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Class Formations III Journal of the Mathematical Society of Japan Vol. 7, Supplement, December,1955 Class formations III By Yukiyosi KAWADA (Received Nov. 25, 1955) This paper is a continuation of our former papers (Kawada [7], Kawada-rate [8]) concerning class formations and their application in algebraic function fields in one variable. After some preliminaries in § 1 we shall consider in § 2 the system of groups { W(K/k)} in a class formation, which was investigated by A. Weil [12] in case of number fields. We shall arrange the formulas for { W(KK/k)} so that we are able to consider their inverse limit groups in § 3. The sections § 4, § 5 are devoted to the application of the results in § 2, § 3 to the case of algebraic function fields. There we shall find the explicit structure of these groups { W(K/k) } and their limit groups using the formulations in [8]. The results of A. Weil [12] were treated cohomology-theoreti- cally first by T. Nakayama and G. Hochschild [5], [10]. Though the results in § 2, § 3 of this paper are not published hitherto in the literature, they would be known by mathematicians working in this field. The author does not claim any priority on these results. It should be mentioned that there are unpublished investigations of E. Artin and J. Tate concerning the structure of the inverse limit groups of { W(K/k) } in case of number fields. Also the explicit structure of { W(k) } in Theorem 6 (~ 5) was suggested by J. Tate. The author wishes to express his hearty thanks to Professor John Tate for his discussions during the preparation of this paper. § I. Preliminaries I. We repeat here some necessary preparations from Part I (Kawada [7]) which we need later. Let ko be a fixed ground field and 12 be a fixed infinite separable normal algebraic extension of k0. Let S be the set of all finite extensions of ko which are contained 454 Y. KAWABA in 2. In the following we denote the fields which belong to S by k, 1,K,Letc. We assume further that a certain abelian group A(K) is as- sociated with each K E S with the following properties : F 1. Let kCK then there exists an isomorphism qiK ,k of A(k) into A(K). F 2. Let kcl cK then the transitivity relation qK,l°glk=q'K,k holds. F 3. Let K/k be a normal extension with the Galois group G=G(K/k)'~ then G acts on A(K) as an operator group such that qK,kA(k)=A(K)G holds.2~ F 4. In CASE I : k ci Kc L, K/k and L/k are both normal with the Galois groins G = G(L/k), H= G(L/K) and F= G(K/k) the relation o- o qi ,x(a) = q'L,x ° (*F,G°-)(a)3) holds for every o E G and a A(K). Furthermore, for every normal extension K/k with G= G(K/k) we assume C 1: H' (G, A(K)) = 0 and C 2 : H2(G, A(K)) Z/[G :1] • Z. We call then {S, A(K) } a class formation. Let {, A(K) } be a class formation. Then we can choose for each normal extension K/k a 2-cocycle fKI/ of G over A(K) with the following properties: (i) the cohomology class of fK/k is a generic class of H2(G, A(K)), (ii) in case I we have inflG, F fKlk^-fLr'k(m= [L: K]), (iii) in CASE II: k cit c K, K/k is normal with G = G(K/k) and H= G(K/l ) we have resH,GfKlk^'fKll •1) We call then that {fK/k} is a system of fundamental (canonical) 2-cocycles in this class formation. Let {K/k} be another system of 2-cocycles. Then {K/k} is also a system of fundamental 2-cocycles if and only if (i) gxlk"'fKlklk~ rKlk and [K: k] are relatively prime; (ii) in case I rLlk=rKlk (mod [K: k]); (iii) In case II rKil=rK/k (mod [K: l]). Now let us consider the group pairing (Z, A)-~A by (n, a)-4a2. 1) We use the notation G(K/k) to denote the Galois group of a normal extension K/k. 2) In a G-group A we assume alz=(ar )d. By AG we mean the set of all elements a e A which are invariant by all o~E G. 3) For a subgroup H of G we mean by 'G,H the injection mapping H-~G and for a normal subgroup H of G and F=G/H we mean by 'F, G the canonical homomorphism G->F. As usual we mean by Z, R and C the modules of all integers, all real numbers and all complex numbers respectively. 4) By res H,G we mean the restriction mapping: H"(G, A)->H"(H, A) for an abelian G-group A in case II, and by inflG, F we mean the inflation mapping (lift) : H'(F, AH)-> II''r(G, A) in case I. Class formations III. 455 Then the cup-multiplication g'-g' U fK/k gives the isomorphism: Hr(G, Z)~Wr+2(G, A(K)) for all rEZ (Theorem of Tate [11]). In particular, for r= -2 112(G, Z)~G/Gc is isomorphic to H°(G, A(K)) A(K)G/NGA(K)5) by the mapping a- (mod G) -~ fK,k(G, a-) (mod NGA(K))6~(Isomorphism theorem). We define the norm-residue symbol (a, K/k)=o- (mod Gc) for qi ,k(a)=fKlk(G, a-) (mod NGA(K)). Let Ak be the maximal abelian extension of k in 2 and we denote by I7(k) the compact Galois group G(Ak/k). The generalized norm-residue symbol (a, k) (a EEA(k)) which takes value in F(k) is defined as the limit of (a, KA/k) for k C K,, C Ak. The definition (a, k) depends on the choice of a system of fundamental 2-cocycles {fKlk}. But if we define (a, k)' by menas of another system {gKlk},then there exists an isomorphism qi of I'(k) onto itself such that (a, k)'= ji o(a, k) holds for all a A(k). The following two axioms are satisfied in case of class field theory and in several other cases of class formations. P. For any extension K/k (k, KE S) there exists a cyclic extension Z/k such that [Z: k] is a multiple of [K: k]. P'. There exists an increasing sequence of fields K1, K2,..., Ku,... in S such that 2= U K,2 holds. n=1 LEMMA1. Suppose that the axiom P is satisfied in S. I f the generalized norm-residue symbols (a, k) (a A(k)) defined by two systems {fKIk} and {gKIk} of fundamental 2-cocycles coincide, then fKIk-gKJk holds for every normal K/k. (PROOF) Let gKlk-fx klk then we have (a, K/k)"CI k= (a, K/k) by our assumption. Especially this implies rZIk-1 (mod [Z: k]) for cyclic extension Z/k. Now for an arbitrary extension K/k take a cyclic extension Z/k such that [Z: k] is a multiple of [K: k]. Then from the relations rKZIk-rZIk (mod [Z: k]) and rKZlk=rKlk (mod [K: k]) follows that rKlk-1 (mod [K: k]). This means fK/k-gK/k,-q. e. d. 2. We remark here some elementary properties of group ex- tensions. Let A, E, G be groups, c, Jt be homomorphisms such that c (1) 1--- A---*E-- *G--*1 5) Gc means the commutator group of G. NG means the norm with respect to G. 6) fKIk(G,o")=HreG1Klk(T, d) 456 Y. KAWADA is an exact sequence. Then we say that E is a group extension of A by G and we denote E(A, G t, v). By a homomorphism µ(X, v) of E(A, G, t, fr) into E'(A', G', ~', fr') we mean a set of homomorphisms x : A -A',A', ,u: E--~ E', v : G- ~ G' such that t 'fr 1-- ~A --~E -*G - --~1 (2) x t 1->A'---E'----G'-----+i is a commutative diagram. In the rest of this section we assume that the group A is abelian. Then G can be considered as an operator group of A by a~ = t-' (u~• t(a) • uc') (o G, a ETA) where {u~} is a system of representatives of { Jr-' (o-)c E; o-E G }. (3) f(o-, T)= t-'(uQuzzc) (a-, T E G) is a 2-cocycle of G over the G-group A. We call such a pair 2-({u}, {f(o-, T)}) a frame of E(A. G, t, tJr). Any two pairs ({u~}, {f(a-, T)}) and ({v6}, {g(a-,T) }) are related by suitable b;s (bQE A) by vQ= t(b6)u6, with g(o-,T) = f(a-, T) bdbTb;V'. For a fixed 2-cocycle {f(o-, T)} there are as many different pairs ({u~}, {f(o-,T)}) as sets {b~; aEG(b6E A)} which satisfies bUb°b6r'=1. The cohomology class of {f (a-, T)} is uniquely determined by E(A, G, c, ijc) and is inde- pendent of the choice of the representatives {u6}. So we call it the 2-cohomology class associated with E. Suppose that two group extensions E(A, G, ~, J.r) and E'(A', G', t', J'), a homomorphism A : A-~A' and an isomorphism v: G-~G' (into) are given such that X(a6)_ x(a)) (a A, o- G) holds.
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