Topics in Cohomology of Groups_Serge Lang.pdf

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VI Chapter V. Augmented Products w Definitions ........................................... 109 w Existence ............................................ 112 w Some properties ...................................... 113 Chapter VI. Spectral Sequences w Definitions ........................................... 116 w The Hoehschild-Serre spectral sequence ............... 118 w Spectral sequences and cup products .................. 121 Chapter VII. Groups of Galois Type (Unpublished article of Tate) w Definitions and elementary properties ................. 123 w Cohomology .......................................... 128 w Cohomological dimension ............................. 138 w Cohomological dimension _< 1 ......................... 143 w The tower theorem ................................... 149 w Galois groups over a field ............................. 150 Chapter VIII. Group Extensions w Morphisms of extensions .............................. 156 w Commutators and transfer in an extension ............ 160 w The deflation ......................................... 163 Chapter IX. Class formations w Definitions ........................................... 166 w The reciprocity homomorphism ...................... 171 w Weil groups .......................................... 178 Chapter X. Applications of Galois Cohomology in Algebraic Geometry (from letters of Tate) w Torsion-free modules ................................. 189 w Finite modules ....................................... 191 w The Tate pairing ..................................... 195 w (0, 1)-duality for abelian varieties ..................... 199 w The full duality ...................................... 201 w Brauer group ......................................... 202 w Ideles and idele classes ............................... 210 w Idele class cohomology ............................... 212
Preface The Benjamin notes which I published (in French) in 1966 on the cohomology of groups provided missing chapters to the Artin-Tate notes on class field theory, developed by cohomological methods. Both items were out of print for many years, but recently Addison- Wesley has again made available the Artin-Tate notes (which were in English). It seemed therefore appropriate to make my notes on cohomology again available, and I thank Springer-Verlag for pub- lishing them (translated into English) in the Lecture Notes series. The most basic necessary background on homological algebra is contained in the chapter devoted to this topic in my Algebra (derived functors and other material at this basic level). This material is partly based on what have now become routine construc- tions (Eilenberg-Cartan), and on Grothendieck's influential paper [Gr 59], which appropriately defined and emphasized 5-functors as such. The main source for the present notes are Tate's private papers, and the unpublished first part of the Artin-Tate notes. The most significant exceptions are: Rim's proof of the Nakayama-Tate theorem, and the treatment of cup products, for which we have used the general notion of multilinear category due to Cartier. The cohomological approach to class field theory was carried out in the late forties and early fifties, in Hochschild's papers [Ho 50a], [Ho 50b], [HoN 52], Nakayama [Na 41], [Na 52], Shafarevich [Sh 46], Well's paper [We 51], giving rise to the Weft groups, and seminars of Artin-Tate in 1949-1951, published only years later [ArT 67]. As I stated in the preface to my Algebraic Number Theory, there
Page 1 and 2: Lecture Notes in Mathematics Editor
Page 3 and 4: Serge Lang Topics in Cohomology of
Page 5: Contents Chapter I. Existence and U
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 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-
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II.3 53 Proposition 3.2. Let G be a
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II.3 55 Theorem 3.8. Let U be of fi
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II.3 57 Proposition 3.11. Let G be
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II.4 59 with {7} in some finite sub
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II.4 Proposition 4.4. If c~ E H~(U,
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III.1 63 Corollary 1.2. Let G be a
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III. 1 65 Corollary 1.6. Hypotheses
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III.1 67 Corollary 1.10. Let A E Mo
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III.2 69 have an injection c : A---
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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=
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IV. 1 77 then we obtain a commutati
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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
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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
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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
Page 123 and 124:
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
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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
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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
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X.3 195 on Z/eZ. HOmGL/k So no horn
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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
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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
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Table of Notation Ap~ : Elements of
Page 227 and 228:
Index Abutment of spectral sequence
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HomG(A, B) 11 Homogeneous standard
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