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

Abstract

Let F be a field containing a primitive pth 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,Fp) as an Fp[G_F/U]-module for all n in N. Previously this structure was known only for n=1, and until recently the structure even of H^1(U,Fp) was determined only for F a local field, a case settled by Borevic and Faddeev in the 1960s. We apply these results to study partial Euler-Poincare characteristics of open subgroups N of the maximal pro-p quotient T of G_F. We extend the notion of a partial Euler-Poincare characteristic to this case and we show that the nth partial Euler-Poincare characteristic Theta_n(N) is determined only by Theta_n(T) and the conorm in H^n(T,Fp).