Topics in Cohomology of Groups_Serge Lang.pdf

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The above argument may be expressed in the form of a useful general lemma. If, in a cube, all the ]aces are commutative except possibly one, and one of the arrows as above is surjective, then this face is also commutative. Next we have to show that ~1 commutes with 6, that is (~P0,cPl) is a 6-morphism. Let 0 ~ A' --* A ~ A" ~ 0 be an exact sequence in 9/. Then there exist morphisms v : A ~ MA, and w : A" ---* XA, making the following diagram commutative: 0 ) A' ~ A , A" ) 0 0 ) A' ' ]VIA, ) XA' ) 0 because MA, is injective. There results the following commutative diagram: H~ '') ,, Ho(XA ,) 9 H'(A') ~o /~) ~ ~(A') FO(XA ,) 9 r I(A') 6F
1.2 9 We have to show that the right square is commutative. Note that the top and bottom triangles are commutative by definition of a 6-functor. The left square is commutative by the hypothesis that c20 is a morphism of functors. The front square is commutative by definition of ~I(A'). We thus find (r = I(A')6HH~ (top triangle) which concludes the proof. = 6FqOoH~ (front square) = ~rF~ (left square) - - 6F~0 (bottom triangle), Finally, let us make explicit what we mean by saying that ~1 depends functorially on ~0- Suppose we have three functors H, F, E defined in degrees 0,1; and suppose given ~0 : H ~ ~ F ~ and ~0 : F ~ ~ E ~ Suppose in addition that the erasing functor erases both H 1 and F 1 . We can then construct ~1 and r by applying the theorem. On the other hand, the composite r = ~0 " H ~ ---+ E ~ is also a morphism, and the theorem implies the existence of a morphism {?1 " H1 --+ E 1 such that ({?0, (71) is a 6-morphism. By uniqueness, we obtain {71 = r o ~I. This is what we mean by the assertion that c21 depends functorially on ~o. w Notation, and the uniqueness theorem in Mod(G) We now come to the cohomology of groups. Let G be a group. As usual, we let Q and Z denote the rational numbers and the integers respectively. Let Z[G] be the group ring over Z. Then
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: 1.1 7 which makes the right square
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 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
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
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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
Page 87 and 88:
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
Page 93 and 94:
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
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
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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
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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
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
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VII.2 133 and takes on only a finit
Page 141 and 142:
VII.2 135 Addition is defined in Iv
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VII.2 137 discrete case. Again, we
Page 145 and 146:
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
Page 153 and 154:
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
Page 161 and 162:
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
Page 171 and 172:
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
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
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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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