Galois module structure of Galois cohomology and partial Euler-Poincaré characteristics

Abstract

Let F be a field containing a primitive p th root of unity, and let U be an open normal subgroup of index p of the absolute Galois group G F of F . Using the Bloch-Kato Conjecture we determine the structure of the cohomology group H n ( U , 𝔽 p ) as an 𝔽 p [ G F /U ]-module for all . Previously this structure was known only for n = 1, and until recently the structure even of H 1 ( U , 𝔽 p ) was determined only for F a local field, a case settled by Borevič and Faddeev in the 1960s. For the case when the maximal pro- p -quotient T of G F is finitely generated, we apply these results to study the partial Euler-Poincaré characteristics of χ n (N) of open subgroups N of T . We show in particular that the n th partial Euler-Poincaré characteristic χ n (N) is determined by only χ n (T) and the conorm in H n ( T , 𝔽 p ).