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Complementary References

A. ADEM and R.J. MILGRAM, Cohomology of Finite Groups, Springer-Verlag 1994 K. BROWN, Cohomology of Groups, Springer-Verlag 1982 S. LANG, Algebraic Number Theory, Addison-Wesley 1970; Springer• Verlag 1986, 2nd edn. 1994 S. MAC LANE, Homology, Springer-Verlag 1963, 4th printing 1995 Table of Notation

Apoo : Elements of A annihilated by a power of p A

Am : Kernel of the homomorphism rnA :: A --+ A such that a 1--+ rna cd : Cohomological dimension r, : Z/pZ G: Character group, Hom(G, Q/Z) }{G : Natural homomorphism of AG onto HO (G, A) or HO (G, A)

JKG : Natural homomorphism of As into H-I(G,A) Galm(G) : Galois modules

Galmp ( G) : Galois modules whose elements are annihilated by a p-power Galmtor(G) : Torsion Galois modules GC : Commutator group, or closure of commutator if G is topological

Gp : p-Sylow subgroup of G Grab: Category of abelian groups 2 h l / 2 : Herbrand quotient, order of H divided by order of HI HG : Functor such that HG(A) = AG HG : Functor such that HG(A) = AG/SGA

I G : Augmentation ideal, generated by the elements a - e, a E G 222

Mo(A) : Functions (sometimes continuous) from G into A Mo: Z[G] 0A Mg : Induced functions Mod(G) : Abelian category of G-modules Mod(Z) : Abelian category of abelian groups sed : Strict cohomological dimension So : The relative trace, from a subgroup U of finite index, to G So : The trace, for a finite group G Tr : Transfer of group theory tr : Transfer of cohomology Z[G] : Group ring Index

Abutment of spectral sequence 117 Admissible subgroup 177 Augmentation 10, 27 Augmented cupping 109, 198 Bilinear map of complexes 84 Brauer group 167, 202, 209 Category of modules 10 Characters 28 Class formation 166 Class module 71 Coerasing functor 5 Cofunctor 4 Cohomological cup functor 76 Cohomological dimension 138 Cohomological period 96 Cohomology ring 89 Complete resolution 23 Conjugation 41, 174 Consistency 173 Cup functor 76 224

Cup product 75 Cyclic groups 32 Deflation, def 164 Delta-functor 3 Double cosets 58 Duality theorems 93, 190, 192, 199 Edge isomorphisms 118 Equivalent extensions 159 Erasable 4 Erasing functor 4, 15, 134 Extension of groups 156 Extreme isomorphisms 118 Factor extension 163 Factor sets 28 Filtered object 116 Filtration 116 Fundamental class 71, 167 G-module 10 G-morphism 11 G-regular 17 Galm(G) 127

Galmp ( G) 138 Galmtor(G) 138 Galois group 151, 195 Galois module 127 Galois type 123 Grab 11 Herbrand lemma 35 Herbrand quotient 35 Hochschild-Serre spectral sequence 118 225

HomG(A, B) 11 Homogeneous standard complex 27 Idele 210 Idele classes 211 Induced representation 52, 134 Inflation in(;/G' 40 Invariant inv., 167 Lifting morphism 38 Limitation theoreem 176 Local component 55 Maximal generator 95 Maximal p-quotient 149 MG(A) 13,19 Mod(G) 10 Mod(R) 10 Mf;(B) 52 Morphism of pairs 38 Multilinear category 73 Nakayama maps 101 Periodicity 95 p-extensive group 144 p-group 50, 126 Positive spectral sequence 118 Profinite group 124, 147 Projective 17 Reciprocity law 198 Reciprocity mapping 173 Regular 17 Restriction resg 39 Semilocal 71 226

Shafarevich-Weil theorem 177 Spectral functor 117 Splitting functor 5 Splitting module 70 Standard complex 26,27 Strict cohomological dimension 138, 193 Supernatural number 125 Sylow group 50, 126, 137 Tate pairing 195 Tate product 109 Tate theorems 23, 70, 98 Tensor product 21 Topological class formation 185 Topological Galois module 185 Trace 12, 15 Transfer of cohomology trc 43 Transfer of group theory Trc 48, 160, 174 Transgression tg 120 Translation 46, 174 Triplet theorem 68, 88 Triplet theorem for cup products 88 Twin theorem 65 Uniqueness theorems 5, 6 Unramified Brauer group 204 Weil group 179, 185