Topics in Cohomology of Groups_Serge Lang.pdf

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40 (c) Inflation. Let A : G ---, G/G' be a surjective homomorphism. Let A E Mod(G). Then A G' is a G/G'-module for the obvious action induced by the action of G, trivia[ on G', and of course A G' is also a G-module for this operation. We have a morphism of inclusion u : A a' ~ A in Mod(G), which induces a homomorphism H~(u) = u~ " H~(G,A a') ~ H~(G,A) for r => 0. We define inflation int~a/c'- H~(G/G',A a') ---, H~(G,A) to be the composite of the functorial morphism H"(G/G',A a' ) .-_, H"(G,A a' ) followed by the induced homomorphism ur for r => 0. Note that inflation is NOT defined for the special cohomology functor when G is finite. In dimension 0, the inflation therefore gives the identity map (A G' )G/G' --+ A G. In dimension r > 0, it is induced by the cochain homomorphism in the standard complex, which to each cochain {f(~h,... ,~r)} with ~i E G/G' associates the cochain {f(al,... ,~rr)} whose values are constant on cosets of G'. We have already observed that if G acts trivially on A, then Hi(G, A) is simply Horn(G, A). Therefore we obtain: Proposition 1.3. Let G' be a normal subgroup o/G and suppose G acts trivially on A. Then the inflatwn ini~a/c'- HI(G/G',A a') ~ HI(G,A) induces the inflation of a homomorphism ~ : G/G' --* A to a homomorphism X " G --~ A. Let G' be a normal subgroup of G. We may consider the asso- ciation FG:A~A G'
II.1 41 as a functor, not exact, from Mod(G) to Mod(G/G'). Inflation is then a morphism of functors (but not a cohomological morphism) Ha~a, o FG, --+ Ha on the category Mod(G). Even though we are not dealing with a cohomological morphism, we can still use the uniqueness theorem to prove certain commutativity formulas, by decomposing the inflation into two pieces. As another special case of Proposition 1.1, we have: Proposition 1.4. Let G', N be subgroups of G with N normal in G, and N contained in G'. Then on Hr(G/N, A N) we have in~GG ' ,IN o resG,/N G/N = resG G, o in~G/N. We also have transitivity, also as a special case of Proposition 1.1. Proposition 1.4. Let G ---* G1 ---* G2 be aurjective group ho- momorphisma. Then ing'o ing~ = ini~a =. (d) Conjugation. Let U be a subgroup of G. For ~ E G we have the conjugate subgroup v = = = The notation is such that U (~) = (U~) ~'. On Mod(G) we have two cohomological functors, Hu and Hu-. In dimension 0, we have an isomorphism of functors A U--~A U~ given by a~r-la. We may therefore extend this isomorphism uniquely to an isomorphism of/-/u with Hu. which we denote by a, and which we call conjugation. Similarly if U is finite, we have conjugation ~, on the special functor Ha --~ Hu~.
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: II.1 39 Proof. Since ~* is a morphi
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
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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