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University LECTURE 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 http://dx.doi.org/10.1090/ulect/047 Residues and Duality for Projective Algebraic Varieties University LECTURE 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 M THE ATI A CA M L 42(4/3 -( N %)3)47 S A O C C I I R E E T !'%7-% Y M A F O 8 U 88 NDED 1 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Jerry L. Bona NigelD.Higson Eric M. Friedlander (Chair) J. T. Stafford 2000 Mathematics Subject Classification. Primary 14Fxx, 14F10, 14B15; Secondary 32A27, 14M10, 14M25. For additional information and 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 and contributions by David A. Cox and Alicia Dickenstein. p. cm. — (University lecture series ; v. 47) Includes bibliographical references and index. ISBN 978-0-8218-4760-2 (alk. paper) 1. Algebraic varieties. 2. Geometry, Projective. 3. Congruences and residues. I. Title. QA564.K86 2009 516.353—dc22 2008038860 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgmentofthesourceisgiven. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. c 2008 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 131211100908 Contents Preface .............................................................vii GlossaryofNotation ................................................ ix 1. LocalCohomologyFunctors ..........................................1 2. LocalCohomologyofNoetherianAffineSchemes .....................6 3. CechCohomologyˇ ...................................................11 4. KoszulComplexesandLocalCohomology ...........................22 5. ResiduesandLocalCohomologyforPowerSeriesRings .............35 6. TheCohomologyofProjectiveSchemes .............................47 7. DualityandResidueTheoremsforProjectiveSpace .................52 8. Traces,ComplementaryModules,andDifferents .....................65 9. The Sheaf of Regular Differential Forms on an Algebraic Variety . 81 10. ResiduesforAlgebraicVarieties. LocalDuality ......................89 11. DualityandResidueTheoremsforProjectiveVarieties .............100 12. CompleteDuality ..................................................110 13. ApplicationsofResiduesandDuality(AliciaDickenstein) ..........115 14. ToricResidues(DavidA.Cox) .....................................128 Bibliography........................................................151 Index...............................................................155 v Preface The present text is an extended and updated version of my lecture notes Residuen und Dualit¨at auf projektiven algebraischen Variet¨aten (Der Regensburger 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 translation and transforming the manuscript into the appropriate LATEX2ε style, to Alicia Dickenstein and David Cox for encouragement and critical comments and for enriching the book by adding two sections, one on applications of algebraic residue theory and the other explaining toric residues and relating them to the earlier text. The main objective of my old lectures, which were strongly influenced by Lip- man’s monograph [71], was to describe local and global duality in the special case of irreducible algebraic varieties over an algebraically closed base field k in terms of dif- 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 ωV/k of regular d-forms. This sheaf is an intrinsically defined subsheaf of d the constant sheaf ΩR(V )/k,whereR(V ) is the field of rational functions on V .We construct ωV/k in § 9 after the necessary preparation. Similarly, for a closed point x ∈ V ,thestalk(ωV/k)x is a canonical choice for the dualizing (canonical) module ωOV,x/k studied in local algebra. We have the residue map d −→ Resx : Hx (ωV/k) k defined on the d-th local cohomology of ωV/k. The local cohomology classes can be written as generalized fractions ω f1,...,fd ∈ O where ω ωOV,x/k and f1,...,fd is a system of parameters of V,x.Usingthe residue map, we get the Grothendieck residue symbol ω Resx . f1,...,fd For a projective variety V the residue map at the vertex of the affine cone C(V ) induces a linear operator on global cohomology d : H (V,ωV/k) −→ k V called the integral. The local and global duality theorems are formulated in terms of Resx and V . There is also the residue theorem stating that “the integral is 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 calculus are formulated and proved, and later generalized to toric residues by David 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 ω Resx f1,...,fd can be considered as intersection invariants, and by a suitable choice of the regular d-form ω, a residue can have many geometric interpretations, including intersection multiplicity, angle of intersection, curvature, and the centroid of a zero-dimensional scheme. The residue theorem then gives a global relation for these local invariants. In this way, classical results of algebraic geometry can be reproved and generalized. 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 and duality to partial differential equations and problems in interpolation and ideal 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 Corollary 11.9 and those at the end of § 12. These articles extend the theory of the book considerably in many directions. This leads to a large bibliography, though it 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 K¨ahler differentials, and they knew basic algebraic geometry. Some of them had profited from the exchange program between the University of Regensburg 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 H¨ubl, Martin Kreuzer, Markus N¨ubler and Gerhard Quarg, all of whom also later worked on algebraic residue theory, much to my 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 f¨ur Mathematik Universit¨at 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´on I (C1428EGA) Buenos Aires, Argentina Glossary of Notation Unless otherwise stated all rings are commutative with 1. A multiplicatively closed set of a ring R always contains 1R. A ring homomorphism R → S maps 1R to 1S. This is also called an algebra S/R. We try to use the standard notation and language of algebraic geometry. N {0, 1, 2,...} N+ {1, 2, 3,...} U(X) set of open subsets of X 1
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