Representations and Cohomology: Volume 2, Cohomology of Groups and ModulesCambridge University Press, 22. kol 1991. - Broj stranica: 288 This is the second of two volumes which will provide an introduction to modern developments in the representation theory of finite groups and associative algebras. The subject is viewed from the perspective of homological algebra and the theory of representations of finite dimensional algebras; the author emphasises modular representations and the homological algebra associated with their categories. This volume concentrates on the cohomology of groups, always with representations in view, however. It begins with a background reference chapter, then proceeds to an overview of the algebraic topology and K-theory associated with cohomology of groups, especially the work of Quillen. Later chapters look at algebraic and topological proofs of the finite generation of the cohomology ring of a finite group, and an algebraic approach to the Steenrod operations in group cohomology. The book cumulates in a chapter dealing with the theory of varieties for modules. Much of the material presented here has never appeared before in book form. Consequently students and research workers studying group theory, and indeed algebra in general, will be grateful to Dr Benson for supplying an exposition of a good deal of the essential results of modern representation theory. |
Sadržaj
Background from algebraic topology | 1 |
12 Homotopy groups | 2 |
13 The Hurewicz theorem | 9 |
14 The Whitehead theorem | 11 |
15 CWcomplexes and cellular homology | 12 |
16 Fibrations and fibre bundles | 16 |
17 Paracompact spaces | 21 |
18 Simplicial sets | 23 |
46 Adem relations | 144 |
47 Serres theorem on products of Bocksteins | 148 |
48 Steenrod operations and spectral sequences | 150 |
Varieties for modules and multiple complexes | 153 |
52 Restriction to elementary abelian subgroups | 155 |
53 Poincare series and complexity | 157 |
54 Varieties and commutative algebra | 161 |
extraspecial 2groups | 169 |
19 The Milnor exact sequence | 27 |
Cohomology of groups | 29 |
22 EilenbergMac Lane spaces | 32 |
23 Principal Gbundles | 35 |
24 Classifying spaces | 37 |
25 Ktheory | 44 |
26 Characteristic classes | 48 |
27 Transfer | 51 |
28 Stable cohomotopy and the Segal conjecture | 56 |
29 Cohomology of general linear groups | 60 |
210 The plus construction and algebraic Ktheory | 68 |
211 Hochschild homology | 73 |
212 Free loops on BG | 77 |
213 Cyclic homology | 80 |
214 Cyclic sets | 85 |
215 Extended centralisers | 90 |
Spectral sequences | 93 |
32 The spectral sequence of a filtered chain complex | 98 |
33 The spectral sequence of a fibration | 104 |
34 The spectral sequence of a double complex | 106 |
35 The spectral sequence of a group extension | 109 |
36 The Kiinneth spectral sequence | 111 |
37 The EilenbergMoore spectral sequence | 112 |
38 The Atiyah spectral sequence | 114 |
39 Products in spectral sequences | 115 |
310 Equivariant cohomology and finite generation | 117 |
The Evens norm map and the Steenrod algebra | 121 |
42 Finite generation of cohomology | 126 |
43 The Bockstein homomorphism | 132 |
44 Steenrod operations | 136 |
45 Proof of the properties | 138 |
56 The Quillen stratification | 172 |
57 Varieties for modules | 176 |
58 Rank varieties | 180 |
59 The modules L | 186 |
510 Periodic modules | 191 |
511 Andrews theorem | 192 |
512 The variety of an indecomposable kGmodule is connected | 194 |
dihedral 2groups | 195 |
514 Multiple complexes | 199 |
515 Gaps in group cohomology | 205 |
516 Isomorphisms in group cohomology | 208 |
517 Poincare duality | 209 |
518 CohenMacaulay cohornology rings | 211 |
Group actions and the Steinberg module | 215 |
62 Gposets | 217 |
63 The Lefschetz Invariant | 218 |
64 Equivariant homotopy | 220 |
65 Quillens lemma | 222 |
66 Equivalences of subgroup complexes | 224 |
67 The generalised Steinberg module | 226 |
a crash course | 228 |
69 Steinberg module inversion and Alperins conjecture | 233 |
Local coefficients on subgroup complexes | 237 |
72 Constructions on coefficient systems | 239 |
73 Chain complexes and homology of coefficient systems | 242 |
74 Symplectic and orthogonal groups | 243 |
75 Smiths theorem and universal coefficient systems | 246 |
251 | |
269 | |
Ostala izdanja - Prikaži sve
Representations and Cohomology: Volume 2, Cohomology of Groups and Modules D. J. Benson Pregled nije dostupan - 1991 |
Representations and Cohomology: Volume 2, Cohomology of Groups and Modules D. J. Benson Pregled nije dostupan - 1991 |
Representations and Cohomology: Volume 2, Cohomology of Groups and Modules D. J. Benson Pregled nije dostupan - 1991 |
Uobičajeni izrazi i fraze
abelian group Algebra base space basepoint Berlin/New York Bockstein chain complex Chevalley group classes coefficient system cohomology of groups cohomology rings composite Corollary corresponding CW-complex cyclic homology D. J. Benson defined DEFINITION degree denote diagram dimension dimensional double complex element example ExtRG fibration finite groups follows FP+¹X functor G-action G-bundle G-poset given group cohomology hence Ho(FM Hochschild complex homomorphism homotopy equivalence Hurewicz inclusion induces isomorphism J. F. Carlson K-theory kG-module lemma long exact sequence map f Math max(A nilpotent Noetherian non-zero obtain p-group p-subgroups paracompact polynomial poset principal G-bundle projective resolution PROOF Proposition Quillen quotient representation resp Section Serre short exact sequence simplex simplicial complex simplicial set spectral sequence Steenrod operations subgroup of G Suppose tensor theorem theory topological space topology trivial variety vector bundle VG(M Volume zero