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Unformatted text preview: DUALITY THEOREMS IN GALOIS COHOMOLOGY OVER NUMBER FIELDS By JOHN TATE 1. Notation and terminology Let X be a Dedekind ring with field of fractions k and let G be a commu- tative group scheme over X. Except in the special case X=T& or C (real or complex field) we put, for all rZ, H'(X,C)=iimH r (G Klk !C Y ), ~K the direct limit taken over all finite Galois extensions K of k in which the integral closure Y of X is unramified over X, where G KJk denotes the Galois group of such an extension, and where G Y denotes the group of points of G with coordinates in Y. For example, if X=k, our notation coincides with that of . For any X, the group H r (X,C) is the r-th cohomology group of the profinite group G t jr =lim G Klk (fundamental group of Spec X) with coefficients in the x-module lim C Y of points of C with coordinates in the maximal unramified extension of X; a general discussion of the cohomology theory of profinite groups can be found in . In the special case X R orC we put H r (B,C)= r (G m ,C) and H r (G,C)=E r (G mg C)~0, where denotes the complete cohomology sequence of the finite group G, in general non trivial in negative dimensions (, Ch. 12). In our applications, X will be a ring associated with an algebraic number field, or with an algebraic function field in one variable over a finite constant field, and the group scheme G will be one of two special types, which we will denote by M and A, respectively. By M we shall always understand (the group scheme of relative dimension zero over X associated with) a finite 6r z-module whose order, | M \ = card M, is prime to the characteristics of the residue class fields of X. By A we shall denote an abelian scheme over X (i.e., an abelian variety defined over k having "non-degenerate reduction" at every prime of X). Underlying our whole theory is the cohomology of the multiplicative group, r m , as determined by class field theory. For any M, we put M' = Horn (M,G m ). By our assumption that \M | is invertible in X, we see that M' is a group scheme of the same type as M, namely the one associated with the 6rjrmodule M' =Hom(Jf ,/Lt) where /u, denotes the group of roots of unity. Moreover we have \M\ =\M'\, and M^(M')'. For any A we put A' = Txt(A,(x m ), the dual abelian variety; then A^(A') f by "biduality". Our aim is to discuss dualities between the cohomology of G and C in both cases, C=M and G=A. Notice that Ext(Jf,6 m ) =0 and Hom(^4,6 m ) =0, so that in each case, 0' denotes the only non-vanishing group in the sequence Ext r ((7,G m ). Thus our results are presumably special cases of a vastly more DUALITY THEOREMS IN GALOIS COHOMOLOGY 289 general hyperduality theorem for commutative algebraic groups envisaged by Grothendieck, involving all the Ext r (C,G m ) simultaneously....
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