and H Skoda Sur la clture intgrale dun idal de germes de fonctions

n

holomorphes en un point de C C R Acad Sci Paris Sr A

M Ho chster and C Huneke Tight closure invariant theory and the BrianonSkoda

theorem J Amer Math So c

C Huneke Uniform bounds in Noetherian rings Invent Math

C Huneke and K Smith Kodaira Vanishing and Tight Closure In preparation

J Lipman Adjoints of ideals in regular local rings Math Res Letters

J Lipman and A Sathaye Jacobian ideals and a theorem of BrianonSkoda Michigan

Math J

J Lipman and B Teissier Pseudorational local rings and a theorem of BrianonSkoda

about integral closure of ideals Michigan Math J

I Swanson Joint reductions tight closure and the BrianonSkoda theorem J Algebra

Authors Address

Craig Huneke Dept of Math Purdue University W Lafayette IN

USA chunekemathlsaumichedu

Elimination by arithmetic circuits The duality to ol

Teresa Krick

Le but de lexp os est de dcrire une mtho de informatique le circuit arithmtique qui

p ermet de fournir des calculs dvaluation p our les p olynmes apparaissant dans les formules

de reprsentation lors de la rsolution des systmes p olynomiaux

Plus prcisment on traitera le problme du Thorme des Zros Eectif et celui de

lappartenance et la reprsentation p our des intersections compltes dans lanneau de p olynmes

K X X

n

Soit I f f K X X un idal dont la varit des zros dans la clture

s n

algbrique de K est vide resp de dimension pure n s et dans ce cas soit f I

Posons d maxfdeg f deg f g

i

O n

Alors il existe un circuit arithmtique de taille sd et de profondeur O n log d qui

p ermet dvaluer des p olynmes a a K X X vriant

s n

a f a f resp f a f a f

s s s s

Ce rsultat p ermet de fournir dans le cas de lappartenance une intersection complte

O n

la b orne optimale d p our les degrs des p olynmes a a et de retrouver cette

s

mme b orne p our le Nullstellensatz due initialement Brownawell CanigliaGalligo

Heintz et Kollr

Si de plus les p olynmes ff f g sont co ecients entiers rationnels on retrouve

s

galement p our le Nullstellensatz les rsultats sur la hauteur des p olynmes ra

tionnels a a de BerensteinYger et on obtient simultanment Elkadi le

s

mme type de b ornes p our lappartenance une intersection complte si H ma jore

O (n)

d

tous les co ecients des p olynmes f et f en valeur absolue alors H ma jore

i

numrateurs et dnominateurs de tous les co ecients des g

i

Les calculs dvaluation avaient t appliqus dans le cadre de llimination ds par

Heintz Morgenstern Schnorr Sieveking et dautres Ils ont ensuite t repris par les deux

premiers Giusti et le group e Fitchas p our les questions cidessus

Dans notre contexte cest loutil de la dualit la trace et celui de lvitation des divisions

de Strassen qui se prtant bien une traduction circuit arithmtique entrane les rsultats

mentionns

On se prop ose aussi de discuter la p ossibilit dobtenir grce cette mme philosophie des

rsultats dun ordre compltement nouveau p our les deux problmes considrs on se place

maintenant dans le cadre dune suite rgulire f f et ventuellement f non diviseur

r r

de zro mo dulo f f Les entres f et f sont reprsentes de mme que le seront les

r i

sorties g par des circuits arithmtiques p ermettant de les valuer Les paramtres mesurant

i

lentre sont d n la taille et la profondeur du circuit p ermettant dvaluer lentre et les

degrs gomtriques anes des varits dnies par les p olynmes f f i r

i

Dans ces conditions les taille et profondeur du circuit p ermettant de fournir les calculs

dvaluation des p olynmes g sont maintenant p olynomiales en tous les paramtres considrs

i

Notons que ceci est imp ossible si lon donne lentre et la sortie par reprsentation dense

et aussi sembleraitil selon une b orne infrieure de HeintzMorgenstern si on utilise les

circuits arithmtiques mais on nintroduit pas le degr gomtrique en tant que paramtre

de mesure de lentre Une consquence de ce fait est que les degrs des p olynmes g est

i

p olynomial en d n et les degrs gomtriques anes des varits dnies par f f Ceci

i

est un travail en cours de Fitchas et Giusti

Des ma jorations p our les degrs de mme type p olynomiales en tous les paramtres

considrs p euvent galement tre obtenues p our les degrs des gnrateurs du radical dun

idal p olynomial dcrit par une suite rgulire sous certaines conditions comme le montre

un rsultat rcent dArmendariz et Solern

Elimination by arithmetic circuits The duality to ol

The aim of the talk is to describ e a computational metho d the arithmetic circuit or straight

line program which allows one to evaluate certain representations of p olynomials which

app ear when one solves p olynomial systems Roughly sp eaking a divisionfree nonscalar

straight line program is a divisionfree algorithm which computes the evaluation of a mul

n

tivariate p olynomial f K X X at any p oint in K introducing if necessary some

n

auxiliary xed constants of K The size of the algorithm is measured by taking K linear

op erations for free and its depth is the maximal number of intermediate p olynomials re

cursively linked see Well consider here the eective Nullstellensatz and the member

ship and representation problems within complete intersection ideals in the p olynomial ring

K X X where K is a eld of characteristic zero

n

Let I f f K X X b e an ideal which denes the empty variety over

s n

the algebraic closure of K resp of pure dimension ns and in this case let f I

Set d maxfdeg f g resp d maxfdeg f deg f g

i i

O n

Under these conditions there exists a straight line program of size sd and depth

O n log d which computes the evaluation of p olynomials a a K X X

s n

satisfying

a f a f r esp f a f a f

s s s s

O n

This result allows one to obtain the optimal b ound d for the degrees of the

p olynomials a a in the membership problem for complete intersections and to

s

recover this same b ound for the Nullstellensatz initially due to Brownawell Caniglia

GalligoHeintz and Kollr

Moreover when the p olynomials f f and f have integer co ecients one also

s

recovers for the Nullstellensatz the b ounds for the heights of the rational p olynomials

a a due to BerensteinYger and gets simultaneous to Elkadis result

s

the same type of b ound for the membership problem for a complete intersection

if H b ounds from ab ove the absolute value of all the co ecients of the p olynomials f

i

O (n)

d

and f then H b ounds the numerators and denominators of all the co ecients

of the a

i

These evaluation programs had b een applied in the elimination framework around

by Heintz Morgenstern Schnorr Sieveking and others They have b een reconsidered more

recently for the questions stated ab ove by the rst two researchers Giusti and the Fitchas

group see In our context it is the trace duality to ol and the avoidance

of divisions of Strassen which admitting a nice straight line program translation

pro duces the results mentioned here

Also we would like to discuss the p ossibility of applying the same philosophy and obtain

ing results of a completely new style for the same two questions Assume that f f form

r

a regular sequence r n and that f is not a zerodivisor mo dulo f f Moreover

r r

supp ose that the ideals f f are reduced for i r We consider the Nullstel

i

lensatz for f f or the membership of f in f f The inputs f f

r r r

and eventually f will b e given no longer by their dense representation but by straight line

programs The output will also b e a straight line program which computes a a The

r

parameters measuring the input are d n the size and the depth of the straight line program

pro ducing the input and the ane geometric degrees of the varieties dened by f f for

i

i r Under these conditions the size and depth of the straight line program pro ducing

the output a a is p olynomial in all the aforementioned parameters Note that this

r

is imp ossible if the input and output are given in the dense representation and also it seems

due to a lower b ound by HeintzMorgenstern if one considers straight line programs

but do es not introduce the geometric degrees as a parameter measuring the input This

is work in progress by FitchasGiusti A consequence of this fact is that the degrees of the

p olynomials a are p olynomial in d n and the degrees of the varieties dened by f f

i i

as it implicitly app ears in

Upp er b ounds of the same type p olynomial in all the aforementioned parameters can also

b e obtained for the degrees of the generators of the radical of a p olynomial ideal describ ed

by a regular sequence under certain conditions as shown in a recent result by Armendariz

Solern

References

IArmendariz PSolern On the computation of the radical of complete intersection

p olynomial ideals to app ear in Pro c AAECC Paris

CBerenstein AYger Eective Bezout identities in Qz z Acta Math

n

MElkadi Bornes p our le degr et les hauteurs dans le problme de division Michigan

Math J

NFitchas MGiusti FSmietanski Sur la complexit du thorme des zros Pro c nd

Int Conf on Approximation and Optimization La Habana to app ear

Jvon zur Gathen Parallel arithmetic computations a survey Pro c th Symp MFCS

Springer LN Comput Sci

MGiusti JHeintz La dtermination des p oints isols et de la dimension dune varit

algbrique p eut se faire en temps simplement p olynomial Pro c Int Meeting on Commutative

Algebra Cortona

MGiusti JHeintz JMorais LPardo When p olynomial equation systems can b e solved

fast to app ear in Pro c AAECC Paris

MGiusti JHeintz JSabia On the eciency of eective Nullstellenstze Computational

Complexity

JHeintz JMorgenstern On the intrinsic complexity of elimination theory J of Com

plexity

TKrick LPardo A computational metho d for diophantine approximation Pro c Eec

tive Metho ds in Santander to app ear

EKunz Khler Dierentials Advanced Lectures in Mathematics Vieweg Verlag

JSabia PSolern Bounds for traces in complete intersections and degrees in the Null

stellensatz AAECC Journal to app ear

VStrassen Vermeidung von Divisionen Crelle J Reine Angew Math

Authors Address

Teresa Krick Dpartement de Mathmatiques Facult des Sciences

de Limoges Av Albert Thomas F Limoges France Univ

Buenos Aires krickmariepolytechniquefr

Generalization of a theorem of Chasles

Ernst Kunz

Joint work with RWaldi

r

P

n i i n

P b e an eective cycle in A Let Z ie N P a a A

i i i i

n

K K

i

r

P

b e the degree of Z The vector sum closed p oints Assume Char K and let N

i

i

r

X

i i n

Z a a A

i

n K

N

i

is called the centroid of Z

b e a smo oth curve of degree d which has d distinct p oints at innity Let C A

K

Let L b e the linear system of lines parallel to a given line not asymptotic to C and let

Z P where P C are the p oints with tangents T C b elonging to L and

L P P P

C T C intersection multiplicity The theorem of Chasles states that the centroid

P P

Z is indep endent of L see T and C p It is called the tangential center of C

L

The theorem is a corollary of the residue theorem as stated in K In fact if C is the

zeroset of a p olynomial F K X X of degree d then Z turns out to b e the centroid

L

F F

of the critical scheme Sp ec K X X

X X

1 2

The theorem b e generalized to reduced curves in higher dimensional ane spaces and

systems L of parallel hyperplanes if the curve is not the union of pairwise disjoint lines The

system L denes a no etherian normalization K x K C and Z is the cycle on C dened

L

by the Dedekind dierent of K C K x Again Z is indep endent of L Our pro of is by

L

generic pro jection to the plane

If the curve is a union of pairwise disjoint lines there is obviously no tangential center

References

C JL Co olidge A Treatise on Algebraic Plane Curves Dover Publications New York

K E Kunz ber den ndimensionalen Residuensatz Jb er d DMV

KW E Kunz and R Waldi Generalization of a Theorem of Chasles Preprint

T A Terquem Dmonstration du thorme de MChasles sur les tangents parallles et

les plans tangents parallles point de moyenne distance

Nouvelles Ann Math IV Paris

Authors Address

Ernst KUNZ Universitt Regensburg NWF I Mathematik D

Regensburg KUNZvaxrzuniregensburgdde

Formal Duality Fundamental Class and the Residue Theorem

1

Joseph Lipman

Grothendieck Duality is based on the existence for certain prop er scheme maps f X Y

of a right adjoint f of the derived functor Rf or at least of its restriction to complexes

with quasicoherent homologyand the compatibility of f with at base change The

R

Rf f Grothendieck residue arose as a lo calized asp ect of the adjunction map

f

Hp Approaching lo cal residues via global duality while ultimately essen

tial to a full understanding is rather indirect and abstract crying out for downtoearth

commutativealgebra constructions Over the past twenty ve years several such treatments

of residues and their relationships have b een worked out in terms of dierential forms or

Ho chschild homology or top ological lo cal elds and asso ciated trace maps see eg HK

Y S and their references

In HK Hbl and Kunz consider certain equidimensional ring maps R S together with

an S ideal J such that SJ is nite over R Under their assumptions there is a nitely gener

ated mo dule consisting of meromorphic dierential dforms of SR dbre dimension

SR

the mo dule of regular dierential dforms implicit in Grothendiecks duality theory but rst

explicated by Kunz see KW At primes where SR is smo oth eg coincides with the

SR

d

relative holomorphic dierentials Hbl and Kunz dene concretely a residue symbol

SR

R

d

H R given roughly by brewise summation of and also a local integral

SR

J

SRJ

residues see HK HK gives the following version of Lo cal Duality where de

R

d

notes J adic completion The pair represents the functor Hom H E R

R

SR

J

S RJ S

of S mo dules E

They then globalize considering a nitetype map f X Y of eg reduced excellent

no etherian schemes with f generically smo oth and equidimensional of bredimension d

together with a closed subscheme Z X nite over Y whose inverse image over any ane

op en subset say U Sp ec R is contained in an ane op en subset of f U say V

Sp ec S The regular dierentials glue together into a coherent sheaf on X whose mo dule

f

R

d

R f O which of sections over any such V is There is then a unique map

f Y

SR

Z

f Z

R

where S and R are as just stated and J S over any ane U is the sheacation of

SRJ

d

is a dening ideal of Z V Here R f the relative lo cal cohomology supp orted in Z is

Z

a derived functor of f b eing the sheaed functor of sections with supp ort in Z

Z Z

These maps b ehave functorially with resp ect to inclusions Z Z

2

The basic lo calglobal relationship is summarized in the Residue Theorem which for

R

d

d

proper f asserts the existence of a canonical map R f O such that

f Y

f

R

d

d

i The corresp onding map via duality H f O is an isomorphism ie

f Y f

f

d

represents the functor HomR f E O of quasicoherent O mo dules E

Y X

R R

nat l

d

d d

with R f R f any Z as ab ove is ii The comp osition of

Z

f Z f

1

Partially supp orted by the National Security Agency

2

stated at several levels of generality in various places in the literature a recent oneinclud in g references

to othersb eing HS

Thus we have via dierentials and residues a canonical realization of and compatibility

b etween lo cal and global duality

In the talk we will outline new pro ofs of the two parts of the Residue Theorem

d

For ii going from concrete back to abstract we replace by H f O thereby b egging

f Y

R

by the homology of a more general but abstract formal integral the question of i and

f Z

ie a map of the form Rf f where f is now a prop er map of formal no etherian schemes

with resp ect to which a generalized version of Grothendieck Duality obtains

This Formal Duality theorem contains eg formal duality la Hartshorne Hp and

a related lo calglobal duality theorem of Lp which includes the classical lo cal duality

theorem Mo dulo i the central Theorem in Lp and the ab ove Lo cal Duality

are also sp ecial cases Formal Duality turns ii into a simple statement ab out functoriality

of the formal integral with resp ect to the canonical map from the formal completion of X

along Z to X itself

As for i one can formalize the ab ove Lo cal Duality to see that the concrete and abstract

integrals b oth provide representations of the same functor and so deduce the existence of a

d

local isomorphism H f O j see ab ove The remaining question is Does this

Y V

SR

local isomorphism comes from a global one

In HS the question is answered armatively by means of a fairly complex pasting

argument generalizing L where Y is the Sp ec of a p erfect eld reducing ultimately

d

via Zariskis Main Theorem and traces to the case X P which is treated in HK

Y

We will indicate a more direct approach The basic relation of holomorphic dierential

dforms to f is encapsulated in a canonical derived category map

d

C f O

f Y

f

the fundamental class of f For smo oth f C is a wellknown isomorphism Vp

f

Thm The characteristic case with singularities was studied by Angniol and El

Zein AE by means of a theorem of Bott on Grassmannians We will dene C quite gen

f

erally via simple derivedcategory formalism and state a trace prop erty relating C and

f g

C g X X a nite map which should provide an answer to the preceding question

f

References

AE B Angniol and F Elzein Complexe dualisant et applications la classe fondamentale

dun cycle Bul l Soc Math France Mmoire

H R Hartshorne Residues and Duality Lecture Notes in Math no SpringerVerlag

New York

H On the de Rham cohomology of algebraic varieties Publications Math IHES

HK R Hbl and E Kunz Integration of dierential forms on schemes J reine angew

Math

HK Regular dierential forms and duality for projective morphisms J reine angew

Math

HS R Hbl and P Sastry Regular dierential forms and relative duality American J

Math

KW E Kunz and R Waldi Regular Dierential Forms Contemporary Math v

American Math So ciety Providence

L J Lipman Desingularization of twodimensional schemes Annals of Math

L Dualizing sheaves Dierentials and Residues on Algebraic Varieties Astrisque

vol So c Math de France

S P Sastry Residues and duality on algebraic schemes Comp ositio Math to app ear

V JL Verdier Base change for twisted inverse image of coherent sheaves Algebraic

Geometry Bombay Oxford Univ Press London

Y A Yekutieli An Explicit Construction of the Grothendieck Residue Complex Astrisque

vol So c Math de France

Authors Address

Joseph Lipman Dept of Mathematics Purdue University W

Lafayette IN USA lipmanmathpurdueedu

ALGORITHME DE GELFANDLERAYSHILOV ET

COURANTSRSIDUS

Salomon Ofman

RSUM

La thorie des rsidus a son origine dans les travaux de Cauchy et surtout p our le multi

rsidu chez Poincar Une thorie satisfaisante base sur la dualit est donne par J Leray

dans L dont lasp ect cohomologique a t montre par F Norguet N cep endant cette

situation est essentiellement formule dans le cadre des formes semimromorphes densemble

p olaire runion dhypersurfaces lisses Une gnralisation de cette thorie au cas des formes

semimromorphes quelconques est faite par F Cole et N Herrera avec les courantsrsidus

Cep endant alors quon a un algorithme simple p our calculer le rsidu dans le cadre de la

thorie de Leray cest lalgorithme de GelfandLerayShilov ou GLS cela est b eaucoup

moins ais dans celui de ColeHerrera Cest une motivation p our tudier la dp endance du

rsidu en fonction dun paramtre an de ramener certains problmes sur les courantsrsidus

des problmes de rsidus cohomologiques

Cep endant cette dp endance nest certainement pas continue comme le met en vidence

lexemple trs simple cidessous en une variable

Soit D resp ectivement le disque unit resp ectivement le cercle unit de C la

fonction gale dans C si est une forme semimromorphe dans D densemble p olaire

C soit Res le courantrsidu au sens de ColeHerrera Pour tout t D fg on a en

C

p osant C ft tg

t

i hRes z dz z t i

C

t

tandis que

i hRes z dz z i

fg

Par contre en utilisant le thorme de Sto ckes p our les ensemble semianalytiques on a cf

O

Thorme Soit une forme direntielle dferme de dimension p dans une var

it analytique Y F f f une famil le de puples de fonctions mromorphes

T t pt

p

1

densembles polaires respectifs C c c si lorsque T tend vers la famil le F

T t pt T

p

1

converge dans un sens naturel vers le puple F on a lim Res F Res F

F T F

T

T

En utilisant les proprits fondamentales des courantsrsidus des formes semimromorphes

on p eut p our certains calculs se ramener par des tronquages adquats au cas du thorme

prcdent puis celui des rsidus cohomologiques de Leray Lalgorithme GLS donne alors

une formule explicite p our lobtention du rsidu

Z

k

K jK j k p

1

p

hRes dU U i u u i K

F

p

c c

p

1

o dsigne le puple form de u une quation minimale de c i p

i i

U u u et une fonction C supp ort compact

p

Un exemple dapplication est une dmonstration trs simple et une gnralisation de la

formule de transformation cf D sous des hypothses convenables essentiellement les

ensembles dannulation des fonctions f forment une intersection complte soit alors

i

une forme direntielle C dans une varit analytique complexe Y F f f et

p

G g g deux puples de fonctions mromorphes sur Y avec G M F o M est une

p

matrice de fonctions holomorphes

Thorme Pour tout puple dentiers I on a

X

I L

Res G Res C detM K L F

G F

pjLjjI j

les C detM K L tant des constantes convenables qui dans le cas lisse sont celles inter

venant dans la formule de changement de variables des drivations de fonctions

Un cas particulier de cette galit est utilis dans O p our la construction de la transfor

mation de Radon gnrale sur les cycles analytiques les varits analytiques

PETIT GLOSSAIRE

Admissible tra jectoire cest une fonction dni sur un voisinage de R valeurs

p

dans R vriant lorsque tant vers et converge vers plus vite que

p i

toute puissance de i f p g

i

Courantrsidu soit une forme direntielle semimromorphe dans un p olydisque

n

D C densemble p olaire contenu dans la runion des hypersurfaces c c dquations

p

resp ectives u Pour toute forme direntielle C supp ort compact et toute tra jec

i

R

p

toire admissible la limite note hRes i ou encore hRes i

U C

T

fz ju z j g

i i

i=1

existe lorsque tend vers est indp endant du choix des quations u et dp end continment

i

de U et C dsignant resp ectivement les puples u u et c c Ainsi Res

p p U

dnit un courantrsidu sur D et par partition de lunit on p eut les dnir sur toute varit

analytique

Dimension dune forme si est une forme direntielle de degr k sur une varit Y

sa dimension est dim Y k

Semimromorphe forme une forme direntielle est semimromorphe si elle scrit

lo calement comme quotient dune forme C par une fonction holomorphe

GELFANDLERAYSHILOV ALGORITHM AND

RESIDUESCURRENTS

ABSTRACT

The origins of the theory of residues are in Cauchys works and esp ecially for multi

residues in Poincars A satisfactory theory based on duality was given by J Leray L

and the cohomological asp ect was shown by F Norguet N nevertheless this situation is

essentially formulated for semimeromorphic dierential forms whose p olar set have smo oth

irreducible parts The generalization to any semimeromorphic forms was done by F Cole

and N Herrera with the residuescurrents But whereas we have a convenient algorithm

to compute the residues inside Lerays theory the socalled GelfandLerayShilov or GLS

algorithm this is much less easy for the ColefHerreras one This is a motivation to

study how the residues are dep ending of a parameter in order to solve some problems on the

residuescurrents using the to ols of cohomological residues

However this dep endence is not continuous as we can see in the very simple following

example in one variable

Let D resp b e the unit disk resp the unit circle in C the function equals to in

C and C ft tg For any t D fg we obtain

t

i hRes z dz z t i

C

t

whereas

i hRes z dz z i

fg

Let Y an analytic manifold of dimension Y and a dclosed dierential of dimension p

in Y Using Sto ckes theorem for semianalytic sets we have however cf O

f Theorem Let F f a puples family of meromorphic functions of

pt T t

p

1

respective polar sets C c c if lim F F in a natural way we have

T t pt T

p

1

T

lim Res F Res F

F T F

T

T

Using some fundamental prop erties of the residuescurrents and appropriate truncating

it is p ossible reduce some problems to the case of the previous theorem and then to the one

of Lerays cohomological residues Then the GLS algorithm gives an explicit formula to

obtain the residu

Z

k

K k jK j p

1

p

hRes dU U i u u i K

F

p

c c

p

1

where is the puple p u a minimal equation of c i p U u u

i i p

and a function C with compact supp ort

An application is a very simple demonstration and generalization of the transform formula

cf D under some suitable essentially the zero sets of the functions f is a complete

i

intersection Let F f f and G g g b e two puples of meromorphic

p p

functions in Y and G M F where M is a matrix of holomorphic functions in Y

Theorem For any puple of integers I we have

X

I L

Res G Res C detM K L F

G F

pjLjjI j

where the C detM K L are suitable constants in the smooth case they are the ones

appearing in the formula of change of variables for the derivatives of functions

A particular case of this equality is used in O for the construction of the Radon trans

formation for the analytic cycles of analytic manifolds

SMALL GLOSSARY

Admissible tra jectory it is an function dened in a neighborho o d of R with

p

values in R such that when tends to and tends to faster than any

p i

p ower of i f p g

i

n

Residucurrent Let b e a semimeromorphic dierential form in the p olydisc D C

of p olar set contained in the union of hypersurfaces c c of resp ective equations u

p i

For any C dierential form with compact supp ort and any admissible tra jectory the

R

p

limit denoted by hRes i or hRes i exists when tends to

U C

T

fz ju z j g

i i

i=1

do es not dep end of the choice of the equations u and dep ends continuously of U and

i

C are resp ectively the puples u u and c c Then Res denes a current

p p U

on D the residucurrent by a partition of the unity we can dene them on any analytic

manifold

Dimension of a form if is a dierential form of degree k on Y its dimension is

dim Y k

Semimeromorphic form a dierential form is semimeromorphic if it is lo cally a quo

tient of a C form by an holomorphic function

QUELQUES RFRENCES BIBLIOGRAPHIQUES

CH N ColeM Herrera Les Courants Rsiduels Associs une Forme Mromorphe

Lect Notes in Math Springer Verlag

DS A DickensteinC Sessa Rsidus de formes mromorphes et cohomologie mo dre

Gomtrie Complexe F NorguetS OfmanJJ Szczeciniarz ed Actualits scientiques et

industrielles Hermann paratre

L J Leray Le calcul direntiel et intgral sur une varit analytique complexe problme

de Cauchy I I I Bul l Soc Math France

N F Norguet Sur la thorie des rsidus CRAS

O S Ofman La transformation de Radon Analytique en dimension quelconque Preprint

Universit Paris paratre

Authors Address

Salomon OFMAN rue de la Glacire Paris France

ofmanmathpjussieufr

Quadratic forms and Bezoutians

MarieFranoise Roy

In the univariate situation quadratic forms based on Bezoutians and residues can b e

dened and their signatures are the Cauchy indices of rational functions Hermite quadratic

form counting the number of real ro ots can b e recovered this way The multivariate situation

is as follows Multivariate Bezoutians and residues give in the complete intersection case a

general construction of quadratic forms Their signature give interesting informations in some

particular cases top ological degree EisenbudLevins results real ro ot counting Hermite

metho d The general case is not clearly understo o d yet and we do not know what is the

multivariate generalization of the Cauchy index

From a computational p oint of view the quadratic forms so obtained have small co e

cients using dual basis shortens the data In general Bezoutians and residues have b een

shown useful in order to control the size of co ecients in several imp ortant problems of com

puter algebra and a b etter knowledge of the algebraic approach should help us to understand

these phenomena more deeply

References

V I Arnold A N Varchenko S M Gusein Zade Singularities of dieren

tiable maps Vol Birkhuser

Becker E Cardinal JP Roy MF and Szafraniec Z Multivariate Be

zoutians Kronecker symbol and EisenbudLevin formula In Eective Metho ds in Al

gebraic Geometry MEGA Progress in Math Birkhuser To app ear

Cardinal JP Dualit et algorithmes itratifs pour la solution des systmes polyno

miaux Thse Universit de Rennes I

Eisenbud D Levine H I An algebraic formula for the degree of a C map germ

Annals of Mathematics

G M KhimshiashviliOn the local degree of a smooth map Soobshch Akad Nauk

Gruz SSR in Russian

Kreuzer M and Kunz E Traces in strict Frobenius algebras and strict complete

intersections J reine angew Math

Kunz E Khler dierentials Vieweg advanced lecture in Mathematics Braun

schweig Wiesbaden

Kunz E ber den ndimensionalen Residuensatz Jahresb ericht der Deutschen

MathematikerVereinigung

Roy MF Basic algorithms in real algebraic geometry From Sturm theorem to the

existential theory of reals De Greuter to app ear

Scheja G and Storch U ber Spurfunktionen bei vol lstndigen Durchschnitten

J reine angew Math

Scheja G and Storch U Quasi FrobeniusAlgebren und lokal vol lstndige

Durchschnitte manuscripta math

Scheja G and Storch U Residuen bei vol lstndigen Durchschnitten Math

Nachr

Authors Address

MarieFrancoise ROY

RMAR Universiti de Rennes I Rennes CEDEX France

costeroyunivrennesfr

Formal completion and duality

Leovigildo Alonso Tarro

In this talk we will explain the results in AJL and some consequences for the algebraic

theory of residues

Let X b e a no etherian separated scheme Let DX b e the derived category of the

category of sheaves of mo dules over X and D X the full sub category of complexes with

q c

quasicoherent homology For any closed subscheme Z of X we can consider the endofunctors

of O Mo d dened over the ob jects by

X

n

F lim HomO I F and

X

Z

n

n

F lim O I F

Z X

n

where I is a coherent O ideal such that Z is the supp ort of O I

X X

The functor is a left exact subfunctor of and it can b e derived on the right

Z

Z

F R F is an isomorphism using injective resolutions Moreover the natural map R

Z

Z

if F D X The functor is not right exact but it has a left derived functor L

q c Z Z

D X DX describable via quasicoherent at resolutions

q c

This two op erations are related by an adjunction formula

RHomR E F RHomE L F E DX F D X

Z q c

Z

which is a sheaed derivedcategory version of GM Thm We note that this is

proved in AJL for a general quasicompact separated scheme X with a mild restriction

over Z The pro of reduces to the case E O establishing an isomorphism b etween

X

RHomR O whose homology can b e called local homologyand L as functors

Z X Z

from D X to DX

q c

Some consequences of this result are local duality as in Gr p Thm and H

p cor a more general localglobal duality L Theorem in p and Hartshornes

ane duality in H p Thm

Furthermore when F D Y coherent homology the previous formula b ecomes

c

RHomR E F RHom E F

Z

b

where X X is the canonical map

Z

Let f X Y b e a prop er map of no etherian schemes where Z and W are closed subsets

of X and Y resp ectively such that f Z W The natural map Rf R f R

Z W

abstract residue obtained via the trace map of Grothendieck duality together with

induces a functorial isomorphism

RHom E f F Rf RHomRf E R F

Z W

for E D X and F D Y

q c

c

This isomorphism dep ends only on the formal completion of X resp Y with resp ect to

Z resp W The underlying reason is that it is p ossible to establish a Grothendieck duality

theory for prop er maps of no etherian formal schemes As an application we can recover

HK p Thm and H p Prop

References

AJL L Alonso Tarro A Jeremas Lp ez and J Lipman Local homology and cohomology

of schemes preprint

GM J C P Greenlees and J P May Derived functors of Iadic completion and local

homology J Algebra

Gr R Grothendieck notes by R Hartshorne Local cohomology Lecture Notes in Math

SpringerVerlag New York

H R Hartshorne Residues and duality Lecture Notes in Math SpringerVerlag New

York

H R Hartshorne Ane duality and coniteness Inventiones Math

H R Hartshorne On the De Rham cohomology of algebraic varieties Publications Math

IHES

HK R Hbl and E Kunz Integration of dierential forms on schemes J Reine u Angew

Math

HK R Hbl and E Kunz Regular dierential forms and duality for pro jective morphisms

J Reine u Angew Math

L J Lipman Desingularization of twodimensional schemes Annals of Math

pp

L J Lipman Dualizing sheaves dierentials and residues on algebraic varieties Aster

isque So c Math France Paris

L J Lipman Notes on derived categories Preprint Purdue University

Authors Address

Leovigildo ALONSO TARRIO Departamento de Algebra Facultade

de Matematicas Universidade de Santiago E Santiago de Com

postela SPAIN lalonsozmatusces

Adeles and the De RhamResidue Complex

Amnon Yekutieli

Introduction

Let me b egin by recalling some notions from complex geometry These will serve as mo dels for

the algebrogeometric constructions which will follow Let M b e an ndimensional complex

pq pq

manifold On it we have the sheaf A of smo oth p q forms and the sheaf D of p q

currents By denition for any U M op en

pq npnq

D U dual space of A U

c

A is a DGA dierential graded algebra and D is a DG A mo dule The map

Z

A D

U

is a quasiisomorphism

Observe that A pulls back under any morphism f M N and D pushes forward when

f is prop er

De RhamResidue Complex

This is the analog of the Dolb eault complex of currents Supp ose X is a nite type scheme

over a p erfect eld k The De RhamResidue Complex app eared b efore in work of Hartshorne

and ElZein under another name the canonical or Cousin resolution of

Xk

First consider the residue complex K It has an explicit construction which go es like

X

b

this For a p oint x X let O O the complete lo cal ring By the theory of

X x

X x

Beilinson completion algebras BCAs there is a dual mo dule K O Set

X x

M

K x K x K O K

X X

X x

X

xX

If x y is a saturated chain of p oints immediate sp ecialization then there is a BCA O

X xy

and a map

K O K O K O

xy X x X xy X y

P

Putting we get our complex

X

xy

Theorem Given a dierential operator DO D M N of O mo dules there is a

X

functorial DO

M D N HomN K

X

which is a map of complexes

pq p q

Set F Hom K

X X

Xk

Corollary F is a complex with operator D d

X

X

Theorem If f X Y is proper then there is a homomorphism of complexes Tr

f

f F F

X Y

The fundamental class C is an easily dened global section of F

X

X

AdeleDe Rham complex

This is our analog of the Dolb eault complex of smo oth forms Given a quasicoherent sheaf

q

on X let A M b e the sheaf of degree q reduced Beilinson adeles For an op en set U

red

Q

q

M M where runs over the length q chains of p oints in U and M is the U A

red

Beilinson completion

pq p q

A Set A A is a DGA D d and A is a quasi

X X

X red

Xk

Xk

isomorphism

Variance for any morphism f X Y there is a DGA homomorphism A f A

Y X

Theorem F is a right DG A module

X X

A Sketch of pro of one has A is obvious Supp ose O The action of

X

red

X

Xk Xk

a a is an adele Fix a chain x x and K x What is a It is nonzero

q

only if is saturated and x x Then there are a BCA O and homomorphisms

X

Tr

K O K O K O

X

X x X x

q

0

and we set a Tra

C are quasi F A Cohomology for X smo oth the maps

X

X X

Xk

isomorphisms Otherwise in char we consider a smo oth formal embedding X X for

example take an embedding X Y Y smo oth and set X Y the formal completion

X

i i DR i

X H X F X H X A and H Then H

X X

i DR

Example Chern Character

Some of this is joint with R Hbl There is an adelic ChernWeil theory It uses the Thom

Sullivan adeles A Here X is smo oth and chark By integration on the simplex we

X

R

get a map of complexes A A Let E b e a lomcally free sheaf On A E one can

X X X

put a connection r and the curvature R r A E ndE Set

X

Z

chE r tr exp R A

X

X This denes the usual Chern character ch K X H

DR

Theorem Suppose Z X is integral of codimension m and E E O is a

Z

lomcally free resolution Then there are adelic connections r on E st

i i

C chE r F

X Z

X

chE r is the alternating sum and moreover

C chE r C F

X m Z

X

References

Be AA Beilinson Residues and adeles Funkt Anal Pril no English

trans in Func Anal Appl no

EZ F El Zein Complexe Dualizant et Applications la Classe Fondamentale dun Cycle

Bull So c Math France Mmoire

Fu W Fulton Intersection Theory Springer Berlin

GH P Griths and J Harris Principles of Algebraic Geometry Wiley New York

Ha R Hartshorne On the De Rham Cohomology of algebraic varieties Publ Math IHES

Hr A Hub er On the ParshinBeilinson Adeles for Schemes Abh Math Sem Univ

Hamburg

HY R Hbl and A Yekutieli Adelic Chern forms and the Bott residue formula preprint

Sa P Sastry Residues and duality on algebraic schemes to app ear Comp ositio Math

SY P Sastry and A Yekutieli On residue complexes dualizing sheaves and lo cal coho

mology mo dules Israel J Math

Ye A Yekutieli An Explicit Construction of the Grothendieck Residue Complex with

an app endix by P Sastry Astrisque

Ye A Yekutieli Smo oth formal embeddings in preparation

Ye A Yekutieli Traces and dierential op erators over Beilinson completion algebras to

app ear Comp ositio Math

Ye A Yekutieli Residues and dierential op erators on schemes preprint

Ye A Yekutieli The action of adeles on the De Rhamresidue complex preprint

Authors Address

Amnon Yekutieli Department of Theoretical Mathematics

The Weizmann Institute of Science Rehovot ISRAEL

amnonwisdomweizmannacil

Integral representation formulas and Multidimensional

residues

Alain Yger

Integral representation formulas of the Bo chnerMartinelli or AndreottiNorguet type pro

vide some sharp estimates for eectivity questions For example if P P is a

n

n n

prop er map from C to C with Lo jasiewicz exp onent and Q CX X the degree

n

of the p olynomial map

QdX dX

n

w w w

n

P w P w

n n

is at most

deg Q D n n

n

where

X

D max deg P

k

j n

k j

When the Lo jasiewicz exp onent is strictly a negative number one can show that if P P

m

are p olynomials such that

kP X k kX k kxk

h h

and that the sheaf I corresp onding to the ideal P P satises depthI then

m

q

P P for any Q

m

m

X

P Q Q

j j

j

deg P Q deg Q nj j max deg P

j j j

j m

where denotes the maximum of lo cal No ether exp onents at all common zero es of the P s

j

We will show in this lecture the role of to BrianonSkoda theorem together with Jacobi

Kronecker formula in order to get economic solutions for the algebraic Nullstellenstze where

the estimates dep end on the ane degree or on the Lo jasiewicz exp onent rather than on

the pro jective degree as in Kollrs approach

The second aim of this lecture is to emphazise the role of the Bo chnerMartinelli or

AndreottiNorguet formulas in the analytic theory of multidimensional residues Recall that

in the lo cal situation where f f are germs of holomorphic functions dening the origin

n

n

as an isolated zero one can dene the lo cal residue symbols for m N as

hd d

n

m

m

1

n

f f

n

R

(n(n1)2

P V

njmj

n

m k

d ds hs s

n

j k

j k k

k k

mi

where s s s denotes any nuplet of C functions in some neighborhho d of fk k g

n

P P

m

m m

1

n

such that s f s f in this neighborhho o d here s s s jmj m

j j j

n

m m m From Kroneckers formula one knows that any germ h in O can b e

n n

expanded as

X

hz d d

n

m

f z hz

m

m

1

n

f f

n

n

mN

where denotes the determinant of a matrix of germs g z such that

ij

n

X

g z z i n f z f

ij j j i i

j

One can write using ideas inspired by analytic theory of currents if is any n test

form d with compact supp ort such that near the origin then if we chose s f kf k

can b e rewritten see for example as

hd d

n

m

m

1

n

f f

n

h i

R

(n(n1)2

m

njmj

njmj

hkf k f df d

n

n

C

mi

df df df the notation meaning that one considers the analytic continuation where

n

of the function of clearly dened for Re and takes its value at The

n current dened by the right hand side of makes sense even if f f do not

n

dene the origin as an isolated zero In fact the function of in the right hand side of is

for any n test form a meromorphic function with p oles in Q its value at the origin

n

n

denes the action of a n current which is as a current annihilated by the ideals I I

n

n

I b eing the integral closure of f f and I the integral closure of I We will show

n

how currents of this form play a role in a substitute for Kroneckers formula in the case

when f f do not dene the origin as an isolated zero We will even assume that the

n

number

P

m

f dX

k

k

k

Q X

kf k

Let us write for p inf n m the formal expression

kf k

p

kf k Q X Q X

kf k

as

p

X

T dX

i i i

p

1

l

i i n

l

p

1

All functions T corresp ond to pcurrentvalued meromorphic functions of

i i

p

1

p

J f mg J p all with p oles in Q with values at the origin p currents T

J

p

p

annihilated as currents by the ideals I or I where I denotes the ideal f f and

m

the notation bar means as b efore one takes the integral closure Assume that m n We can

represent the germs h in O mo dulo the ideal as follows Take representants of f f

n m

dened in some neighborho o d of the origin let b e a C test function with compact

n

supp ort in equal to near some op en subset such that let b e a C

valued function dened in a neighborho o d of S upp and such that z

for z and S upp

n

X

z d z

j j

j

n

X

g z g z d i m

i ij j

j

Then if h is some holomorphic function in then see section one can write h in

as

h i E D

V P P

p

m

np

hz z T h g

J f1 mg

n

j

j J p

J

i

#J =p

P

m

h z f z

j j

j

where h h are holomorphic functions in The algebraic understanding of these tech

m

niques all based on the approach of multidimensional residues with the Bo chnerMartinelli

or AndreottiNorguet formulas remains to b e cleared In particular one would like to in

p

J f mg J p involved in the division formula terpret the role of all currents T

J

We would like also to p oint out that the transformation law in its usual version or

generalized version can b e also obtained this was done in a recent work by J Y Boyer

as an easy consequence of Bo chnerMartinelli and Norguet formulas The crucial role of this

transformation law in the algebraic theory of residues is a further reason for such to ols to b e

understo o d from the algebraic p oint of view

References

C A Berenstein and A Yger Eective Bzout identities in Qz z Acta Math

n

C A Berenstein R Gay A Vidras and A Yger Residue currents and Bzout identities

Progress in Mathematics Birkhuser

P Griths and J Harris Principles of algebraic geometry WileyInterscience New

York

A M Kytmanov A transformation formula for Grothendieck residues and some of its

applications Sib erian Math Journal

C A Berenstein and A Yger Une formule de Jacobi et ses consquences Annales de

lEcole Norm Sup Paris

A Dickenstein R Gay C Sessa and A Yger Analytic functionals annihilated by ideals

preprint

Authors Address

Alain Yger Laboratoire de Mathmatiques Pures Universit Bor

deaux I cours de la libration Talence FRANCE

ygermathunivbordeauxfr

List of participants

Ab dellah AL AMRANI UFR de Mathmatiques et dInformatique Universit

Louis Pasteur rue Ren Descartes F Strasbourg cedex France

Leovigildo ALONSO TARRIO Departamento de Alxebra Facultade de Matem

aticas Universidade de Santiago E Santiago de Compostela SPAIN

lalonsozmatusces

Francesco AMOROSO Universit di Pisa

amorosogaussdmunipiit

Ab dalah ASSI Universit dAngers France

assitontonunivangersfr

Carlos BERENSTEIN Institute for Systems Research Dept of Mathemat

ics University of Maryland College Park MD USA

carlossrcumdedu

Isab el BERMEJO Departamento de Matematica Fundamental Facultad de

Matematicas Universidad de La Laguna La Laguna Tenerife Spain

ibermejoulles

Djilali BOUDIAF Universit de Poitiers Dpartement de Mathmatiques

Avenue du Recteur Pineau POITIERS France

boudiafmathrsunivpoitiersfr

JeanYves BOYER cit Barthelemy BORDEAUX France

pellemathubordeauxfr

JeanPaul BRASSELET CIRM Luminy Case Marseille Cedex

France

jpbcirmunivmrsfr

Jo l BRIANON Dpartement de Mathmatique Universit de Nice Parc

Valrose Nice Cedex France

Pierrette CASSOUNOGUS Laboratoire de Mathmatiques Pures Univer

sit Bordeaux I cours de la libration Talence France

thnufrbdxcribxubordeauxfr

Marc CHARDIN cole polytechnique F Palaiseau France Cedex

chardingagepolytechniquefr

Vincent COSSART cole polytechnique Universit de Versailles F

Palaiseau France Cedex

cossartgagepolytechniquefr

Alicia DICKENSTEIN Univ Buenos Aeres

alidickmatedmubaar

Pierre DOLBEAULT rue des Cordelires Paris France

pidoccrjussieufr

David EISENBUD Institut Henri Poincar Univ Brandeis USA

eisenbudihpjussieufr

Mohamed ELKADI Dpartement de Mathmatiques Universit de Nice

Parc Valrose Nice

elkadimathunicefr

Fouad ELZEIN Universit de Nantes dpt de Mathmatiques rue de la

Houssinire Nantes France

elzeinmathpjussieufr

Adelina FABIANO Dipartimento di matematica Universit della Calabria

Arcavacata di Rende Cosenza

pucciccuscunicalit

Andrei GABRIELOV Cornell MSI College Av Ithaca NY

USA

andreimsiadmincitedu

Andre GALLIGO Dpartement de Mathmatiques Universit de Nice Parc

Valrose Nice Cedex

galligosophiainriafr

Bernard GLEYSE INSA de ROUEN BP Mont StAignan Cedex

BernardGleyseinsarouenfr

Ngo c Minh HOANG LIFL UA CNRS Universit de Lille I I

Villeneuve dAscq France

hoangliflliflfr

Alain HENAUT Laboratoire de Mathmatiques Pures Universit Bordeaux

I cours de la libration Talence France

henautmathubordeauxfr

IChiau HUANG Institute of Mathematics Academia Sinica Nankang Taipai

Taiwan

maichiauccvaxsinicaedutw

Reinhold HBL Universitat Regensburg NWF I Mathematik D

Regensburg Germany

HUEBLvaxrzuniregensburgdde

Craig HUNEKE Dept of Math Purdue University

chunekemathlsaumichedu

Ana JEREMIAS LOPEZ Departamento de Alxebra Facultade de Matemat

icas Universidade de Santiago E Santiago de Compostela Spain

jeremiaszmatusces

JeanPierre JOUANOLOU UFR de Mathmatiques et dInformatique Uni

versit Louis Pasteur rue Ren Descartes F Strasbourg cedex

jouanolomathustrasbgfr

Askold KHOVANSKII Dm Ulianova st apt Moscow

Russia

ashkoldmathtorontoedu

Theresa KRICK Dpartement de Mathmatiques Facult des Sciences de

Limoges Av Albert Thomas F Limoges France

krickmariepolytechniquefr Krickmatedmubaar

Ernst KUNZ Universitt Regensburg NWF I Mathematik D Re

gensburg

KUNZvaxrzuniregensburgdde

Yves LEGRANDGERARD quipe de Logique Mathmatique Tour couloir

pice Universit Paris place Jussieu Paris France

ylglogiquejussieufr

Monique LEJEUNEJALABERT Universit de Grenoble France

lejeunefouriergrenetfr

Joseph LIPMAN Dept of Mathematics Purdue University W Lafayette

IN USA

lipmanmathpurdueedu

Danielle MATTHYS rue de la Corniche Orvault France

Michel MEO meopuccinigrenetfr

David MOND Mathematics Institute University of Warwick Coventry

CV AL England

mondmathswarwickacuk

Marcel MORALES Universit de Grenoble Institut Fourier Mathma

tiques BP Saint Martin dHres France

moralesfouriergrenetfr

Guillermo MORENO SOCAS Laboratoire GAGE Centre de Mathma

tique cole polytechnique F PALAISEAU cedex FRANCE

morenogagepolytechniquefr

Bernard MOURRAIN INRIA Projet SAFIR Route des Lucioles

Valbonne France

mourrainsophiainriafr

Salomon OFMAN rue de la Glacire Paris France

ofmanmathpjussieufr

Keith PARDUE Mathematics Department University of Toronto Toronto

ON MS A CANADA

parduemathtorontoedu

Mikael PASSARE Department of Mathematics Stockholm University S

Stockholm SWEDEN

passarematematiksuse

PAUGAM Michel Dpartement de mathmatiques Universit de Caen F

Caen cedex

paugammathunicaenfr

Thierry PELLE Laboratoire de Mathmatiques Pures Universit Bordeaux

I cours de la libration Talence France

pellemathubordeauxfr

Michel PETITOT Laboratoire des Signaux et Systmes LSS cole SUP

ELEC Plateau du Moulon F GifsurYvette cedex France

petitotliflfr

Patrice PHILIPPON Universit Paris UMR Problmes Diophantiens

T e tage case place Jussieu PARIS cedex

pphccrjussieufr

Olivier PILTANT Centre de Mathmatiques cole polytechnique F

Palaiseau cedex France

piltantorpheepolytechniquefr

Valeri POUCHNIA Institute for Systems Research Dept of Mathematics

University of Maryland College Park MD USA

valerimathumdedu

JeanCharles POURTIER Batiment B Apt residence les vergers rue

du Docteur Roux EYSINES France

pellemathubordeauxfr

Gemma PUCCI Dipartimento di matematica Universit della Calabria

Arcavacata di Rende Cosenza

fabianoccuscunicalit

MarieFrancoise ROY IRMAR Universit de Rennes I Rennes CEDEX

France

costeroyunivrennesfr

Uwe STORCH Fakultt und Institut fr Mathematik Ruhr Universitt

Bochum Bochum Germany

UweStorchrzruhrunibochumde

Aviva SZPIRGLAS Dpartement de Mathmatiques Universit Paris XI I I

ParisNord Villetaneuse France

avivamathunivparisfr

Bernard TEISSIER cole Normale Superieure rue dUlm PARIS

Cedex France

teissierdmiensfr

Bernd ULRICH Michigan State University Lansing USA

ulrichmathmsuedu

Alekos VIDRAS Departement of Mathematics and Statistics University

of Cyprus Nicosia P O Box Cyprus

msvidraspythagorasmasucyaccy

Amnon YEKUTIELI Dept of Theoretical Mathematics Weizmann Insti

tute of Science Rehovot ISRAEL

amnonwisdomweizmannacil

Alain YGER Laboratoire de Mathmatiques Pures Universit Bordeaux

I cours de la libration Talence France

ygermathunivbordeauxfr

Contents

Djilali Boudiaf

Eduardo Cattani David Cox and Alicia Dickenstein

Reinhold Hbl

Craig Huneke

Teresa Krick

Ernst Kunz

Joseph Lipman

Salomon Ofman

MarieFranoise Roy

Leovigildo Alonso Tarro

Amnon Yekutieli

Alain Yger