(Springer monographs in mathematics) jean pierre serre (auth ) galois cohomology springer berlin

Springer Monographs in Mathematics Springer-Verlag Berlin Heidelberg GmbH Jean-Pierre Serre Galois Cohomology Translated from the French by Patrick Ion , Springer ]ean-Pierre Serre College de France rued'Ulm 75005 Paris France e-mail: serre@dmi.ens.fr Patrick ion (Translator) Mathematical Reviews P O Box 8604 Ann Arbor, MI 48017-8604 USA Library of Congress Cataloging-in-Publication Data applied for Die Deutache Bibliothek - C1P-Einheitsaufnahme SerIe Jean-Pierre: Galois cohomology I Jean-Pierre Serre Transl from the French by Patrick Ion Corr printing - Berlin; Heide1berg ; New York; Barcelona ; Hong Kong ; London ; MUan ; Paris ; Tokyo : Springer 2002 (Springer monographs in mathematics) Einheitssacht.: Cohomologie galoisienne ISBN 978-3-642-63866-4 DOI 10.1007/978-3-642-59141-9 ISBN 978-3-642-59141-9 (eBook) Corrected Second Printing 2002 of the First English Edition of 1997 Mathematics Subject Classification (2000): 12B20 ISSN 1439-7382 ISBN 978-3-642-63866-4 ThiI work is subject to copyright AU righu are reserved, whether the whole or part of the material is concerned specifica\lythe rightl oftranslation reprinting reuse ofillustrationl.recitation broadcasting, reproduction on microfilm or in any other way and storage in data banks Duplication of this publication or parts thereof is permitted on1y under the provisions of the German Copyright Law of September 1965 in its current version and permission for use must a1ways be obtained from Springer-Verl.g Violations are Iiable for prosecution under the German Copyright Law http://WWW.springer.de © Springer-ver\ag BerIio Heidelberg 1997 OriginaJ\ypub\ishedby Springer-Ver\ag BerIio Heidelberg New York in 1997 The use of general descriptive names registered names trademarks etc in this publication doel not inlply even in the absence of a specific statement that such namel are exempt from the relevant protective Iaws and regu1ations and therefore free for general use SPIN: 10841416 4113142LK - 432 o - Printed on acid-free paper Foreword This volume is an English translation of "Cohomologie Galoisienne" The original edition (Springer LN5, 1964) was based on the notes, written with the help of Michel Raynaud, of a course I gave at the College de France in 1962-1963 In the present edition there are numerous additions and one suppression: Verdier's text on the duality of profinite groups The most important addition is the photographic reproduction of R Steinberg's "Regular elements of semisimple algebraic groups", Publ Math LH.E.S., 1965 I am very grateful to him, and to LH.E.S., for having authorized this reproduction Other additions include: - A proof of the Golod-Shafarevich inequality (Chap I, App 2) - The "resume de cours" of my 1991-1992 lectures at the College de France on Galois cohomology of k(T) (Chap II, App.) - The "resume de cours" of my 1990-1991 lectures at the College de France on Galois cohomology of semisimple groups, and its relation with abelian cohomology, especially in dimension (Chap III, App 2) The bibliography has been extended, open questions have been updated (as far as possible) and several exercises have been added In order to facilitate references, the numbering of propositions, lemmas and theorems has been kept as in the original 1964 text Jean-Pierre Serre Harvard, Fall 1996 Table of Contents Foreword V Chapter I Cohomology of profinite groups §1 Profinite groups 1.1 Definition 1.2 Subgroups 1.3 Indices 1.4 Pro-p-groups and Sylow p-subgroups 1.5 Pro-p-groups 3 §2 Cohomology 2.1 Discrete G-modules 2.2 Cochains, cocycles, cohomology 2.3 Low dimensions 2.4 Functoriality 2.5 Induced modules 2.6 Complements 10 10 10 11 12 13 14 §3 Cohomological dimension 3.1 p-cohomological dimension 3.2 Strict cohomological dimension 3.3 Cohomological dimension of subgroups and extensions 3.4 Characterization of the profinite groups G such that cdp ( G) :5 3.5 Dualizing modules 17 17 18 19 21 24 §4 Cohomology of pro-p-groups 4.1 Simple modules 4.2 Interpretation of Hl: generators 4.3 Interpretation of H2: relations 4.4 A theorem of Shafarevich 4.5 Poincare groups 27 27 29 33 34 38 VIII Table of Contents §5 Nonabelian cohomology 5.1 Definition of HO and of HI 5.2 Principal homogeneous spaces over A - a new definition of HI(G,A) 5.3 Twisting 5.4 The cohomology exact sequence associated to a subgroup 5.5 Cohomology exact sequence associated to a normal subgroup 5.6 The case of an abelian normal subgroup 5.7 The case of a central subgroup " 5.8 Complements 5.9 A property of groups with cohomological dimension :s: 45 45 46 47 50 51 53 54 56 57 Bibliographic remarks for Chapter I 60 Appendix J Tate - Some duality theorems 61 Appendix The Golod-Shafarevich inequality " 66 The statement 66 Proof 67 Chapter II Galois cohomology, the commutative case §1 Generalities 71 1.1 Galois cohomology 71 1.2 First examples 72 §2 Criteria for cohomological dimension 2.1 An auxiliary result 2.2 Case when p is equal to the characteristic 2.3 Case when p differs from the characteristic 74 74 75 76 §3 Fields of dimension::::;1 3.1 Definition 3.2 Relation with the property (C I ) 3.3 Examples of fields of dimension :s: 78 78 79 80 §4 Transition theorems 4.1 Algebraic extensions 4.2 Transcendental extensions 4.3 Local fields 4.4 Cohomological dimension of the Galois group of an algebraic number field 4.5 Property (C r ) 83 83 83 85 87 87 Table of Contents IX §5 p-adic fields 5.1 Summary of known results 5.2 Cohomology of finite Gk-modules 5.3 First applications 5.4 The Euler-Poincare characteristic (elementary case) , 5.5 Unramified cohomology 5.6 The Galois group of the maximal p-extension of k 5.7 Euler-Poincare characteristics 5.8 Groups of multiplicative type 90 90 90 93 93 94 95 99 102 §6 Algebraic number fields 6.1 Finite modules - definition of the groups P'(k, A) 6.2 The finiteness theorem 6.3 Statements of the theorems of Poitou and Tate 105 105 106 107 Bibliographic remarks for Chapter II 109 Appendix Galois cohomology of purely transcendental extensions110 An exact sequence 110 The local case 111 Algebraic curves and function fields in one variable 112 The case K = k(T) 113 Notation 114 Killing by base change 115 Manin conditions, weak approximation and Schinzel's hypothesis 116 Sieve bounds 117 Chapter III Nonabelian Galois cohomology §1 Forms 1.1 Tensors 1.2 Examples 1.3 Varieties, algebraic groups, etc 1.4 Example: the k-forms of the group SLn 121 121 123 123 125 §2 Fields of dimension:::; 2.1 Linear groups: summary of known results 2.2 Vanishing of Hi for connected linear groups 2.3 Steinberg's theorem 2.4 Rational points on homogeneous spaces 128 128 130 132 134 §3 Fields of dimension:::; 139 3.1 Conjecture II 139 3.2 Examples 140 X Table of Contents §4 Finiteness theorems 4.1 Condition (F) 4.2 Fields of type (F) 4.3 Finiteness of the cohomology of linear groups 4.4 Finiteness of orbits 4.5 The case k = R 4.6 Algebraic number fields (Borel's theorem) 4.7 A counter-example to the "Hasse principle" 142 142 143 144 146 147 149 149 Bibliographic remarks for Chapter III 154 Appendix Regular elements of semisimple groups (by R Steinberg) 155 Introduction and statement of results 155 Some recollections 158 Some characterizations of regular elements 160 The existence of regular unipotent elements 163 Irregular elements 166 Class functions and the variety of regular classes 168 Structure of N 172 Proof of 1.4 and 1.5 176 Rationality of N 178 10 Some cohomological applications 184 11 Added in proof 185 Appendix Complements on Galois cohomology Notation The orthogonal case Applications and examples Injectivity problems The trace form Bayer-Lenstra theory: self-dual normal bases Negligible cohomology classes 187 187 188 189 192 193 194 196 Bibliography 199 Index 209 194 I1I.Appendix Complements on Galois cohomology where Q and Q' are of rank (this is possible according to [154], App I), we have Q' = (2d) ® Q, where d is the discriminant of E (Le of qE)' 2) Suppose one has Wl(qE) = and W2(QE) = O One may ask whether qE is isomorphic to the unit form (1, , I) (as would be the case if the rank were < 6) This is true if k is a number field (or a rational function field over a number field) It is in general false § Bayer-Lenstra theory: self-dual normal bases Let G be a finite group We are interested in the G-Galois algebras over k, L e in the G-torsors over k, G being considered as an algebraic group of dimension over k Such an algebra L is determined, up to a nonunique isomorphism, by a continuous homomorphism ipL : Gal(ks/k) -+ G When ip L is surjective, L is a field, and it is a Galois extension of k with Galois group isomorphic to G In [9], E Bayer and H Lenstra are interested in the case when L has a selfdual normal basis ("an SDNB"); this means that there exists an element x of L such that qdx) = and that x is orthogonal (relative to qL) to every gx, E G, =I- (Thus, the gx form a "normal basis" of L, and this basis is its own dual with respect to qL') One can give a cohomological criterion for the existence of a SDNB: if UG denotes the unitary group of the involutory algebra k[G], there is a canonical embedding of G into UG(k); by composing ipL with this embedding one obtains a homomorphism Gal(ks/k) > Ua(k), and this homomorphism may be viewed as a l-cocycle of Gal(ks/k) in Ua(k s ) The class CL of this co cycle is an element of Hl(k, Ua ) One has CL = if and only if L has an SDNB From this criterion, combined with (4.4), Bayer-Lenstra deduced the following theorem: (6.1) - If there exists an extension of k of odd degree over which L acquires an SDNB, then L has an SDNB over k In particular: (6.2) - If G is of odd order, every Galois G-algebra has an SDNB Here are some other results about SDNB, obtained in collaboration with E Bayer, cf [11]: Let L be a Galois G-algebra, and let ipL : Gal(ks/k) > G be the corresponding homomorphism If x is an element of Hn(G, Z/2Z), its image under will be denoted by XL Bayer-Lenstra theory: self-dual normal bases 195 (6.3) In order that L have an SDNB, it is necessary that XL = for every X E Hl(G, Z/2Z) (i.e., the image of Gal(ks/k) in G is in all the index-2 subgroups of G) This condition is sufficient if the cohomological2-dimension of Gal( ks / k) is :::; (i e., if the Sylow 2-groups of Gal(ks/k) are free pro-2-groups) (6.4) - Suppose that k is a number field In order that L have an SDNB, it is necessary that d Cv) = for every real place v of k (Cv denoting the complex conjugation with respect to an extension of v to ks ) This condition is sufficient if Hl(G, Z/2Z) = H2(G, Z/2Z) = o (6.5) The case where a Sylow 2-group of G is elementary abelian Let S be a Sylow 2-group of G Suppose that S is an elementary abelian group of order 2r , r ~ Ij the order of G is 2r m, with m odd (6.5.1) - There exists an r-fold Pfister form qi, and, up to isomorphism, only one, such that (2r) ® qL ~ m ® ql (a direct sum of m copies of qi) This form is an invariant of the Galois algebra L It is the unit form if L has an SDNB Conversely: (6.5.2) - Suppose that the normalizer N of S acts transitively on S - {I} The following are equivalent: (i) L has an SDNB (ii) The form qL is isomorphic to the unit form of rank 2rm (iii) The form ql is isomorphic to the unit form of rank 2r When r is small enough, this result can be translated into cohomological terms Indeed, the hypothesis that N act transitively on S - {I} implies that there exists an element x of Hr(G, Z/2Z) whose restriction to any subgroup of order of G is i- 0, and such an element is unique, up to the addition of a "negligible" cohomology class (cf § below) The corresponding element XL of Hr(k, Z/2Z) is an invariant of the Galois algebra L (6.5.3) - Suppose that r :::; The conditions (i), (ii), (iii) in (6.5.2) are then equivalent to: (iv) XL = in Hr(k, Z/2Z) The hypothesis r :::; could be dropped if the conjectures in §2.3 were proved Examples 1) Suppose that r = and that N acts transitively on S-{l}j this is so when G = A , A5 or PSL2 (F q) with q == (mod 8) The group H2( G, Z/2Z) contains a single element x i- OJ let be the corresponding extension of G by Z/2Z It follows from (6.5.3) that L has an SDNB if and only if the homomorphism tpL : Gal(ks/k) + G lifts to a homomorphism in Such a lifting corresponds to a Galois a-algebra Lj one can show it is possible to arrange that L also has an SDNB a a 196 III.Appendix Complements on Galois cohomology 2) Take as G the group SL2(Fs) or the Janko group J • The hypotheses in (6.5.2) and (6.5.3) are then satisfied with r = The group H3(G, Z/2Z) contains a single element x =F 0, and one sees that L has an SONB if and only if XL = in H3(k, Z/2Z) Remark The property that a G-Galois algebra L have an SONB can be translated into "Galois twisting" terms as follows: Let V be a finite-dimensional vector space over k, equipped with a family q = (qi) of quadmtic tensors (of type (2,0), (1,1), or (0,2), it doesn't matter which) Suppose that G acts on V and fixes every qi One may then twist (V, q) by the G-torsor corresponding to L In this way one obtains a k-form (V, q)L of (V, q) One can prove: (6.6) If L has an SONB, (V,q)L is isomorphic to (V,q) Moreover, this property chamcterizes the Galois algebras which have an SONB (Note that such a statement would be false for cubic tensors.) § Negligible cohomology classes Let G be a finite group and C a G-module An element x in Hq(G, C) is said to be negligible (from the Galois standpoint) if, for every field k, and every continuous homomorphism cp : Gal(ks/k) -+ G, we have cp*(x) = (This amounts to saying that XL = for every G-Galois algebra L.) Example If a and b are two elements of Hl(G, Z/2Z), the cup-product ab(a + b) is a negligible element of H (G, Z/2Z) Here are some results about these classes: (7.0) - If q = 1, no nonzero element of Hq(G,C) is negligible The same is true if q = and G acts trivially on C (7.1) - For every finite group G there exists an integer q(G) such that any cohomology class of G of odd order and dimension q > q( G) is negligible This result does not extend to classes of even order Indeed, no cohomology class (other than 0) of a cyclic group of order is negligible, as one sees by taking k=R (7.2) - Suppose that G is elementary abelian of order 2r Ifx E Hq(G,Z/2Z), the following properties are equivalent: Negligible cohomology classes 197 (a) x is negligible (b) The restriction of x to any subgroup of order is O (c) x belongs to the ideal of the algebra H*(G, Z/2Z) generated by the cupproducts ab(a + b), where a and b run over Hl(G, Z/2Z) {There are analogous results when G is elementary abelian of order pT (p and C = Z/pZ.) f 2), (7.3) - Suppose that G is isomorphic to a symmetric group Sn Then: (a) If N is odd, every element of Hq{G, Z/NZ), q ~ 1, is negligible (b) In order that an element of Hq{G, Z/2Z) be negligible, it is necessary and sufficient that its restrictions to the subgroups of G of order vanish Bibliography [I] A Albert - Structure of Algebras, A.M.S Colloquium Publ 24, Providence, 1961 [2] A Albert and N Jacobson - On reduced exceptional simple Jordan algebras, Ann of Math 66 (1957), 400-417 [3] J Arason - Cohomologische Invarianten quadratischer Formen, J Algebra 36 (1975), 446-491 [4] " " - A proof of Merkurjev's theorem, Canadian Math Soc Conference Pmc (1984), 121-130 [5] E Artin and O Schreier - Eine Kennzeichnung der reeH abgeschlossenen Karper, Hamb Abh (1927), 225-231 (= E Artin, C.P 21) [6] E Artin and J Tate - Class Field Theory, Benjamin Publ., New York, 1967 [7] M Artin, A Grothendieck and J-L Verdier - Cohomologie Etale des Schemas (SGA 4), Lect Notes in Math 269-270-305, Springer-Verlag, 1972-1973 [8] J Ax - Proof of some conjectures on cohomological dimension, Proc A.M.S 16 (1965), 1214-1221 [9] E Bayer-Fluckiger and H.W Lenstra, Jr - Forms in odd degree extensions and self-dual normal bases, Amer J Math 112 (1990), 359-373 [lOJ E Bayer-Fluckiger and R Parimala - Galois cohomology of classical groups over fields of cohomological dimension ~ 2, Invent Math 122 (1995), 195229 [111 E Bayer-Fluckiger and J-P Serre - Torsions quadratiques et bases normales autoduales, Amer J Math 116 (1994), 1-63 [12] F van der Blij and T.A Springer - The arithmetics of octaves and of the group G2 , Indag Math 21 (1959), 406-418 [13J A Borel - Groupes lineaires algebriques, Ann of Math 64 (1956), 20-82 (= Oe 39) [14] " " - Some finiteness properties of adele groups over number fields, Publ Math I.H.E.S 16 (1963), 5-30 (= Oe.60) [15] " " - Arithmetic properties of linear algebraic groups, Proc Int Congress Math Stockholm (1962), 10-22 (= Oe.61) [16] " " - Linear Algebraic Groups, 2nd edition, Springer-Verlag, 1991 [17] A Borel and Harish-Chandra - Arithmetic subgroups of algebraic groups, Ann of Math 75 (1962) 485-535 (= A Borel, Oe 58) [18] A Borel and J-P Serre - Theoremes de finitude en cohomologie galoisienne, Comm Math Helv 39 (1964), 111-164 (= A Borel, Oe.64) 200 Bibliography [19J A Borel and T.A Springer - Rationality properties of linear algebraic groups, Proc Symp Pure Math A.M.S (1966), 26-32 (= A Borel, Oe 76); II, Taho/ru Math J 20 (1968), 443-497 (= A Borel, Oe 80) [20J A Borel and J Tits - Groupes roouctifs, Publ Math I.H.E.S 27 (1965), 55-150 (= A Borel, Oe 66); Complements, ibid 41 (1972), 253-276 (= A Borel, Oe 94) [21J Z.L Borevic and LR Safarevic - Number Theory (in Russian), 3rd edition, Moscow, 1985 [22J F Bruhat and J Tits - Groupes algebrique simples sur un corps local, Proc Con! 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Birkhauser, Boston, 1982 [186] II II - Basic Number Theory, Springer-Verlag, 1967 [187] E Witt - Theorie der quadratischen Formen in beliebigen Korpern, J Crelle 176 (1937), 31-44 [188] V.1 Yanchevski'l- K-unirationality of conic bundles and splitting fields of simple central algebras (in Russian), Dokl Akad Nauk S.S.S.R 29 (1985), 1061-1064 [189] H Zassenhaus - The Theory of Groups, 2nd ed., Chelsea, New York, 1949 Index (1.1.5) = chap I, §1.5 associated (profinite group - to a discrete group) 1.1.1 Bayer-Lenstra (theory) III.App.2.6 Borel - (subgroup) 111.2.1 - (theorem of -) I1I.4.6 Cartan (subgroup) 111.2.1 cocycle (of G in a G-group) 1.5.1 cohomology - (exact sequence) 1.5.4 - (of a profinite group) 1.2.2 - set (first -) 1.5.1 condition (F) I1I.4.1 conjecture I 111.2.3 conjecture II 111.3.1 corestriction 1.2.4 DemuSkin (group) 1.4.5 DemuSkin-Labute (classification theorem) 1.4.5 dimension ~ (field of -) 11.3.1 dimension (cohomological- of a profinite group) 1.3.1 discrete (G-module) 1.2.1 dualizing (module) 1.3.5 Euler-Poincare (characteristic) 11.5.4 1.4.1, finiteness (theorem) 11.6.2 form 111.1 free (pro-p-group) 1.1.5 Galois cohomology 11.1.1,111.1.1 G-group 1.5.1 G-set 1.5.1 Golod-Shafarevich (theorem) 1.4.4 good (group) 1.2.6 Hasse (principle) I1I.4.7 Hasse-Witt (invariant) I1I.3.2 index (of a closed subgroup) induced (module) 1.2.5 lifting (property) 1.5.9 1.1.3 Manin conditions II.App Merkurjev-Suslin (theorem) 11.4.5 Milnor (conjectures) III.App.2.2.3 multiplicative type (groups of -) 11.5.8 negligible (cohomology class) III.App.2.7 order (of a profinite group) 1.1.3 p-adic (field) 11.5 parabolic (subgroup) 111.2.1 p-cohomological dimension 1.3.1 p-completion (of a discrete group) 1.3.1 p-dimension (cohomological-) 1.3.1 p-extension (maximal- of a field) 11.2 Poincare (group) 1.4.5 Poitou-Tate (theorems) 11.6.3 principal (homogeneous space) 1.5.2 profinite (group) 1.1.1 projective (profinite group) 1.5.9 pronilpotent (profinite group) 1.5.9 pro-p-group 1.1.4 property (C ) 11.3.2 property (C r ) 11.4.5 quasi-split (semisimple group) 111.2.2 radical (of an algebraic group) 111.2.1 rank - (of a pro-p-group) 1.4.2 - (of a free pro-p-group) 1.1.5 - (of a normal subgroup) 1.4.3 residue (of a cohomology class) II.App residue formula II.App 210 Index restriction 1.2.4 Schinzel (hypothesis) II.App section (of a projection onto a quotient) 1.1.2 self-dual (normal basis) III.App.2.6 semisimple (algebraic group) 111.2.1 Shapiro-Faddeev (theorem) 1.2.5 Shafarevich (theorem) 1.4.4 simply connected (semisimple group) I1I.3.1 split - (extension) 1.3.4 - (group) I1I.2.2 Springer (theorem) 111.2.4 Steinberg (theorem) 111.2.3 Stiefel-Whitney (classes) I1I.App.2.2.1 strict (cohomological dimension) 1.3.2 supernatural (number) 1.1.3 Sylow (subgroups of a profinite group) 1.1.4 Tate (theorems) 11.5.1,11.5.7 torsor 1.5.2 tower (class field -) 1.4.4 trace form III.App.2.5 twisting 1.5.3, I1I.1.3 unipotent (algebraic group) 111.2.1 unramified (module) 11.5.5 weak approximation II.App Zp-algebra (of a pro-p-group) 1.1.5 [...]... extension of a group G satisfying A2 (a) Assume first that N is finite Let 1 be the centralizer of N in E Show that 1 is of finite index in Ej deduce th~t 1/(1 n N) satisfies A2 [apply 1, (d)], since there exists subgroup Eo of finite index in E such that Eo n N = {1} (b) Assume from now on that N is finitely generated Show (using (a )) that every subgroup of N of finite index contains a subgroup of the form... profinite group G, and let A E C H The induced module A* = MlJ (A) is defined as the group of continuous maps a* from G to A such that a*(hx) = h· a*(x) for h E H,x E G The group G acts on A* by (ga *)( x) = a*(xg) If H = {I}, one writes MG(A)j the G-modules obtained in this way are called induced ("co-induced" in the terminology of [145 ]) If to each a* E MlJ (A) one associates its value at the point... H/(HnU )) , where U runs over the set of open normal subgroups of G It is also the km of the indices (G : V) for open V containing H Proposition 2 (i) If K C H C G are profinite groups, one has (G: K) = (G: H) (H: K) (ii) If (Hi) is a decreasing filtration of closed subgroups of G, and if H = nHi , one has (G: H) = km(G: Hi) (iii) In orner that H be open in G, it is necessary and sufficient that (G : H)... obtains a homomorphism MlJ (A) t A which is compatible with the injection of H into G (cf §2.4)j hence the homomorphisms Hq(G,MlJ(A )) ~ Hq(H,A) Proposition 10 The homomorphisms Hq(G, MlJ (A )) above are isomorphisms t Hq(H, A) defined One first remarks that, if BE CG, one has HomG(B, MlJ (A )) = Hom H (B, A) This implies that the functor MlJ transforms injective objects into injective objects Since,... pro finite group G One has cdp(H) scdp{H) ~ ~ cdp(G) scdp{G) with equality in each of the following cases: (i) (G : H) is prime to p (ii) H is open in G, and cdp{G) < +00 We will consider only cdp, since the argument is analogous for scdp If A is a discrete torsion H-module, MlJ (A) is a discrete torsion G-module and Hq{G,MlJ{A» = Hq{H,A), whence obviously the inequality cdp{H) ~ cdp{G) The inequality... hypothesis "H is open in G" by ''the exponent of pin (G : H) is finite" 2) With the same notation as in prop 15, assume that the exponent of p in (G : H) is not zero (Le cdp(GjH) #- 0) Show that one has the inequality scdp(G) :::; cdp(H) + scdp(GjH) 3) Let n be an integer Assume that for each open subgroup H of G, the p-primary components of Hn+l(H, Z) and Hn+2(H, Z) are zero Show that scdp ( G) ::; n [If... element of N) 6 1.§1 Profinite groups Let us show (i): if U is an open normal subgroup of G, set Gu = GjU, Hu = Hj(H n U), Ku = Kj(K n U) One has Gu : :) Hu : :) K u , from which (G u : Ku) = (Gu : Hu)· (Hu : Ku) By definition, lcm(Gu : Ku) = (G : K) and lcm(Gu : Hu) = (G: H) On the other hand, the H n U are cofinal with the set of normal open subgroups of H; it follows that lcm(Hu : Ku) = (H: K), and from... morphism f : F(I) - G(I) the family = (f(Xi )) ' The fact that the correspondence obtained in this way is bijective is clear (g ,) Remark Along with F(I) one may define the group F8(I) which is the projective limit of the L(I)jM for those M just satisfying a) This is the p-completion of L(I); the morphisms of F8(I) into a pro-p-group are in one-to-one correspondence with arbitrary families (gi)iEI of elements... of f(G) in G' is prime to p (2p) For any p-primary G'-module A, the homomorphism Hl(G',A) -+ Hl(G,A) is injective [Reduce this to the case where G and G' are pro-p-groups.] (b) Show the equivalence of: ( 1) f is surjective ( 2) For any G'-module A, the homomorphism H 1 (G',A) -+ Hl(G,A) is injective ( 3) Same assertion as in ( 2), but restricted to finite G'-modules A 2.5 Induced modules 13 2.5 Induced... 14 6) One thus has Hl(G, ZjpZ) 1= 0, whence cdp(G) 2: 1 Corollary 3 If cdp { G) 1= 0,00, the exponent of p in the order of G is infinite 20 1.§3 Cohomological dimension Here again, one may assume G is a pro-p-group If G were finite, part (ii) of the proposition would show cdp(G) = cdp({l }) = 0, in contradiction to our hypothesis Therefore G is infinite Corollary 4 Assume cdp ( G) = n is finite In order .. .Springer Monographs in Mathematics Springer- Verlag Berlin Heidelberg GmbH Jean- Pierre Serre Galois Cohomology Translated from the French by Patrick Ion , Springer ]ean -Pierre Serre College... G .) = dim Hi(G) (Notice that We make use of the exact sequence in §2.6, and of H (F(n )) = 0-+ HI(G) -+ HI(F(n )) -+ HI(R)G ~ H2(G) -+ o One finds: This exact sequence shows that HI(R)G and H2(G)... Tokyo : Springer 2002 (Springer monographs in mathematics) Einheitssacht.: Cohomologie galoisienne ISBN 978-3-642-63866-4 DOI 10.1007/978-3-642-59141-9 ISBN 978-3-642-59141-9 (eBook) Corrected