A Finiteness Theorem in the Galois Cohomology of Algebraic Number Fields

Abstract

In this note we show that if $k$ is an algebraic number field with algebraic closure $\overline k$ and $M$ is a finitely generated, free ${{\mathbf {Z}}_l}$-module with continuous $\operatorname {Gal} (\overline k /k)$-action, then the continuous Galois cohomology group ${H^1}(k, M)$ is a finitely generated ${{\mathbf {Z}}_l}$-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.