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University LECTURE Series

Volume 47

Residues and 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

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American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Jerry L. Bona NigelD.Higson Eric M. Friedlander (Chair) J. T. Stafford

2000 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. , 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 of Regular Differential Forms on an . . . . 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 field k in terms of dif- ferential forms and their residues. Although the of a d-dimensional algebraic variety V is only unique up to , there is a canonical choice, the sheaf ωV/k of regular d-forms. This sheaf is an intrinsically defined subsheaf of d the 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)

ωOV,x/k studied in local algebra. We have the residue d −→ Resx : Hx (ωV/k) k defined on the d-th local of ωV/k. The 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 V the residue map at the 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 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 , 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 . 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 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 , 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 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 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 of a R always contains 1R. A 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 of X 1 Ab(X) of abelian sheaves on X 1

sx of a section s at x 1 Supp(s) support of a section s 1 F (U) group of sections on an U 1

Fx of F at x 1

ΓY (X, F ) group of sections with support in Y 1 i F HY (X, ) i-th cohomology with support in Y 1 i F Hx(X, ) i-th local cohomology at a point x 1 M sheaf associated to a module M 1 F flasque sheaf associated to F 2

j!F extension of a sheaf by zero 2 F | U restriction of a sheaf to an open set U 2 Mod(X) category of sheaves of modules on a ringed X 4 Hi(X, F ) i-th global cohomology 4 j∗F direct image sheaf 5

RM an S-module considered as an R-module via R → S 6 Spec R R 6 D(f) non-vanishing set of f on an affine scheme 7

Mf localization of a module at f 7 Ann annihilator of an element or ideal or module 7 V (a) vanishing set of an ideal a on an affine scheme 8 V (f) vanishing set of a set f of polynomials 8 Γa(M) sections of a module with support in V (a)8

ix x GLOSSARY OF NOTATION

i Ha(M) i-th cohomology with support in V (a)8 i Hm(M) i-th local cohomology at a m 8

JY of Y 9 Or O X of r copies of X 9 k(x) residue field of OX,x 9 Pd A d-dimensional projective space over a ring A 11 C•(U, F ) alternating Cechˇ complex with respect to a covering U 11 Hˇ •(U, F ) Cechˇ cohomology 11 C•(U, F ) normalized Cechˇ complex 11 C •(U, F ) sheafified Cechˇ complex 12 dim (Krull) of a ring or module or scheme 16 C•[1] complex with degree shift by 1 16 Rd of Rd 22 K•(t, M) homological Koszul complex 22 K•(t, M) cohomological Koszul complex 23 H•(t, M) of K•(t, M)23 H•(t, M) homology of K•(t, M)23

MS localization with respect to S 23 ∆ transition determinant 26

µx multiplication by x 27 S(R) socle of a zero-dimensional local or graded ring R 29 lim direct (injective) 29 −→ M/ t M → Hd M Φ t canonical map ( ) m( )32 m generalized fraction of m over denominator set t 32 t • ΩR/k algebra of differential forms of R/k 35 ResR residue for a complete reduced R 39, 93

ρt canonical map induced by the residue 40, 89 f(0) constant term of a power series f 41

J, Jf Jacobian determinant (of a set f of polynomials) 43 lim inverse (projective) limit 45 ←− S+ homogeneous maximal ideal of a graded ring S 47 Proj S projective scheme of a graded ring S 47

D+(f) non-vanishing set of a homogeneous element f ∈ S on Proj S 47

M(f) homogeneous localization of a graded module 47 M(n) graded module shifted by n 47 M ∗ sheaf associated to a module 47 GLOSSARY OF NOTATION xi dimk dimension of a k- 49 Γ∗F graded module associated to a sheaf 49 Γ(X, F ) module of global sections 50 χ(F ) of F 50 Supp(F ) support of a sheaf F 50 PF Hilbert polynomial of F 50 pa(X) genus of X 51 p ΩX/A sheaf of p-forms on a scheme X/A 52 R(X) field of rational functions on X/A an integral scheme 52 X integral for a projective variety X 55, 101 δF duality isomorphism 55, 94, 102 R, M completion of a local ring R or an R-module M 56

Resx residue at a closed point x 56, 95 X(k)setofk-rational points of X 57

V+(I) zero-set of a homogeneous ideal I in Proj S 58 Ad k affine space over k 58 G(f)=Gf degree form of a polynomial f 58

TP (V ) of a variety V at P 61 grad f gradient of a polynomial f 63  ,  standard scalar product 63 H∞ hyperplane at infinity 63 f = ∂f partial derivative of f with respect to X 63 Xi ∂Xi i tr(ϕ) trace of an endomorphism 65 trS/R trace map of an algebra 65

ωS/R canonical (dualizing) module 65, 84 Q(R) full ring of quotients of a ring R 66

CS/R Dedekind complementary module 66 dD(S/R) Dedekind different 66 fT/S conductor of an algebra T/S 67 k(SP) residue field of SP 68 µ(M) minimal number of generators of M 68 µP(M) minimal number of generators of MP 68 rP Cohen-Macaulay type at P 68 Max S set of maximal ideals of S 69

δ canonical multiplication map S ⊗R S → S 70 dN (S/R)Noetherdifferent 70 dK (S/R)K¨ahler different 70 t ∆x B´ezoutian 72 xii GLOSSARY OF NOTATION

x τt associated to the B´ezoutian 72 t dx generalized Jacobian 74

ωV/k sheaf of regular differential forms (also called canonical or dualizing sheaf) 84 Reg V set of regular points of V 84

cV/k fundamental class of a variety 85 F ∗, F ∗∗ (double dual) of a sheaf F 85 Pic V of a variety V 86

CV/W complementary module of a finite V → W 87

µx(X) multiplicity of X at x 96 ∆(X) centroid of a zero-dimensional affine scheme 96

pg(V ) geometric genus of a projective variety 103 Coh(V ) category of coherent sheaves on V 110 i δF i-duality isomorphism 111 Sing(X) singular locus of X 112 δ(X) singularity degree of X 113 c(X) conductor degree of X 113 d+1 |µ| µ0 + ···+ µd, µ =(µ0,...,µd) ∈ N 115 d+1 µ! µ0! ···µd!, µ =(µ0,...,µd) ∈ N 115 ∂µ ( ∂ )µ0 ···( ∂ )µd , µ =(µ ,...,µ ) ∈ Nd+1 115 ∂X0 ∂Xd 0 d µ µ0 ··· µd ∈ Nd+1 X X0 Xd , µ =(µ0,...,µd) 116 PF (∂) differential operator associated to F = {F0,...,Fd} 115

ev0 evaluation at zero 116 polysol(I(∂)) polynomial solutions of the differential equations associated to an ideal I 116 P(V ) projective space associated to a vector space V 117 X(∆) associated to a polytope ∆ 119

∆f (X, Z)representativeofB´ezoutian of f = {f1,...,fd} 120 deg f degree of f in a set of variables X 123 X (f) radical of the ideal (f) 127

Ω0 homogeneous d-form used for residues 128, 136 Pd T the torus of k or a toric variety 129, 131 G multiplicative group 129 m G Ω 0 homogeneous generalized fraction 129, 138 F0 ···Fd N,M dual lattices 130 σ∨ dual of a convex cone σ 130 σ(1) set of 1-dimensional faces of σ 130 GLOSSARY OF NOTATION xiii

Aσ semigroup algebra of σ 130 χm character associated to m ∈ M 130, 131

Vσ affine toric variety corresponding to σ 131 ω the d-form dt1 ··· dtd or the t1 td (d +1)-form dt0 ··· dtd 131, 132 t0 td Int σ∨ of σ∨ 131 C(∆) cone associated to a lattice polytope ∆ 131

S∆ semigroup algebra of a lattice polytope ∆ 132

L∆ quotient field of S∆ 132

I∆ dualizing ideal corresponding to ∆ 132

Resf toric residue (first and second versions) 133 uρ minimal generator of ρ ∩ N 133

Dρ divisor associated to minimal generator ρ 133 S total coordinate ring of the toric variety 133 div(χm) divisor of the character χm 134 Cl(X) group of torus-invariant Weil divisors modulo linear equivalence 134

∆d standard d-simplex 134 β sum of the degrees of the variables 135 0 d − δ the critical degreee i=0 deg Fi β0 138 ResF toric residue (homogeneous versions) 138, 140

Σ∆ normal fan of ∆ 138

Resf global residue 144 ∆(f) Newton polytope of a Laurent series f 144 σ ± ∆F polynomial of toric residue 1 148

Bibliography

[1] J. Alexander and A. Hirschowitz, Polynomial interpolation in several variables, J. Alg. Geom. 4, 201–222 (1995) [2] L. Alonso Tarrio, A. Jeremias L´opes and J. Lipman, Studies in Duality on Noetherian Formal Schemes and Non-noetherian Ordinary Schemes, Contemp. Math. 244,Amer.Math.Soc., Providence, RI (1999) [3] V. Batyrev and D. Cox, On the of projective hypersurfaces in toric varieties, Duke Math. J. 1994, 293–338 (1994) [4] and E. Materov, Mixed toric residues and Calabi-Yau complete intersections,in Calabi-Yau Varieties and Mirror Symmetry, N. Yui and J. D. Lewis, eds., Fields Institute Communications 38, Amer. Math. Soc., Providence, 3–27 (2003) [5] and , Toric residues and mirror symmetry, Moscow Math. J. 2, 435–475 (2002) [6] C. Berenstein and A. Yger, Residue calculus and effective Nullstellensatz, Amer. J. Math. 121, 723–769 (1999) [7] G. Biernat, On the Jacobi-Kronecker formula for a polynomial mapping having zeros at infinity, Bull. Soc. Sci. Lett.L´  od´zS´er. Rech. D´eform. 14, 103–114 (1992/93) [8] M. P. Brodmann and R. Y. Sharp, Local Cohomology: An Algebraic Introduction with Geo- metric Applications, Cambridge Univ. Press, Cambridge (1998) [9] W. Bruns, J. Gubeladze and N. V. Trung, Normal polytopes, triangulations, and Koszul algebras, J. Reine Angew. Math. 485 (1997), 123–160. [10] and J. Herzog, Cohen-Macaulay Rings, Cambridge Univ. Press, Cambridge (1993) [11]E.Cattani,D.CoxandA.Dickenstein,Residues in toric varieties, Compositio Math. 108, 35–76 (1997) [12] and A. Dickenstein, A global view of residues in the torus, J. Pure Appl. Algebra 117/118, 119–144 (1997) [13] , and B. Sturmfels, Computing multidimensional residues,In:Algorithms in Algebraic Geometry and Applications, L. Gonzalez-Vega and T. Recio, eds., Progress in Mathematics 143,Birkh¨auser, Basel-Boston-Berlin, 135–164 (1996) [14] , and , Residues and resultants, J. Math. Sci. Univ. Tokyo 5, 119–148 (1998) [15] C. Ciliberto, Geometric aspects of polynomial interpolation in more variables and of Waring’s problem,In:European Congress of Mathematics, Vol. I (Barcelona 2000),C.Casacuberta, R. M. Mir´o-Roig, J. Verdera and S. Xamb´o-Descamps, eds., Progress in Mathematics 201, Birkh¨auser, Basel-Boston-Berlin, 289–316 (2001) [16] B. Conrad, Grothendieck Duality and Base Change, Lecture Notes in Math. 1750, Springer, Berlin, 2000 [17] D. Cox, The homogeneous coordinate ring of a toric variety, J. Algebraic Geom. 4, 17–50 (1995) [18] , Toric residues, Ark. Mat. 34, 73–96 (1996) [19] , Toric varieties and toric resolutions,inResolution of Singularities,H.Hauser,J. Lipman, F. Oort and A. Quiros, eds., Birkh¨auser, Basel-Boston-Berlin, 259–284 (2000) [20] and A. Dickenstein, Codimension theorems for complete toric varieties, Proc. Amer. Math. Soc. 133, 3153–3162 (2005) [21] C. D’Andrea and A. Khetan, Macaulay style formulas for toric residues,Compos.Math. 141, 713–728 (2005) [22] E. D. Davis, Complete intersections of codimension 2 in Pr:TheB´ezout-Jacobi-Segre theo- rem revisited. Rend. Sem. Mat. Univ. Politecn. Torino 43, 333-353 (1985)

151 152 BIBLIOGRAPHY

[23] A. Dickenstein and I. Emiris (Eds.), Solving polynomial equations. Foundations, algorithms, and applications, Algorithms and Computation in Mathematics, 14, Springer-Verlag, Berlin (2005) [24] A. Dickenstein, N. Fitchas, M. Giusti, C. Sessa, The membership problem for unmixed polyno- mial ideals is solvable in single exponential time, Special Volume, Proc. AAECC-7, Discrete Applied Mathematics 33, 73–94 (1991) [25] A. Dickenstein and C. Sessa, An effective residual criterion for the membership problem in C[z1, ··· ,zn]., J. Pure Appl. Algebra 74, 149–158 (1991) [26] and , Duality methods for the membership problem,In:Effective methods in algebraic geometry (Castiglioncello, 1990),Progr.Math.94,Birkh¨auser Boston, Boston, MA, 89–103 (1991) [27] D. Eisenbud, Commutative algebra, with a view toward algebraic geometry, Graduate Texts in Math. 150, Springer-Verlag, New York, (1995) [28] , C. Huneke and W. Vasconcelos, Direct methods for primary decomposition,Invent. Math. 110, 207–235 (1992) [29] M. Elkadi and B. Mourrain Introduction `alar´esolution des syst`emes polynomiaux, Math´ematiques & Applications (Berlin) 59, Springer, Berlin (2007) [30] J. Emsalem and A. Iarrobino, Inverse system of a symbolic power I,J.Algebra174, 1080– 1090, (1995) [31] S. Ertl and R. H¨ubl, The Jacobian formula for Laurent polynomials, Univ. Jagellonicae Acta Math. 37, 51–67 (1999) [32] W. Fulton, Introduction to toric varieties, Princeton Univ. Press, Princeton (1993) [33] A. V. Geramita, Inverse Systems of Fat Points: Waring’s Problem, Secant Varieties and Veronese Varieties and Parametric Spaces of Gorenstein Ideals,In:The Curves Seminar at Queen’s, Vol. X, A. V. Geramita, ed., Queen’s Papers in Pure and Applied Mathematics, 102, Queen’s University, Kingston, ON, 3–114 (1996) [34] A. Grothendieck, Sur quelques points d’alg`ebre homologique,Tˆohoku Math. J. 9, 119–221 (1957) [35] , The cohomology of abstract algebraic varieties,In:Intern. Conf. of Math. at Edin- burgh 1958, Cambridge Univ. Press, Cambridge, 103–118 (1960) [36] , Local Cohomology (Notes by R. Hartshorne), Lectures Notes in Math. 41, Springer, Heidelberg (1967) [37] P. Griffiths and J. Harris, Principles of Algebraic Geometry, John Wiley & Sons, New York (1979) [38] R. Hartshorne, Residues and Duality, Lecture Notes in Math. 20, Springer, Heidelberg (1966) [39] , Algebraic Geometry, Springer, Heidelberg (1977) [40] H. Hasse, Theorie der Differentiale in algebraischen Funktionenk¨orpern mit vollkommenem Konstantenk¨orper, J. Reine Angew. Math. 174, 55–64 (1934) [41] I.-C. Huang, Application of residues to combinatorial identities, Proc. Amer. Math. Soc. 125, 1011–1017 (1997) [42] , An explicit construction of residual complexes,J.Algebra225, 698–739 (2000) [43] , Inverse relations and Schauder bases, J. Combinatorial Theory 97, 203–224 (2002) [44] , Residue methods in combinatorial analysis,In:Local cohomology and its applications (Guanajuato 1999), Lect. Notes in Pure and Appl. Math. 226, Dekker, New York, 255–342 (2002) [45] R. H¨ubl, Graded duality and generalized fractions, J. Pure and Appl. Algebra 141, 225–247 (1999) [46] and E. Kunz, Regular differential forms and duality for projective ,Jour. reine ang. Math. 410, 84–108 (1990) [47] and , On the intersection of algebraic curves and hypersurfaces,Math.Z. 227, 263–278 (1998) [48] and P. Sastry, Regular differential forms and relative duality, Amer. J. Math. 115, 749–787 (1993) [49] C. Huneke, Hyman Bass and ubiquity: Gorenstein rings Algebra, K-theory, groups, and education (New York, 1997), 55–78, Contemp. Math. 243, Amer. Math. Soc., Providence, RI, (1999) [50] G. Humbert, Application g´eom´etrique d’un th´eor`eme de Jacobi,J.Math.4 I, 347–356 (1845) BIBLIOGRAPHY 153

[51] A. Iarrobino, Inverse System of a Symbolic Power II. The Waring Problem for Forms,J. Algebra 174, 1091–1110 (1995) [52] , Inverse system of a symbolic power. III. Thin algebras and fat points,Compositio Math. 108, 319–356 (1997) [53] K. G. Jacobi, Theoremata nova algebraica circa systema duarum aequationem inter duas variabiles propositarium, J. Reine Angew. Math. 14, 281–288 (1835) [54] J.-P. Jouanolou, Le formalisme du r´esultant,Adv.Math.90, 117–263 (1991) [55] , Singularit´es rationnelles du r´esultant,In:Algebraic geometry (Proc. Summer Meet- ing, Univ. Copenhagen, Copenhagen, 1978), Lecture Notes in Math. 732, Springer, Berlin, 183–213 (1979) [56] K. Kaveh, Note on the cohomology ring of spherical varieties and volume polynomial (preprint), arXiv:math/0312503 [57] M. Kersken, Cousin-Komplexe und Nennersysteme,Math.Z.182, 389–402 (1983) [58] , Regul¨are Differentialformen,Manuscriptamath.46, 1–25 (1984) [59] A. Khetan and I. Soprounov, Combinatorial construction of toric residues, Ann. Inst. Fourier (Grenoble) 55, 511–548 (2005) [60] A. G. Khovanskii, Newton polyhedra and the Euler-Jacobi formula, Russian Math. Surveys 33, 237–238 (1978) [61] , Newton polyhedra and toroidal varieties, Anal. Appl. 11, 289–296 (1977) [62] S. L. Kleiman, Relative duality of quasi-coherent sheaves,CompositioMath.41, 39–60 (1980) [63] M. Kreuzer and E. Kunz, Traces in strict Frobenius algebras and strict complete intersections, J. Reine Angew. Math. 381, 181–204 (1987) [64] E. Kunz, K¨ahler Differentials, Advanced Lectures in Math., Vieweg-Verlag, Wiesbaden (1986) [65] , On a formula of Beniamino Segre, Univ. Jagellonicae Acta Math. 39, 17-24 (2001) [66] , Geometric applications of the residue theorem on algebraic curves,In:Algebra, Arithmetic and Geometry with Applications, Papers from Shreeram S. Abhyankar’s 70th birthday conference, C. Christensen, G. Sundaram, A. Sathaye, C. Bajaj, eds., Springer, New York, 565–589 (2003) [67] , Introduction to Plane Algebraic Curves, Translated from the 1991 German edition byRichardG.Belshoff,Birkh¨auser Boston, Boston, MA (2005) [68] and R. Waldi, Regular Differential Forms,Contemp.Math.79,Amer.Math.Soc., Providence, RI (1988) [69] and , Deformations of zerodimensional intersection schemes and residues, Note di Mat. 11, 247–259 (1991) [70] and , Generalization of a theorem of Chasles, Arch. Math. 65, 384–390 (1995) [71] J. Lipman, Dualizing Sheaves, Differentials and Residues on Algebraic Varieties,Ast´erisque 117 (1984) [72] , Lectures on local cohomology and duality,In:Local cohomology and its applications (Guanajuato 1999), Lect. Notes in Pure and Appl. Math. 226, Dekker, New York, 39–89 (2002) [73] , Residues and Traces of Differential Forms via Hochschild Homology, Contemp. Math. 61, Amer. Math. Soc., Providence, RI (1987) [74] , Notes on Derived and Grothendieck Duality, Lecture Notes in Math., Springer, New York, to appear [75] , S. Nayak and P. Sastry, Variance and Duality for Cousin Complexes on Formal Schemes, Contemp. Math. 325, Amer. Math. Soc., Providence, RI (2005) [76] and P. Sastry, Regular differentials and equidimensional scheme maps, J. Algebraic Geometry 1, 101–130 (1992). A sign error is corrected in Correction to: “Regular differentials and equidimensional scheme maps”, J. Algebraic Geometry 14, 593–599 (2005) [77] F. S. Macaulay, The Algebraic Theory of Modular Systems, with a new introduction by Paul Roberts, Cambridge Univ. Press, Cambridge, (1994) (Originally published 1916) [78] , Modern algebra and polynomial ideals,Proc.Camb.Philos.Soc.30, 27–46 (1934) [79] H. Matsumura, Commutative Algebra (second edition), Benjamin, Reading, MA (1980) [80] B. Mourrain, Isolated points, duality and residues, Algorithms for algebra (Eindhoven, 1996), J. Pure Appl. Algebra 117/118, 469–493 (1997) [81] M. Nagata, On the 14-th problem of Hilbert, Amer. J. Math. 81, 766–772, 1959 [82] M. Nagata, On rational surfaces. II, Mem. Coll. Sci. Univ. Kyoto Ser. A Math. 33, 271–293 (1960/1961) 154 BIBLIOGRAPHY

[83] E. Noether, Idealdifferentiation und Differente, J. Reine Angew. Math. 188, 1–21 (1950) [84] T. Oda, Convex Bodies and Algebraic Geometry, Springer, New York (1988) [85] G. Quarg, Uber¨ Durchmesser, Mittelpunkte und Kr¨ummung projektiver algebraischer Va- riet¨aten, Thesis, Regensburg (2001) [86] A. M. Riley, R. Y. Sharp and H. Zakeri, Cousin complexes and generalized fractions,Glasgow Math. J. 26, 51–67 (1985) [87] M. Rosenlicht, Equivalence relations on algebraic curves, Ann. Math. 56, 169–191 (1952) [88] G. Scheja and U. Storch, Residuen bei vollst¨andigen Durchschnitten,Math.Nachr.91 (1979), 157–170. [89] , Uber¨ Spurfunktionen bei vollst¨andigen Durchschnitten, J. Reine Angew. Math. 278/279, 174–190 (1975) [90] B. Segre, Sui teoremi di B´ezout, Jacobi et Reiss, Ann. di Mat. 26, 1–26 (1947) [91] , Some Properties of Differentiable Varieties and Transformations (With Special Ref- erence to the Analytic and Algebraic Case), second edition, Ergebnisse der Math. und ihrer Grenzgebiete, Springer, New York (1971) [92] J.-P. Serre, Sur la cohomologie des vari`et`es alg´ebriques, J. de Maths. Pures et Appl. 36,1–16 (1957) [93] , Groupes alg´ebriques et corps de classes, second edition, Actualit´es scientifiques et industrielles, Hermann, Paris (1975) [94] R. Y. Sharp, Dualizing complexes for commutative noetherian rings, Math. Proc. Cambridge Phil. Soc. 78, 369–386 (1975) [95] and H. Zakeri, Local cohomology and modules of generalized fractions,Mathematika 29, 296–306 (1982) [96] and , Modules of generalized fractions,Mathematika29, 32–41 (1982) [97] B. Siebert, An update on (small) quantum cohomology,In:Mirror symmetry, III (Montreal, PQ, 1995), AMS/IP Stud. Adv. Math. 10, Amer. Math. Soc., Providence, RI, 279–312 (1999) [98] I. Soprounov, Global residues for sparse polynomial systems, J. Pure Appl. Algebra 209, 383–392 (2007) [99] , Toric residue and combinatorial degree, Trans. Amer. Math. Soc. 357, 1963–1975 (2005) [100] B. Sturmfels, Solving systems of polynomial equations, CBMS Regional Conference Series in Mathematics 97, Amer. Math. Soc., Providence, RI (2002) [101] J. Tate, Genus change in inseparable extensions of fields, Proc. Amer. Math. Soc. 3, 400–406 (1952) [102] V. A. Timorin, An analogue of the Hodge-Riemann relations for simple convex polyhedra, Russian Math. Surveys 54, 381–426 (1999) [103] H. Wiebe, Uber¨ homologische Invarianten lokaler Ringe, Math. Ann. 179, 257–274 (1969) [104] A. Yekutieli, An Explicit Construction of the Grothendieck Residue Complex, With an Appendix by P. Sastry,Ast´erisque 202 (1992) [105] , Residues and differential operators on schemes,DukeMath.J.95, 305–341 (1998) [106] A. P. Yuzhakov, On the computation of the complete sum of residues relative to a polynomial mapping in Cn,SovietMath.Dokl.29, 321–324 (1984) [107] O. Zariski and P. Samuel, Commutative algebra. Vol. 1, with the cooperation of I. S. Cohen. Corrected reprinting of the 1958 edition, Graduate Texts in Mathematics 28, Springer-Verlag, New York-Heidelberg-Berlin (1975) Index

abelian sheaf, 1 Cechˇ cohomology action of the torus on an affine toric cochain, cocycle, coboundary, 11 variety, 131 Cechˇ complex acyclic object for a δ-, 3 alternating, normalized, 11 affine sheafified, 12 cone, 47, 101 under refinement, 19 morphism, 20 Cechˇ , 12 residue theorem, 58 centroid (center of mass), 96 toric variety, 131 character, 131 analytic presentation Chow group, 134 as a complete intersection, 72 classical formulas of combinatorics, 46 Apollonius of Perga, 99 Cohen-Macaulay applications point, variety, 86 of residues and duality, 106–109, 112–127 type, 68 of residues in combinatorics, 46 cohomology of the residue theorem, 61–64, 96–99 algebra, 119 of toric residues, 146–148 of the direct image, 20 approach of Scheja-Storch, 79 of the punctured spectrum, 10 , 51 with support in Y ,1 for integral curves, 51 with values in a sheaf, 1 associated sheaf, 1 complementary module, 66 augmented algebra, 74 and completion, 67 complete duality, 46, 111 complete intersection base change, 65, 75 algebra, 71 B´ezoutian, 72, 120 ideal, 115 generalized, 70 subscheme, 61 B´ezout’s formula, 97 complexity, 124, 125 conductor, 67 canonical degree, 113 divisor, 109 cone of a lattice polytope, 131 flasque resolution, 2 constant coefficient differential operator, sheaf, 84 115 trace, 65 constant sheaf, 1 trace of differential forms, 87, 88 Cousin complex, 114 canonical homomorphism criterion of Serre for affineness, 8 ˇ from Cech to , 13 critical degree, 138 from local to global cohomology, 4 curvature of algebraic curves and canonical module, 40, 65, 84 hypersurfaces, 64 of an ideal, 77, 116, 117 of a (local) algebra, 84 Dedekind complementary module Cartier different, 66 divisor, 108 Dedekind’s formula for the conductor and operator, 39 the different, 67

155 156 INDEX degree extension of a sheaf by zero, 2 of a generalized fraction, 42 extension rule for generalized fractions, 33 of a zero-dimensional scheme, 96 exterior of the socle, 60, 77 algebra, 22 degree form, 58 differentiation, 52 δ-functor, 3, 11 Ext groups of sheaves, 110 denominator of a generalized fraction, 32 description of cohomology by generalized facet normal, 134 fractions, 48 field diameter, 99 of Laurent series, 36 of complete intersection curves, 99 of rational functions, 52 of subschemes (cycles) in affine space, 99 finiteness theorem for cohomology, 49 differential finite derivation, 35 forms, 35, 52 Fitting ideal, 70 operator, 115 flasque differentiation module, 1 in the Cechˇ complex, 11 resolution, 2 in the Koszul complex, 22, 23 sheaf, 1 of differential forms, 35 flasque sheaf associated to a sheaf, 2 differents and Jacobians, 72 form of maximal degree (degree form), 58 direct image of a sheaf, 5 Frobenius double complex, 14 algebra, 77 double dual, 85 map, 38 dual fundamental class of a variety, 86 of a lattice, 130 of a sheaf, 85 generalized of a polyhedral cone, 130 B´ezoutian, 70 duality theorem fraction, 32 for Pd,55 Jacobian, 74 for projective varieties, 102 generator of the socle, 29 dualizing germ of a section, 1 complex, 114 geometric genus, 103 form, 119 global module, 40, 65, 84 membership, 123 sheaf, 85 residue, 143 Gorenstein effective computation, 125 algebra, 69 embedding the semigroup algebra into the duality, 77, 78 total coordinate ring, 134 ideal, 77, 116, 117 endomorpphism ring of the canonical point, variety, 86 module, 112 gradient, 63 equality of generalized fractions, 32, 34 Grothendieck residue symbol, 39 ´etale, 66 group of sections with support in Y ,1 Euler characteristic, 50 Hilbert polynomial, 51 derivation, 100 homogeneous cohomology class sequence, 53 generalized fraction, 41, 48 Euler-Cramer paradox, 62 homogenization, 54 Euler-Jacobi in a toric geometry, 135 formula, 62 H¨ubl, R., 63 vanishing condition, 122 hyperplane at infinity, 63 toric generalization, 145, 147 Euler’s formula, 54 incremental quotient, 119 evaluation at the origin, 116 injective exact differentials, 37 object, 1 exactness of the Koszul complex, 26 OX -module, 2 exchange lemma R-module, 6 for separating systems of parameters, 82 sheaf, 3 INDEX 157 integral normal for Pd,55 lattice polytope, 132 for a projective variety, 101 fan of a polytope, 138 interpolation problem, 119, 127 numerator of a generalized fraction, 32 inverse systems, 117, 127 belonging to a divisor, 108 permutation of (quasi-)regular sequences, irregularity of a surface, 103 26 polyhedral cone, 130 Jacobian polynomial solutions determinant, 43, 126 of differential equations, 116 formula, 62 presentation as a complete intersection, 71 ideal, 70 principal part of a Laurent series, 36 projective K¨ahler , 63 different, 70 toric variety, 132 differential forms, 35 punctured spectrum, 9 Koszul complex, 23 quantum cohomology ring, 119 lattice, 130 quasicoherent sheaf, 8 lattice polytope, 131 quasiregular sequence, 26 Laurent quotient property of residues, 40 expansion of a differential, 107 series, 36 ramification index, locus, point, 109 induced by the residue, 40 rational differential form, 52 Lipman, J., 86 reducedness of zero-dimensional complete local cohomology intersections, 61 at a point x,1 refinement of a covering, 19 and completion, 56 regular differential forms, 84 local duality theorem at regular points, 84 for power series rings, 45 at Cohen-Macaulay (Gorenstein) points, for complete reduced local rings, 94 86 on integral varieties, 95 at complete intersection points, 87 local criterion of flatness, 75 ontheaffinecone,101 on toric varieties, 131, 137, 139 Macaulay’s inverse systems, 117, 127 regular differentials in the sense of Matlis dual, 127 Rosenlicht, 87 maximal Cohen-Macaulay module, 31 regular sequence, 26 membership Reiss relation, 64 problem, 123 relation in the radical of a complete intersection, between the arithmetic and the 124 geometric genus, 103 in zero-dimensional ideals, 124 between the Cechˇ and the Koszul midpoint, 99 complex, 29 minimal number of generators of the between toric and local residues, 143 canonical module, 68 to residues on smooth curves, 96 Minkowski sum, 144 representative of the B´ezoutian, 120 module of differentials, 35, 52 residual complex, 114 M-quasiregular sequence, 26 residue M-regular sequence, 26 at a closed point of a variety, 95 multiplicative group, 131 at a point x,56 multiplicity of a variety at a point, 96 at a singular point of a curve, 107 at the vertex of the affine cone, 56, 101 Nagata’s conjecture, 127 for convergent power series, 46 Newton, 99 for power series, 39 polytope, 144 map, 39, 93 Noether different, 70 of an exact differential, 37 Noether normalization, 81, 123 on smooth curves, 96 of an analytic algebra, 82 sum, 57 Noether position, 123 symbol, 39, 79, 115 158 INDEX residue formula G. Humbert, 64 for the multiplicity, 96 Pappus, 62 for centroids, 97 Pascal, 62 Reiss, 64 residue theorem Riemann-Roch for projective curves, 108 affine version, 58 toric d for P ,57 affine variety, 131 for projective varieties, 104 Euler-Jacobi vanishing theorem, 145, 147 for projective curves, 107 homogeneous coordinates, 133 residues and duality for proper varieties, projective variety, 132 106 quotient property, 148 restriction property of residues, 40 residue, 133, 138, 140 retraction, 74 residues and resultants, 149 Riemann-Hurwitz genus formula, 109 transformation law, 140 right-, 1 torsion of differential forms, 86 , 1 torus, 129 torus-invariant Weil divisor, 133 scheme at infinity, 63 total section of a sheaf, 1 complex of a double complex, 14 semigroup algebra, 130 coordinate ring, 133 of a lattice polytope, 132 ramification number, 109 separable Noether normalization, 81 sum of residues, 124 of analytic algebras, 82 trace in the graded case, 100 mapofanalgebra,65 separated scheme, 16 of an endomorphism, 65 separating system of parameters, 82 trace formula Serre’s finiteness conditions, 49 for residues, 93, 95, 104 sheaf of Hasse, 96 associated to a module, 1, 47 transformation law for residues, 40 of differential forms, 52 transition determinant, 26 of rational d-forms, 52 transitive law of regular d-forms, 84 for the complementary module and the of Zariski d-forms, 86 different, 67 simplicial toric variety, 143 for dualizing sheaves, 88 singularity degree, 113 for the trace, 65 skyscraper sheaf, 9 transitive properties of regular differential socle forms, 87, 106 degree, 60, 77, 117 transversal intersection, 61 of a zero-dimensional local or graded ring, 29 universal standard derivation, 35, 52 d covering of Pk,57 module of differentials, 35 d-simplex, 134 universally finite scalar product, 63 derivation, 35 trace, 65 module of differentials, 35 strict complete intersection, 119 support vertex of the affine cone, 47, 101 of a divisor, 108 very ample sheaf, 49 of a section, 1 volume polynomial, 119

T -acyclic object, 3 Waring’s problem, 127 tangent, 63 Weierstraß preparation theorem, 91 tangent hyperplane, 61 weighted case, 61, 121 tangential center, 99 Zariski differentials, 86 Tate trace, 73 zero-dimensional complete intersection, 58 theorem of B´ezout, 97 Cayley-Bacharach, 62 Chasles, 99 Titles in This Series

47 Ernst Kunz (with the assistance of and contributions by David A. Cox and Alicia Dickenstein), Residues and duality for projective algebraic varieties, 2008 46 Lorenzo Sadun, Topology of tiling spaces, 2008 45 Matthew Baker, Brian Conrad, Samit Dasgupta, Kiran S. Kedlaya, and Jeremy Teitelbaum (David Savitt and Dinesh S. Thakur, Editors), p-adic geometry: Lectures from the 2007 Arizona Winter School, 2008 44 Vladimir Kanovei, Borel equivalence relations: structure and classification, 2008 43 Giuseppe Zampieri, Complex analysis and CR geometry, 2008 42 Holger Brenner, J¨urgen Herzog, and Orlando Villamayor (Juan Elias, Teresa Cortadellas Ben´ıtez, Gemma Colom´e-Nin, and Santiago Zarzuela, Editors), Three Lectures on Commutative Algebra, 2008 41 James Haglund, The q, t-Catalan numbers and the space of diagonal harmonics (with an appendix on the combinatorics of Macdonald polynomials), 2008 40 Vladimir Pestov, Dynamics of infinite-dimensional groups. The Ramsey–Dvoretzky– Milman phenomenon, 2006 39 , The moduli problem for plane branches (with an appendix by Bernard Teissier), 2006 38 Lars V. Ahlfors, Lectures on Quasiconformal Mappings, Second Edition, 2006 37 Alexander Polishchuk and Leonid Positselski, Quadratic algebras, 2005 36 Matilde Marcolli, Arithmetic , 2005 35 Luca Capogna, Carlos E. Kenig, and Loredana Lanzani, Harmonic measure: Geometric and analytic points of view, 2005 34 E. B. Dynkin, Superdiffusions and positive solutions of nonlinear partial differential equations, 2004 33 Kristian Seip, Interpolation and sampling in spaces of analytic functions, 2004 32 Paul B. Larson, The stationary tower: Notes on a course by W. Hugh Woodin, 2004 31 John Roe, Lectures on coarse geometry, 2003 30 Anatole Katok, Combinatorial constructions in ergodic theory and dynamics, 2003 29 Thomas H. Wolff (IzabellaLaba  and Carol Shubin, editors), Lectures on harmonic analysis, 2003 28 Skip Garibaldi, Alexander Merkurjev, and Jean-Pierre Serre, Cohomological invariants in , 2003 27 Sun-Yung A. Chang, Paul C. Yang, Karsten Grove, and Jon G. Wolfson, Conformal, Riemannian and Lagrangian geometry, The 2000 Barrett Lectures, 2002 26 Susumu Ariki, Representations of quantum algebras and combinatorics of Young tableaux, 2002 25 William T. Ross and Harold S. Shapiro, Generalized , 2002 24 Victor M. Buchstaber and Taras E. Panov, Torus actions and their applications in topology and combinatorics, 2002 23 Luis Barreira and Yakov B. Pesin, Lyapunov exponents and smooth ergodic theory, 2002 22 Yves Meyer, Oscillating patterns in image processing and nonlinear evolution equations, 2001 21 Bojko Bakalov and Alexander Kirillov, Jr., Lectures on tensor categories and modular functors, 2001 20 Alison M. Etheridge, An introduction to superprocesses, 2000 19 R. A. Minlos, Introduction to mathematical statistical physics, 2000 18 Hiraku Nakajima, Lectures on Hilbert schemes of points on surfaces, 1999 TITLES IN THIS SERIES

17 Marcel Berger, during the second half of the twentieth century, 2000 16 Harish-Chandra, Admissible invariant distributions on reductive p-adic groups (with notes by Stephen DeBacker and Paul J. Sally, Jr.), 1999 15 Andrew Mathas, Iwahori-Hecke algebras and Schur algebras of the symmetric group, 1999 14 Lars Kadison, New examples of Frobenius extensions, 1999 13 Yakov M. Eliashberg and William P. Thurston, Confoliations, 1998 12 I. G. Macdonald, Symmetric functions and orthogonal polynomials, 1998 11 Lars G˚arding, Some points of analysis and their history, 1997 10 Victor Kac, Vertex algebras for beginners, Second Edition, 1998 9 Stephen Gelbart, Lectures on the Arthur-Selberg trace formula, 1996 8 Bernd Sturmfels, Gr¨obner bases and convex polytopes, 1996 7 Andy R. Magid, Lectures on differential , 1994 6 Dusa McDuff and Dietmar Salamon, J-holomorphic curves and quantum cohomology, 1994 5 V. I. Arnold, Topological invariants of plane curves and caustics, 1994 4 David M. Goldschmidt, Group characters, symmetric functions, and the Hecke algebra, 1993 3 A. N. Varchenko and P. I. Etingof, Why the boundary of a round drop becomes a curve of order four, 1992 2 Fritz John, Nonlinear wave equations, formation of singularities, 1990 1 Michael H. Freedman and Feng Luo, Selected applications of geometry to low-dimensional topology, 1989 This book, which grew out of lectures by E. Kunz for students with a background in algebra and algebraic geometry, develops local and global duality theory in the special case of (possibly singular) algebraic varieties over algebraically closed base fields. It describes duality and residue theorems in terms of Kähler differential forms and their resi- dues. The properties of residues are introduced via local cohomology. Special emphasis is given to the relation between residues to classical results of algebraic geometry and their generalizations. The contribution by A. Dickenstein gives applications of residues and duality to polynomial solutions of constant coefficient partial differential equations and to problems in interpolation and ideal membership. D. A. Cox explains toric residues and relates them to the earlier text. The book is intended as an introduction to more advanced treatments and further applica- tions of the subject, to which numerous bibliographical hints are given.

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