Group Cohomology and Algebraic Cycles

246 Pages·2014·1.348 MB·English

by  Totaro B.

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CAMBRIDGE TRACTS IN MATHEMATICS
GeneralEditors
B. BOLLOBA´ S, W. FULTON, A. KATOK, F. KIRWAN,
P. SARNAK, B. SIMON, B. TOTARO
204 GroupCohomologyandAlgebraicCycles
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CAMBRIDGE TRACTS IN MATHEMATICS
GENERAL EDITORS
B.BOLLOBA´S,W.FULTON,A.KATOK,F.KIRWAN,P.SARNAK,
B.SIMON,B.TOTARO
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CUUS2102-BOOK CUUS2102/Totaro 9781107015777 January8,2014 9:28
Group Cohomology and
Algebraic Cycles
BURT TOTARO
UniversityofCalifornia,LosAngeles
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Contents
Preface pagexi
1 GroupCohomology 1
1.1 Definitionofgroupcohomology 1
1.2 Equivariantcohomologyandbasiccalculations 3
1.3 Algebraicdefinitionofgroupcohomology 6
2 TheChowRingofaClassifyingSpace 8
2.1 TheChowgroupofalgebraiccycles 8
2.2 TheChowringofaclassifyingspace 11
2.3 TheequivariantChowring 16
2.4 Basiccomputations 18
2.5 Transfer 22
2.6 Becker-GottliebtransferforChowgroups 24
2.7 Groupsincharacteristicp 26
2.8 Wreathproductsandthesymmetricgroups 28
2.9 Generallineargroupsoverfinitefields 30
2.10 QuestionsabouttheChowringofafinitegroup 31
3 DepthandRegularity 35
3.1 Depthandregularityintermsoflocalcohomology 35
3.2 Depthandregularityintermsofgeneratorsandrelations 41
3.3 Duflot’slowerboundfordepth 46
4 RegularityofGroupCohomology 49
4.1 Regularityofgroupcohomologyandapplications 49
4.2 ProofofSymonds’stheorem 50
5 GeneratorsfortheChowRing 56
5.1 BoundingthegeneratorsoftheChowring 56
5.2 Optimalityofthebounds 59
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viii Contents
6 RegularityoftheChowRing 62
6.1 BoundingtheregularityoftheChowring 62
6.2 Motiviccohomology 70
6.3 Steenrodoperationsonmotiviccohomology 72
6.4 Regularityofmotiviccohomology 73
7 Boundsfor p-Groups 79
7.1 InvarianttheoryofthegroupZ/p 80
7.2 Wreathproducts 82
7.3 BoundsfortheChowringandcohomologyofap-group 85
8 TheStructureofGroupCohomologyandtheChowRring 87
8.1 Thenormmap 88
8.2 Quillen’stheoremandYagita’stheorem 90
8.3 Yagita’stheoremoveranyfield 97
8.4 Carlson’stheoremontransfer 99
9 CohomologymodTransfersisCohen-Macaulay 104
9.1 TheCohen-Macaulayproperty 104
9.2 Theringofinvariantsmodulotraces 109
10 BoundsforGroupCohomologyandtheChowRingModulo
Transfers 113
11 TransferredEulerClasses 122
11.1 BasicpropertiesoftransferredEulerclasses 123
11.2 GeneratingtheChowring 126
12 DetectionTheoremsforCohomologyandChowRings 127
12.1 Nilpotenceingroupcohomology 128
12.2 ThedetectiontheoremforChowrings 131
13 Calculations 140
13.1 TheChowringsofthegroupsoforder16 141
13.2 Themodularp-group 150
13.3 CentralextensionsbyG 154
m
13.4 TheextraspecialgroupE 156
p3
13.5 Calculationsofthetopologicalnilpotencedegree 162
14 GroupsofOrder p4 174
14.1 ThewreathproductZ/3(cid:3)Z/3 174
14.2 Geometricandtopologicalfiltrations 176
14.3 Groupsoforderp4forp ≥5 178
14.4 Groupsoforder81 180
14.5 A1-dimensionalgroup 182