Topics in Cohomology of Groups_Serge Lang.pdf

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42 Proposition 1.5. If or E U then a, is the identity on Hu (reap. Hu if U is finite). Proof. The assertion is true in dimension O, whence in all di- mensions. Let f : A --* B be a U-morphism with A, B E Mod(G). Then f~ = [cr -1If = f[~r] : A ---* B is a U~-morphism. The fact that a, is a morphism of functors shows that o = o rv(f) as morphisms on H(U, A) (and similarly for H replaced by H if U is finite). If U is a normal subgroup of G, then or, is an automorphism of Hu (resp. Hu if U is finite). In other words, G acts on Hu (or Hu). Since we have seen that or, is trivial if ~ E U it follows that actually G/U acts on Hu (resp. Hu). Proposition 1.7. Let V C U be subgroups of G, and let ~r E G. Then a, ores =res . oct. on Hu (resp. Hu if U is finite). Proposition 1.8. Let V C U be subgroups of G o f finite index, and let a E G. Suppose V normal in U. Then in ~ o~r, =(r. oin IV on H(U/V, AV), with A E Mod(G). Both the above propositions are special cases of Proposition 1.1. (e) The transfer. Let U be a subgroup of G, of finite index. The trace gives a morphism of functors H~] ~ U~ by the formula S U 9 A U ---, A G, and similarly in the special case when G is finite, H~r --~ H~ by S U . AU/SuA -., Aa/SaA.
II.1 43 The unique extension to the cohomological functors will be denoted by tr U, and will be called the transfer. The following proposition is proved by verifying the asserted commutativity in dimension 0, and then applying the uniqueness theorem. In the case of inflation, we decompose this map in its two components. Proposition 1.9. Let V C U C G be subgroups o/finite index in G. Then on Hv (resp. Hv) we have (I) tra g o tr y :tra v. (2) ~.otr V =tr vocr. for cr C a. (3) If V is normal in a, then on H~(U/V, AV) with r >= 0 we have . U/V =tr Uoin~U/v. inf~a/v o r, rG/V The next result is particularly important. Proposition 1.10. Let U be a subgroup of finite index in G. Then on Ha (resp. Ha) we have tr U ores a = (G" U), where (G :U) on the right abbreviates (G" U)H, i.e. multipli- cation by the index on the cohomology functor. Proof Again the formula is immediate in dimension O, since restriction is just inclusion, and so the trace simply multiplies elements by (G : U). Then the proposition follows in general by applying the uniqueness theorem. Corollary 1.11. Suppose G finite of order n. Then for all r E Z and A E Mod(C) we have nH~(G,A) = 0. Proofi Take U = {e} in the proposition and use the fact that g~(e,A) =0. Corollary 1.12. Suppose G finite, and A E Mod(G) finitely generated over Z. Then H"(G, A) is a finite group for all r E Z. Proof. First Hr(G, A) is finitely generated, because in the standard complex, the cochains are determined by their values on the
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 and 18: 1.2 11 for all a, r E G and a, b E
Page 19 and 20: 1.2 13 Lemma 2.4. Let A,B, C be G-m
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: II.1 41 as a functor, not exact, fr
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
Page 75 and 76: 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
Page 81 and 82: 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
Page 87 and 88: IV.1 81 Proposition 1.8. Let G be a
Page 89 and 90: IV.2 83 Corollary 1.12. Suppose G f
Page 91 and 92: IV.2 85 Lemma 2.2. Let G be a group
Page 93 and 94: IV.3 87 and the rest of the proof i
Page 95 and 96: IV.5 89 injective, resp. surjective
Page 97 and 98: IV.5 91 Proof. We consider the thre
Page 99 and 100:
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)
Page 105 and 106:
IV.7 99 Then a~ is an isomorphism f
Page 107 and 108:
IV.8 101 The bilinear map on top is
Page 109 and 110:
IV.8 103 There results a pairing of
Page 111 and 112:
IV.8 105 then (a,r)(.) = (r- e)b is
Page 113 and 114:
IV.8 107 Using the cocycle relation
Page 115 and 116:
w Definitions CHAPTER V Augmented P
Page 117 and 118:
V.1 111 as well as bilinear maps A'
Page 119 and 120:
V.3 113 Theorem 2.1. Let 91 be a mu
Page 121 and 122:
V.8 115 Note that the coboundary ma
Page 123 and 124:
VI.1 117 A spectral sequence in A i
Page 125 and 126:
vI.2 119 with G/N acting on Hq(N, A
Page 127 and 128:
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
Page 135 and 136:
VII.2 129 p ut: C"(G,A) =0 if r=0 C
Page 137 and 138:
VII.2 131 Indeed, if G # e, then on
Page 139 and 140:
VII.2 133 and takes on only a finit
Page 141 and 142:
VII.2 135 Addition is defined in Iv
Page 143 and 144:
VII.2 137 discrete case. Again, we
Page 145 and 146:
VII.3 139 Since one sees that H~(G,
Page 147 and 148:
VII.3 141 Corollary 3.6. Let Gp be
Page 149 and 150:
VII.4 143 Proposition 3.12. Let G b
Page 151 and 152:
VII.4 145 From the definition of th
Page 153 and 154:
VII.4 147 Next, we connect cohomolo
Page 155 and 156:
VII.5 149 Corollary 4.5. Let G be a
Page 157 and 158:
VII.6 151 be the Galois group. If K
Page 159 and 160:
VII.6 153 Theorem 6.4. Let k be a f
Page 161 and 162:
vii.6 155 whence the lemma follows.
Page 163 and 164:
VIII. 1 157 a coboundary. Hence the
Page 165 and 166:
VIII. 1 159 One sees that F is uniq
Page 167 and 168:
VIII.2 161 Proposition 2.2. One ha~
Page 169 and 170:
viii.3 163 and an isomorphism 0 ~ A
Page 171 and 172:
VIII.3 165 Corollary 3.2. Let G be
Page 173 and 174:
IX.1 167 H2(V/U, Au). It is by defi
Page 175 and 176:
IX.1 169 Since the restriction is s
Page 177 and 178:
IX.2 171 where a, a' denote the fun
Page 179 and 180:
IX.2 173 with the ordinary functor
Page 181 and 182:
IX.2 175 These properties are calle
Page 183 and 184:
IX.2 177 Theorem 2.6. Let G be abel
Page 185 and 186:
IX.3 179 for every open subgroup U
Page 187 and 188:
IX.3 181 of subextension, and its c
Page 189 and 190:
IX.3 183 We consider the cube: A u
Page 191 and 192:
IX.3 185 immediately from the defin
Page 193 and 194:
IX.3 187 WT 6. The factor group AU/
Page 195 and 196:
X.1 189 sion < n if HT(G,A) = 0 for
Page 197 and 198:
X.2 191 is an ismorphism. It follow
Page 199 and 200:
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
Page 205 and 206:
X.4 199 and let a E A(K) represent
Page 207 and 208:
X.5 201 Identifying H2(G, ~2") with
Page 209 and 210:
X.6 203 By Hilbert's Theorem 90 it
Page 211 and 212:
X.6 205 Thus H2(f~(V)*)/H~(a(V) *)
Page 213 and 214:
X.6 207 Thus we have proved our red
Page 215 and 216:
X.6 209 where the @0 on the left me
Page 217 and 218:
X.7 211 We recall our Picard groups
Page 219 and 220:
X.8 213 and pairings giving rise to
Page 221 and 222:
X.8 and a pairing defined by which
Page 223 and 224:
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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