Representations of Elementary Abelian p-Groups and Vector Bundles

348 Pages·2016·2.18 MB·English

by  David J. Benson

CAMBRIDGE TRACTS IN MATHEMATICS
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208RepresentationsofElementaryAbelianp-GroupsandVectorBundles
CAMBRIDGE TRACTS IN MATHEMATICS
GENERAL EDITORS
B. BOLLOBÁS, W. FULTON, F. KIRWAN,
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Representations of Elementary Abelian
p-Groups and Vector Bundles
DAVID J. BENSON
UniversityofAberdeen
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DOI:10.1017/9781316795699
(cid:2)c DavidJ.Benson2017
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Firstpublished2017
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LibraryofCongressCataloging-in-PublicationData
Names:Benson,D.J.(DavidJ.),1955–
Title:Representationsofelementaryabelianp-groupsandvectorbundles/
DavidJ.Benson,UniversityofAberdeen.
Description:Cambridge:CambridgeUniversityPress,[2017]|
Series:Cambridgetractsinmathematics;208|
Includesbibliographicalreferencesandindex.
Identifiers:LCCN2016046127|ISBN9781107174177(alk.paper)
Subjects:LCSH:Abelianp-groups.|Abeliangroups.|Vectorbundles.
Classification:LCCQA180.B462017|DDC512/.25–dc23
LCrecordavailableathttps://lccn.loc.gov/2016046127
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Contents
Preface pagexi
Introduction xiii
1 ModularRepresentationsandElementaryAbelianGroups 1
1.1 Introduction 1
1.2 RepresentationType 1
1.3 ShiftedSubgroups 3
1.4 TheLanguageofπ-Points 5
1.5 TheStableModuleCategory 6
1.6 TheDerivedCategory 8
1.7 SingularityCategories 10
1.8 CohomologyofElementaryAbelian p-Groups 12
1.9 Chouinard’sTheorem,Dade’sLemmaandRankVarieties 15
1.10 Carlson’s Lζ Modules,andaMatrixVersion 18
1.11 DiagramsforModules 20
1.12 TensorProducts 24
1.13 Duality 26
1.14 SymmetricandExteriorPowers 28
1.15 SchurFunctions 29
1.16 SchurFunctors 31
1.17 RadicalLayersofkE 33
1.18 TwistedVersionsofkE 35
2 CyclicGroupsofOrder p 38
2.1 Introduction 38
2.2 ModulesforZ/p 38
2.3 TensorProducts 39
2.4 GaussianPolynomials 41
2.5 GeneralisedGaussianPolynomialsandaHookFormula 42
v
vi Contents
2.6 λ-RingsandRepresentationsof SL(2,C) 45
2.7 TheRepresentationRingofZ/p 47
2.8 SymmetricandExteriorPowersofJordanBlocks 50
2.9 SchurFunctorsfor SL(2,C)andZ/p 52
3 BackgroundfromAlgebraicGeometry 54
3.1 AffineSpaceandAffineVarieties 54
3.2 GenericPointsandClosedPoints 55
3.3 ProjectiveSpaceandProjectiveVarieties 57
3.4 TangentSpaces 59
3.5 PresheavesandSheaves 60
3.6 StalksandSheafification 62
3.7 TheLanguageofSchemes 63
3.8 SheavesofModules 64
3.9 CoherentSheavesonProjectiveVarieties 65
3.10 CohomologyofSheaves 69
4 JordanType 72
4.1 Nilvarieties 72
4.2 MatricesandTangentSpaces 75
4.3 ATheoremofGerstenhaber 76
4.4 DominanceOrderandNilpotentJordanTypes 78
4.5 GenericandMaximalJordanType 83
4.6 TensorProducts 86
5 ModulesofConstantJordanType 89
5.1 IntroductionandDefinitions 89
5.2 HomogeneousModules 94
5.3 AnExactCategory 98
5.4 EndotrivialModules 101
5.5 WildRepresentationType 103
5.6 TheConstantImageProperty 106
5.7 TheGenericKernel 110
5.8 TheSubquotientRad−1K(M)/Rad2K(M) 112
5.9 TheConstantKernelProperty 114
5.10 TheGenericImage 116
5.11 W-Modules 118
5.12 ConstantJordantypewithOneNon-ProjectiveBlock 119
5.13 Rickard’sConjecture 120
5.14 ConsequencesandVariations 123
5.15 FurtherConjectures 127
6 VectorBundlesonProjectiveSpace 129
6.1 DefinitionsandFirstProperties 129
6.2 TestsforVectorBundles 131
Contents vii
6.3 VectorBundlesonProjectiveSpace 134
6.4 TheTangentBundleandtheEulerSequence 136
6.5 HomogeneityandUniformity 136
6.6 MonadsandSubquotients 138
6.7 TheNullCorrelationBundle 139
6.8 TheExamplesofTango 140
6.9 CohomologyofProjectiveSpace 141
6.10 DifferentialFormsandBott’sTheorem 142
6.11 Simplicity 143
6.12 Hilbert’sSyzygyTheorem 146
7 ChernClasses 149
7.1 ChernClassesofGradedModules 149
7.2 ChernClassesofCoherentSheavesonPr−1 151
7.3 SomeComputations 154
7.4 RestrictionofVectorBundles 155
7.5 ChernNumbersofTwistsandDuals 157
7.6 ChernRoots 159
7.7 PowerSums 160
7.8 TheHirzebruch–Riemann–RochTheorem 166
7.9 ChernNumbersandtheFrobeniusMap 169
8 ModulesofConstantJordanTypeandVectorBundles 172
8.1 Introduction 172
8.2 TheOperatorθ 173
8.3 TheActionofθ onFibres 174
8.4 TheFunctorsFi andFi,j 176
8.5 TwistsandSyzygies 181
8.6 ChernNumbersofF (M) 183
i
8.7 TheConstruction: p =2 185
8.8 TheConstruction: pOdd 187
8.9 ProofoftheRealisationTheorem 189
8.10 Functoriality 192
8.11 TensorProducts 192
8.12 NegativeTateCohomology 194
8.13 TheBGGCorrespondence 195
9 Examples 199
9.1 Modulesfor(Z/2)2 199
9.2 Modulesfor(Z/p)2 201
9.3 LargerRank 203
9.4 Nilvarieties 204
9.5 TheTangentandCotangentBundles 205
9.6 TheNullCorrelationBundle, p =2 207
viii Contents
9.7 TheNullCorrelationBundle, pOdd 208
9.8 InstantonBundles 210
9.9 Schwarzenberger’sBundles 212
9.10 TheExamplesofTango 214
9.11 TheHorrocks–MumfordBundle 215
9.12 AutomorphismsoftheHorrocks–MumfordBundle 218
9.13 RealisingtheHorrocks–MumfordBundle 223
9.14 TheHorrocksParentBundleandtheTangoBundle 226
10 RestrictionsComingfromChernNumbers 232
10.1 MatricesofConstantRank 232
10.2 CongruencesonChernNumbers 235
10.3 RestrictionsonStableJordanType, pOdd 237
10.4 EliminatingMoreStableJordanTypes 238
10.5 RestrictionsonJordanTypefor p =2 240
10.6 ApplyingHirzebruch–Riemann–Rochfor p = 2:The
Casem =0 244
10.7 BypassingHirzebruch–Riemann–Roch 246
10.8 ApplyingandBypassingHirzebruch–Riemann–Rochfor
p =2:TheCase1≤m ≤r −3 246
10.9 NilvarietiesofConstantJordanType[p]n for p ≥3 249
10.10 NilvarietieswithaSingleJordanBlock 251
10.11 BabylonianTowers 253
11 Orlov’sCorrespondence 255
11.1 Introduction 255
11.2 MaximalCohen–MacaulayModules 257
11.3 TheOrlovCorrespondence 259
11.4 TheFunctors 261
11.5 AnExample 263
11.6 TheBidirectionalKoszulComplex 264
11.7 ABimoduleResolution 267
11.8 TheAdjunction 269
11.9 TheEquivalence 269
11.10 TheTrivialModule 270
11.11 ComputerAlgebra 272
11.12 Cohomology 272
11.13 TwistedVersionsofkE 274
12 PhenomenologyofModulesoverElementaryAbelian p-Groups 276
12.1 Introduction 276
12.2 ModuleConstructions 278
12.3 OddPrimesAreMoreDifficult 280
12.4 RelativeCohomology 281