Residues and duality for projective algebraic varieties

177 Pages·2008·1.149 MB·English

by  Ernst Kunz

University
L
ECTURE
Series
Volume 47
Residues and Duality for
Projective Algebraic Varieties
Ernst Kunz
with the assistance of and contributions by
David A. Cox and Alicia Dickenstein
American Mathematical Society

Residues and Duality for
Projective Algebraic Varieties

University
L
ECTURE
Series
Volume 47
Residues and Duality for
Projective Algebraic Varieties
Ernst Kunz
with the assistance of and contributions by
David A. Cox and Alicia Dickenstein
ERMIACAN M(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:3)A(cid:8)T(cid:9)H(cid:10)E(cid:8)(cid:11)M(cid:7)(cid:12)A(cid:5)T(cid:10)IC(cid:3)(cid:6)(cid:7)(cid:6)(cid:8)(cid:4)ALSOYCTIE
FOUNDED 1888
American Mathematical Society
Providence, Rhode Island
EDITORIAL COMMITTEE
Jerry L. Bona Nigel D. Higson
Eric M. Friedlander (Chair) J. T. Stafford
2000 Mathematics Subject Classification. Primary14Fxx, 14F10, 14B15;
Secondary 32A27, 14M10,14M25.
For additional informationand updates on this book, visit
www.ams.org/bookpages/ulect-47
Library of Congress Cataloging-in-Publication Data
Kunz,Ernst,1933–
Residues and duality for projective algebraic varieties / Ernst Kunz ; with the assistance of
andcontributionsbyDavidA.CoxandAliciaDickenstein.
p.cm. —(Universitylectureseries;v.47)
Includesbibliographicalreferencesandindex.
ISBN978-0-8218-4760-2(alk.paper)
1.Algebraicvarieties. 2.Geometry,Projective. 3.Congruencesandresidues. I.Title.
QA564.K86 2009
516.3(cid:1)53—dc22 2008038860
Copying and reprinting. Individual readers of this publication, and nonprofit libraries
actingforthem,arepermittedtomakefairuseofthematerial,suchastocopyachapterforuse
in teaching or research. Permission is granted to quote brief passages from this publication in
reviews,providedthecustomaryacknowledgmentofthesourceisgiven.
Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublication
is permitted only under license from the American Mathematical Society. Requests for such
permissionshouldbeaddressedtotheAcquisitionsDepartment,AmericanMathematicalSociety,
201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by
[email protected].
(cid:1)c 2008bytheAmericanMathematicalSociety. Allrightsreserved.
TheAmericanMathematicalSocietyretainsallrights
exceptthosegrantedtotheUnitedStatesGovernment.
PrintedintheUnitedStatesofAmerica.
(cid:1)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines
establishedtoensurepermanenceanddurability.
VisittheAMShomepageathttp://www.ams.org/
10987654321 131211100908
Contents
Preface .............................................................vii
Glossary of Notation ................................................ix
1. Local Cohomology Functors ..........................................1
2. Local Cohomology of Noetherian Affine Schemes .....................6
3. Cˇech Cohomology ...................................................11
4. Koszul Complexes and Local Cohomology ...........................22
5. Residues and Local Cohomology for Power Series Rings .............35
6. The Cohomology of Projective Schemes .............................47
7. Duality and Residue Theorems for Projective Space .................52
8. Traces, Complementary Modules, and Differents .....................65
9. The Sheaf of Regular Differential Forms on an Algebraic Variety ....81
10. Residues for Algebraic Varieties. Local Duality ......................89
11. Duality and Residue Theorems for Projective Varieties .............100
12. Complete Duality ..................................................110
13. Applications of Residues and Duality (Alicia Dickenstein) ..........115
14. Toric Residues (David A. Cox) .....................................128
Bibliography........................................................151
Index...............................................................155
v

Preface
The present text is an extended and updated version of my lecture notes
Residuen und Dualita¨t auf projektiven algebraischen Varieta¨ten (DerRegensburger
Trichter 19 (1986)), based on a course I taught in the winter term 1985/86 at
the University of Regensburg. I am grateful to David Cox for helping me with the
translationandtransformingthemanuscriptintotheappropriateLATEX2ε style,to
AliciaDickensteinandDavidCoxforencouragementandcriticalcommentsandfor
enriching the book by adding two sections, one on applications of algebraic residue
theoryandtheotherexplainingtoricresiduesandrelatingthemtotheearliertext.
The main objective of my old lectures, which were strongly influenced by Lip-
man’smonograph[71],wastodescribelocalandglobaldualityinthespecialcaseof
irreduciblealgebraicvarietiesoveranalgebraicallyclosedbasefieldkintermsofdif-
ferential forms and their residues. Although the dualizing sheaf of a d-dimensional
algebraic variety V is only unique up to isomorphism, there is a canonical choice,
the sheaf ω of regular d-forms. This sheaf is an intrinsically defined subsheaf of
V/k
the constant sheaf Ωd , where R(V) is the field of rational functions onV. We
R(V)/k
construct ω in § 9 after the necessary preparation. Similarly, for a closed point
V/k
x∈V, the stalk (ω ) is a canonical choice for the dualizing (canonical) module
V/k x
ωOV,x/k studied in local algebra. We have the residue map
Res :Hd(ω )−→k
x x V/k
definedonthed-th localcohomology ofω . The localcohomology classes canbe
V/k
written as generalized fractions
(cid:1) (cid:2)
ω
f ,...,f
1 d
where ω ∈ ωOV,x/k and f1,...,fd is a system of parameters of OV,x. Using the
residue map, we get the Grothendieck residue symbol
(cid:1) (cid:2)
ω
Res .
x f ,...,f
1 d
For a projective variety V the residue map at the vertex of the affine cone C(V)
induces a linear operator on global cohomology
(cid:3)
:Hd(V,ω )−→k
V/k
V
called the inte(cid:4)gral. The local and global duality theorems are formulated in terms
of Res and . There is also the residue theorem stating that “the integral is
x V
the sum of all of the residues.” Specializing to projective algebraic curves gives
the usual residue theorem for curves plus a version of the Serre duality theorem
expressed in terms of differentials and their residues. Basic rules of the residue
calculusareformulatedandproved,andlatergeneralizedtotoricresiduesbyDavid
vii
viii PREFACE
Cox. Because of the growing current interest in performing explicit calculations in
algebraic geometry, we hope that our description of duality theory in terms of
differential forms and their residues will prove to be useful.
The residues (cid:1) (cid:2)
ω
Res
x f ,...,f
1 d
canbe considered as intersection invariants, and by a suitable choice of the regular
d-formω,aresiduecanhavemanygeometricinterpretations,includingintersection
multiplicity,angleofintersection,curvature,andthecentroidofazero-dimensional
scheme. The residue theoremthen gives a global relation forthese localinvariants.
Inthisway, classicalresultsofalgebraicgeometrycanbereprovedandgeneralized.
It is part of the culture to relate current theories to the achievements of former
times. This point of view is stressed in the present notes, and it is particularly sat-
isfying that some applications of residues and duality reach back to antiquity (the-
orems of Apollonius and Pappus). Alicia Dickenstein gives applications of residues
anddualitytopartialdifferentialequationsandproblemsininterpolationandideal
membership.
Since the book is introductory in nature, only some aspects of duality the-
ory can be covered. Of course the theory has been developed much further in
the last decades, by Lipman and his coworkers among others. At appropriate
places, the text includes references to articles that appeared after the publication
of Hartshorne’s Residues and Duality [38]; see for instance the remarks following
Corollary11.9andthose attheendof§12. These articlesextendthetheoryof the
bookconsiderably inmany directions. Thisleadstoalargebibliography, thoughit
is likely that some important relevant work has been missed. For this, I apologize.
The students in my course were already familiar with commutative algebra,
including Ka¨hler differentials, and they knew basic algebraic geometry. Some of
themhadprofitedfromtheexchangeprogrambetweentheUniversityofRegensburg
and Brandeis University, where they attended a course taught by David Eisenbud
out of Hartshorne’s book [39]. Similar prerequisites are assumed about the reader
of the present text. The section by David Cox requires a basic knowledge of toric
geometry.
I want to thank the students of my lectures who insisted on clearer exposition,
especially Reinhold Hu¨bl, Martin Kreuzer, Markus Nu¨bler and Gerhard Quarg, all
of whom also laterworked on algebraic residue theory, much tomy benefit and the
benefit of this book. Thanks are also due to the referees for their suggestions and
comments and to Ina Mette for her support of this project.
July 2008 Ernst Kunz
Fakult¨at fu¨r Mathematik
Universita¨t Regensburg
D-93040 Regensburg, Germany
David A. Cox
Department of Mathematics & Computer Science
Amherst College
Amherst, MA 01002,USA
Alicia Dickenstein
Departamento de Matem´atica, FCEN
Universidad de Buenos Aires
Cuidad Universitaria-Pabell´onI
(C1428EGA)Buenos Aires, Argentina