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COHOMOLOGY OF VECTOR BUNDLES AND SYZYGIES The central theme of this book is an exposition of the geometric technique of calculating syzygies. It is written from the point of view of commutative algebra; without assuming any knowledge of representation theory, the calculation of syzygies of determinantal varieties is explained. The starting pointisadefinitionofSchurfunctors,andthesearediscussedfrombothan algebraicandageometricpointofview.Thenachapteronvariousversions ofBott’stheoremleadstoacarefulexplanationofthetechniqueitself,based on a description of the direct image of a Koszul complex. Applications to determinantal varieties follow. There are also chapters on applications of thetechniquetorankvarietiesforsymmetricandskewsymmetrictensorsof arbitrarydegree,closuresofconjugacyclassesofnilpotentmatrices,discrim- inants, and resultants. Numerous exercises are included to give the reader insightintohowtoapplythisimportantmethod. CAMBRIDGETRACTSINMATHEMATICS GeneralEditors B.BOLLOBAS,W.FULTON,A.KATOK,F.KIRWAN, P.SARNAK 149 Cohomology of Vector Bundles and Syzygies JerzyWeyman NortheasternUniversity Cohomology of Vector Bundles and Syzygies    Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge  , United Kingdom Published in the United States by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521621977 © Jerzy Weyman 2003 This book is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2003 ISBN-13 978-0-511-06601-6 eBook (NetLibrary) ISBN-10 0-511-06601-5 eBook (NetLibrary) ISBN-13 978-0-521-62197-7 hardback ISBN-10 0-521-62197-6 hardback Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this book, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. ToKatarzyna Contents Preface pagexi 1 IntroductoryMaterial 1 1.1 MultilinearAlgebraandCombinatorics 1 1.2 HomologicalandCommutativeAlgebra 12 1.3 DeterminantsofComplexes 27 2 SchurFunctorsandSchurComplexes 32 2.1 SchurFunctorsandWeylFunctors 32 2.2 SchurFunctorsandHighestWeightTheory 49 2.3 PropertiesofSchurFunctors.CauchyFormulas,Littlewood– RichardsonRule,andPlethysm 57 2.4 TheSchurComplexes 66 ExercisesforChapter2 78 3 GrassmanniansandFlagVarieties 85 3.1 ThePlu¨ckerEmbeddings 85 3.2 TheStandardOpenCoveringsofFlagManifolds andtheStraighteningLaw 91 3.3 TheHomogeneousVectorBundlesonFlagManifolds 98 ExercisesforChapter3 104 4 Bott’sTheorem 110 4.1 TheFormulationofBott’sTheoremfortheGeneral LinearGroup 110 4.2 TheProofofBott’sTheoremfortheGeneral LinearGroup 117 4.3 Bott’sTheoremforGeneralReductiveGroups 123 ExercisesforChapter4 132 5 TheGeometricTechnique 136 5.1 TheFormulationoftheBasicTheorem 137 5.2 TheProofoftheBasicTheorem 141 5.3 TheProofofPropertiesofComplexes F(V)• 146 ix x Contents 5.4 TheG-EquivariantSetup 149 5.5 TheDifferentialsinComplexes F(V)•. 152 5.6 DegenerationSequences 154 ExercisesforChapter5 156 6 TheDeterminantalVarieties 159 6.1 TheLascouxResolution 160 6.2 TheResolutionsofDeterminantalIdeals inPositiveCharacteristic 168 6.3 TheDeterminantalIdealsforSymmetricMatrices 175 6.4 TheDeterminantalIdealsforSkewSymmetricMatrices 187 6.5 ModulesSupportedinDeterminantalVarieties 195 6.6 ModulesSupportedinSymmetric DeterminantalVarieties 209 6.7 ModulesSupportedinSkewSymmetric DeterminantalVarieties 213 ExercisesforChapter6 218 7 HigherRankVarieties 228 7.1 BasicProperties 228 7.2 RankVarietiesforSymmetricTensors 234 7.3 RankVarietiesforSkewSymmetricTensors 239 ExercisesforChapter7 245 8 TheNilpotentOrbitClosures 251 8.1 TheClosuresofConjugacyClassesofNilpotentMatrices 252 8.2 TheEquationsoftheConjugacyClasses ofNilpotentMatrices 263 8.3 TheNilpotentOrbitsforOtherSimpleGroups 278 8.4 ConjugacyClassesfortheOrthogonalGroup 283 8.5 ConjugacyClassesfortheSymplecticGroup 296 ExercisesforChapter8 309 9 ResultantsandDiscriminants 313 9.1 TheGeneralizedResultants 314 9.2 TheResultantsofMultihomogeneousPolynomials 318 9.3 TheGeneralizedDiscriminants 328 9.4 TheHyperdeterminants 332 ExercisesforChapter9 355 References 359 NotationIndex 367 SubjectIndex 369 Preface Thisbookisdevotedtothegeometrictechniqueofcalculatingsyzygies.This techniqueoriginatedwithGeorgeKempfandwasfirstusedsuccessfullyby AlainLascouxforcalculatingsyzygiesofdeterminantalvarieties.Sincethen ithasbeenappliedinstudyingthedefiningidealsofvarietieswithsymme- triesthatplayacentralroleingeometry:determinantalvarieties,closuresof nilpotentorbits,anddiscriminantandresultantvarieties. Thecharacterofthemethodmakesitcomparabletothesymbolicmethodin classicalinvarianttheory.Itworksinonlyalimitednumberofcases,butwhen itdoes,itgivesacompleteanswertotheproblemofcalculatingsyzygies,and thisanswerishardtogetbyothermeans. Even though the basic idea is more than 20 years old, this is the first booktreatingthegeometrictechniqueindetail.Thishappensbecauseauthors usingthegeometricmethodhaveusuallybeeninterestedmoreinthespecial casestheystudiedthaninthemethoditself,andtheyhaveusedonlytheaspects ofthemethodtheyneeded.Thereforethebasictheoremsfromchapter5stem fromtheeffortsofseveralmathematicians. The possibilities offered by the geometric method are not exhausted by theexamplestreatedinthebook.Themethodcanbefruitfullyappliedtoany representationofalinearlyreductivegroupwithfinitelymanyorbitsandwith actionssuchthattheorbitscanbedescribedexplicitly.Thevarietiestreated inchapters7and9showthatthescopeofthemethodisnotlimitedtosuch actions. The book is written from the point of view of a commutative algebraist. Wedeveloptherudimentsofrepresentationtheoryofgenerallineargroupin some detail in chapters 2–4. At the same time we assume some knowledge ofcommutativealgebraandalgebraicgeometry,includingsheavesandtheir cohomology—forexample,thenotionscoveredinchaptersIIandIIIofthe book [H1] of Hartshorne. The notions of Cohen–Macaulay and Gorenstein xi