A Finiteness Theorem in the Galois Cohomology of Algebraic Number Fields

In this note we show that if k is an algebraic number field with algebraic closure I and M is a finitely generated, free Zrmodule with continuous Ga\(k/k)action, then the continuous Galois cohomology group Hl(k, M) is a finitely generated Z,-module under certain conditions on M (see Theorem 1 below). Also, we present a simpler construction of a mapping due to S. Bloch which relates torsion algebraic cycles and étale cohomology. The Galois cohomology of an algebraic number field is well-known to be important in the study of the arithmetic of the field and of algebraic varieties defined over the field. Unfortunately, this cohomology is not always well-behaved; for example, the Galois cohomology group H1(k,Zl(l))=: lim Hl(Gal{k/k),ur) n (k an algebraic closure of k) is not a finitely generated Zrmodule. However, there are many important cases in which one knows that the group Hl(k, M) is a finitely generated Zrmodule when M is a finitely generated Zrmodule. In this note we prove a theorem to this effect when M is the /-adic étale cohomology group H'(X,Xi(j)) modulo torsion of the extension to k of a smooth, proper algebraic variety defined over k and i # 2j, 2(j 1). One may view this theorem as a generalization of the weak Mordell-Weil Theorem for the /c-points of an abelian variety; indeed, when M is the Täte module of such a variety our theorem basically comes down to the weak Mordell-Weil Theorem. Also, this theorem gives a much simpler proof of the weak Mordell-Weil Theorem for H°(X, K2)/K2k as proved in [7]We note that a stronger version of our main theorem was proven in the thesis of P. Schneider (see Remark 2 following Theorem 1 for more details). Our excuses for publishing this note are first, that Schneider's result has not been published and second, that our proof is quite short and self-contained. We hope that the applications of the theorem will justify this redundancy. Received by the editors November 14, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 11R34, 14C15.19E08; Secondary 19E15. 743 ©1987 American Mathematical Society 0002-9947/87 $1.00 + $.25 per page License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use