Topics in Cohomology of Groups_Serge Lang.pdf

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12 Corollary 2.2. If G = {e} then H$(A) = 0/or all r > O. Proof. Define HG by letting H~(A) = A e and H~(A) = 0 for r # 0. Then it is immediately verified that He is a cohomologica/ functor, to which we can apply the uniqueness theorem. Corollary 2.3. Let n E Z and let nA " A --* A be the morphism a ~-+ na for a E A. Then H~(nA) =nH (where H stands for Hb(A)). Proof. Since the coboundary 8 is additive, it commutes with multiplication by n, and aga/n we can apply the uniqueness theorem. The existence of the functor HG will be proved in the next sec- tion. We say that G operates trivially on A if A = A a, that is cra -- a for all a E A and ~ C G. We always assume that G operates trivially on Z, Q, and Q/Z. We define the abehan group AG = A/IAa. This is the factor group of A by the subgroup of elements of the form (or - e)a with cr E G and a C A. The association A ~-+ AG is a functor from Mod(G) into Grab. Let U be a subgroup of finite index in G. We may then define the trace S~: A U --* A G by the formula SU(a) = E 6a, where {c} is the set of left cosets of U in G, and 6 is a representative of c, so that G= [.3 ~U. If U = {e}, then G is finite, and in that case the trace is written SG, so SG(a) = E (;a. ~6G For the record, we state the following useful lemma. C C
1.2 13 Lemma 2.4. Let A,B, C be G-modules. Let U be a subgroup of finite indez in G. Let A-L B--L C-% D be morphisms in Mod(G), had suppose that u, w are G-morphisms while v is a U-morphism. Then Proof. Immediate. S (wvu) = ws (v)u We shall now describe some embedding functors in Mod(G). These will turn out to erase some cohomological functors to be defined later. Indeed, injective or projective modules will not suffice to erase cohomology, for several reasons. First, when we change the group G, an injective does not necessarily remain injective. Second, an exact sequence 0 ---* A ~ J ---* A" --~ 0 with an injective module J does not necessarily remain exact when we take its tensor product with an arbitrary module B. Hence we shall consider another class of modules which behave better in both respects. Let G be a group and let B be an abelian group, i.e. a Z- module. We denote by MG(B) or M(G, B) the set of functions from G into B, these forming an abelian group in the usual way (adding the values). We make Ms(B) into a G-module by defining an operation of G by the formulas = for a. We have trivially (ar)(f) = a(rf). Furthermore: Proposition 2.6. Ze~ G ~ be a subgroup of G and let G = U x~G' be a coset decomposition. For f E M(G,B) let Ot f~ be the f~nction in M(G',B) such tha~ f~(y) = f(x~y) for y E G'. Then the map Ot
Page 1 and 2: Lecture Notes in Mathematics Editor
Page 3 and 4: Serge Lang Topics in Cohomology of
Page 5 and 6: Contents Chapter I. Existence and U
Page 7 and 8: Preface The Benjamin notes which I
Page 9 and 10: CHAPTER I Existence and Uniqueness
Page 11 and 12: I.l 5 We have similar notions on th
Page 13 and 14: 1.1 7 which makes the right square
Page 15 and 16: 1.2 9 We have to show that the righ
Page 17: 1.2 11 for all a, r E G and a, b E
Page 21 and 22: 1.2 15 Corollary 2.6. Let G' be a s
Page 23 and 24: 1.2 17 Let K1, K2, K be commutative
Page 25 and 26: 1.2 19 The first one is just the on
Page 27 and 28: 1.3 21 so the cohomology sequence a
Page 29 and 30: 1.3 23 Theorem 3.5. Let G be a fini
Page 31 and 32: 1.3 25 Thus we have defined G-modul
Page 33 and 34: 1.3 27 where the symbol ~j means th
Page 35 and 36: 1.4 by letting One verifies at once
Page 37 and 38: 1.4 31 We end our explicit computat
Page 39 and 40: 1.5 33 be a short exact sequence in
Page 41 and 42: 1.5 35 Now let A E Mod(G) be arbitr
Page 43 and 44: CHAPTER II Relations with Subgroups
Page 45 and 46: II.1 39 Proof. Since ~* is a morphi
Page 47 and 48: II.1 41 as a functor, not exact, fr
Page 49 and 50: II.1 43 The unique extension to the
Page 51 and 52: II. 1 45 Corollary 1.15. Suppose G
Page 53 and 54: II.1 Proposition 1.16. Let A E Mod(
Page 55 and 56: II.1 49 Proof. Since Z[U] is natura
Page 57 and 58: II.2 51 Theorem 2.1. Let Gp be a p-
Page 59 and 60: II.3 53 Proposition 3.2. Let G be a
Page 61 and 62: II.3 55 Theorem 3.8. Let U be of fi
Page 63 and 64: II.3 57 Proposition 3.11. Let G be
Page 65 and 66: II.4 59 with {7} in some finite sub
Page 67 and 68: II.4 Proposition 4.4. If c~ E H~(U,
Page 69 and 70:
III.1 63 Corollary 1.2. Let G be a
Page 71 and 72:
III. 1 65 Corollary 1.6. Hypotheses
Page 73 and 74:
III.1 67 Corollary 1.10. Let A E Mo
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III.2 69 have an injection c : A---
Page 77 and 78:
III.3 71 A' by means of a cocycle {
Page 79 and 80:
CHAPTER IV Cup Products w Erasabili
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IV. 1 75 Let now E1 = (EC~),...,E=
Page 83 and 84:
IV. 1 77 then we obtain a commutati
Page 85 and 86:
IV. 1 79 is uniquely determined by
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IV.1 81 Proposition 1.8. Let G be a
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IV.2 83 Corollary 1.12. Suppose G f
Page 91 and 92:
IV.2 85 Lemma 2.2. Let G be a group
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IV.3 87 and the rest of the proof i
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IV.5 89 injective, resp. surjective
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IV.5 91 Proof. We consider the thre
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IV.5 93 between H-I(Q/Z) and the el
Page 101 and 102:
IV.6 95 where the vertical maps are
Page 103 and 104:
IV.6 97 Then resg(0 has order (U e)
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IV.7 99 Then a~ is an isomorphism f
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IV.8 101 The bilinear map on top is
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IV.8 103 There results a pairing of
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IV.8 105 then (a,r)(.) = (r- e)b is
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IV.8 107 Using the cocycle relation
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w Definitions CHAPTER V Augmented P
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V.1 111 as well as bilinear maps A'
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V.3 113 Theorem 2.1. Let 91 be a mu
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V.8 115 Note that the coboundary ma
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VI.1 117 A spectral sequence in A i
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vI.2 119 with G/N acting on Hq(N, A
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VI.3 121 Theorem 2.5. Suppose Hr(N,
Page 129 and 130:
CHAPTER VII Groups of Galois Type (
Page 131 and 132:
VII. 1 12~ the maps fi. The interse
Page 133 and 134:
VII. 1 127 the intersection of a to
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VII.2 129 p ut: C"(G,A) =0 if r=0 C
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VII.2 131 Indeed, if G # e, then on
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VII.2 133 and takes on only a finit
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VII.2 135 Addition is defined in Iv
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VII.2 137 discrete case. Again, we
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VII.3 139 Since one sees that H~(G,
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VII.3 141 Corollary 3.6. Let Gp be
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VII.4 143 Proposition 3.12. Let G b
Page 151 and 152:
VII.4 145 From the definition of th
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VII.4 147 Next, we connect cohomolo
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VII.5 149 Corollary 4.5. Let G be a
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VII.6 151 be the Galois group. If K
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VII.6 153 Theorem 6.4. Let k be a f
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vii.6 155 whence the lemma follows.
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VIII. 1 157 a coboundary. Hence the
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VIII. 1 159 One sees that F is uniq
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VIII.2 161 Proposition 2.2. One ha~
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viii.3 163 and an isomorphism 0 ~ A
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VIII.3 165 Corollary 3.2. Let G be
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IX.1 167 H2(V/U, Au). It is by defi
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IX.1 169 Since the restriction is s
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IX.2 171 where a, a' denote the fun
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IX.2 173 with the ordinary functor
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IX.2 175 These properties are calle
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IX.2 177 Theorem 2.6. Let G be abel
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IX.3 179 for every open subgroup U
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IX.3 181 of subextension, and its c
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IX.3 183 We consider the cube: A u
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IX.3 185 immediately from the defin
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IX.3 187 WT 6. The factor group AU/
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X.1 189 sion < n if HT(G,A) = 0 for
Page 197 and 198:
X.2 191 is an ismorphism. It follow
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X.2 193 Theorem 2.3. We have scd(Gk
Page 201 and 202:
X.3 195 on Z/eZ. HOmGL/k So no horn
Page 203 and 204:
X.3 197 Let u E UK, and a E ZO,K. W
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X.4 199 and let a E A(K) represent
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X.5 201 Identifying H2(G, ~2") with
Page 209 and 210:
X.6 203 By Hilbert's Theorem 90 it
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X.6 205 Thus H2(f~(V)*)/H~(a(V) *)
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X.6 207 Thus we have proved our red
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X.6 209 where the @0 on the left me
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X.7 211 We recall our Picard groups
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X.8 213 and pairings giving rise to
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X.8 and a pairing defined by which
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218 [Ka 69] [KaS 56] [KaT 55] [La 5
Page 225 and 226:
Table of Notation Ap~ : Elements of
Page 227 and 228:
Index Abutment of spectral sequence
Page 229 and 230:
HomG(A, B) 11 Homogeneous standard
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