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On the Structure of Galois Groups As Galois Modules

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On the Structure of Galois Groups As Galois Modules ON THE STRUCTURE OF GALOIS GROUPS AS GALOIS MODULES Uwe Jannsen Fakult~t f~r Mathematik Universit~tsstr. 31, 8400 Regensburg Bundesrepublik Deutschland Classical class field theory tells us about the structure of the Galois groups of the abelian extensions of a global or local field. One obvious next step is to take a Galois extension K/k with Galois group G (to be thought of as given and known) and then to investigate the structure of the Galois groups of abelian extensions of K as G-modules. This has been done by several authors, mainly for tame extensions or p-extensions of local fields (see [10],[12],[3] and [13] for example and further literature) and for some infinite extensions of global fields, where the group algebra has some nice structure (Iwasawa theory). The aim of these notes is to show that one can get some results for arbitrary Galois groups by using the purely algebraic concept of class formations introduced by Tate. i. Relation modules. Given a presentation 1 + R ÷ F ÷ G~ 1 m m of a finite group G by a (discrete) free group F on m free generators, m the factor commutator group Rabm = Rm/[Rm'R m] becomes a finitely generated Z[G]-module via the conjugation in F . By Lyndon [19] and m Gruenberg [8]§2 we have 1.1. PROPOSITION. a) There is an exact sequence of ~[G]-modules (I) 0 ÷ R ab + ~[G] m ÷ I(G) + 0 , m where I(G) is the augmentation ideal, defined by the exact sequence (2) 0 + I(G) ÷ Z[G] aug> ~ ÷ 0, aug( ~ aoo) = ~ a . o6G o~6G 110 b) % Rmab ~-- ~[G]m-I • ~ as ~[G]-module. c) For a second presentation 1 ÷ R n + E n ÷ G ÷ 1 one has R ab ~9 Z[G] m T Rab ~9 Z[G] n n m and therefore (3) R ab ~ R ab • Z [G] n-m (n > m) n,p m,p p -- ' for every prime p, if we set R m,pab = Zp ®~ R mab (~p the ring of integers in the field ~p of p-adic numbers). In particular the G-structure of R ab only depends on m; for R ab m,p m itself and minimal m this is still an open problem, see [8]. One has R ab ~ I(G) ®zI(G) for m = (G:I)-I, and R ab ~[G] m-I ~ Z for cyclic G m m (~,~,Zp and ~p are always equipped with the trivial G-action). If G is a p-group, R ab is just the factor commutator group of R in any m~p m presentation 1 + R m ÷ F m +G ÷ i of G by a free pro-p-group on m free generators. If the order of G is prime to p, one has <p ~Zp[G]m-l~Zp. Tate has shown (see [16]) that R ab is a class formation module for m G, i.e. , (4) H i(U, Rab'm) ~ H i-2 (U,~) for all subgroups U of G and all i 6Z (here and in the following we take the modified (Tate) cohomology groups), where the isomorphism is obtained by taking cupproduct with the restriction of a generating element of H2(G, R ab'm) ~ Z/(G:I)Z. It turns out that R abm has to be regarded as a standard object with this property - all other class formation modules only differing by "projective kernels": 1.2. THEOREM. Let G be a finite group, G a p-Sylow subgroup and M a p finitely generated ~. [G]-module with the property P H 1 (Gp, M) = 0 (*) H2(Gp, M) ~ ~p/(Gp: l)Zp H 2(G, M) ~ Zp/(G: l)Zp. a) There is an exact sequence (5) 0 ÷ X ÷ R ab ÷ M + 0 m,p for some m 6 IN and some projective Zp[G]-module X. 111 b) If M is torsion free (as Zp-mOdule), the sequence (5) splits, so M @ X ~ R ab m,p c) There is an exact sequence (6) 0 ÷ M + M' + I (G) ÷ 0 P with a cohomologically trivial Zp[G]-module M' and Ip(G) = Zp ® I(G). Proof. The proof of a) is nearly as in [13]I: Let 0 ÷M+ E+G÷I be the group extension corresponding to a generating element of H2(G,M), and choose a homomorphism ~: F + E with dense image (m m suitable). This induces a surjection ~: R ab ÷ M, let X = ker ~. m,p From the long exact sequence of cohomology under G~w we get H2(Gp,X) = 0 = H3(Gp,X), so X is cohomologically trivial, i.e., projective, as X is torsion free. b) is clear (compare [13] 1.5), and M' is defined by the exact commutative diagram I (G) = I (G) P P T 0 ---> X ---> ~p[G] m --> g' --~ 0 Ii T T 0 --~ X --~ R ab ---> M --> 0 , m,p where the middle column is given by l.l.a). 2. Cohomolo~ically trivia! Zp!G]-modules. We fix the following notations. For a finitely generated Z [G]- P module M, Tor(M) will denote the ~p-torsion submodule of M, M* = Hom(M,@ /Z ) is the Pontrjagin dual of M (with the operation -P 1 p (of) (m) = f(o- m)), and dG(M) is the minimal number of ~p[G]- generators for M. Tensor products now are taken over Zp, if not denoted otherwise. For a pro-finite group A (abelian or not) A(p) is the maximal pro-p-quotient. A Zp[G]-module P is projective iff it is torsion free and cohomo- logically trivial, and then determined by the structure of ~pOP as ~p[G]-module by a theorem of Swan [23] 6.4. This generalizes to 112 2.1. THEOREM ([12] 1.2.). For cohomologically trivial, finitely generated ~ [G]-modules M and M' the following statements are P equivalent: i) M ~ M'. ii) Tor(M) ~ Tor(M') and ~p@ M ~ ~pO M'. Furthermore we have the following construction, which follows from [12] 1.8-1.10. 2.2. LEMMA. Let N be a finite ~ [C~]-module. P a) There is a presentation (exact sequence) ~p[G] ~ f--~ ~p[G] m -->N* --~ 0 if and only if there is an exact sequence 0 ÷ =p[G] m -->f+ =piG] Z + M + 0 for the cohomologically trivial ~ [G]-module M with Tor(M) ~ N and £-m P ~p @ M ~ ~p [G] b) In the above statement, f+ can be chosen to be the transpose of f in the following sense: If f is given b w the matrix (~i4)3 with ~ij 6 ~p[G], f+ is then given by the matrix (~.i), where + is the anti-involution of Zp[G] given by ( [ a o) + = [ a ~-i ~6G o6G o 3. Applications to number fields. Let K/k be a finite Galois extension of local or global number fields with Galois group G (function fields can be treated similarly). Fix a prime p and let K be i) the maximal p-extension of K, if k is local, ii) the maximal p-extension of K unramified outside S, if k is global; here S is a finite set of non-archimedean primes of K closed under the action of G and containing all primes above p and all primes ramified in K/k. For every field k c_ L c_ K we set G L = GaI(K/L), and we want to consider the finitely generated Zp[G]-module G Kab • Notations: for any field L, ~L is the group of roots of unity in L, and ~n = {~ 6 U~I~n = i} for an algebraic closure ~ of L. 113 i) For local fields the only interesting case is where p equals the residue characteristic. The following theorem generalizes the results for p-groups due to Borevi6, K. Wingberg and the author (see [2],[3], [13] and [25]): 3.1. THEOREM. Let k be of degree n over Qp. a) G is generated by n+2 elements, and there is an exact sequence ab ÷ ab + 0 (7) 0 + Zp[G] + Rn+2, p G K . b) If K is regular (i.e., ~p ~ K), G is generated by n+l elements, and there is an isomorphism ab ~ Rab (8) GK n+l,p Proof. We only show a), because b) is similar, using the splitting of (7). As the reciprocity map induces an isomorphism between G ab and the projective limit over the groups KX/K xpn for all n, ~K-abKhas the property (*), and using the p-adic logarithm we get an isomorphism ab ~ ~p[G]n @ @p (9) ~p ® G K Let R be defined by the exact commutative diagram I (G) = I (G) P P 1 T 0 + Zp[G] + Z p[ G ]n+2 ÷ M' ÷ 0 ab 0 ~ Zp [G] ~ R ÷ G K ÷ 0 where the right column is given by 1.2.c) and the middle row exists by 2.2., because M' is cohomologically trivial, Tor(M') ~ ~K(p) is cyclic and ~p ~ M' ~ ~p[G] n+l by (9). If we can show that G is generated by n+2 elements , we are done, because then R ~ Rn+2,ab p by applying Schanuel's lemma to the middle column and l.l.a). For this we may assume that the ramification group of G is abelian, by Burnside's theorem on p-groups. If L is the fixed field of the ramification group, G is then a quotient of the middle group in the extension (I0) 1 ÷~ ÷ Gk/[GL,G L] ÷ G ÷ i, 114 where G = Gal(L/k) is generated by 2 elements.
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