Title:

Galois module structure of Galois cohomology and partial Euler-Poincar\'e characteristics

Authors:

N. Lemire, University of Western Ontario

J. Minac, University of Western Ontario

J. Swallow, Davidson College

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
    $H^n(U,\Fp)$ as an $\Fp[G_F/U]$-module for all $n\in\mathbb{N}$.
    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 Borevi\v{c} 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-Poincar\'e characteristics of $\EP_n(N)$
    of open subgroups $N$ of $T$. We show in particular that the
    $n$th partial Euler-Poincar\'e characteristic $\EP_n(N)$ is
    determined by only $\EP_n(T)$ and the conorm in $H^n(T,\Fp)$.

Status: preprint