Cohomology of vector bundles and syzygies

384 Pages·2003·1.566 MB·English

by  Jerzy Weyman

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 commuta-
tive 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
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© Jerzy Weyman 2003
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without the written permission of Cambridge University Press.
First published in print format 2003
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ISBN-13 978-0-521-62197-7 hardback
ISBN-10 0-521-62197-6 hardback
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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