Abstract for MSRI Preprint 2005-005

When is Galois cohomology free or trivial?

Nicole Lemire, Jan Minac and John Swallow

Let $p$ be a prime and $F$ a field, perfect if $p \gt 2$, containing a primitive $p$th root of unity. Let $E/F$ be a cyclic extension of degree $p$ and $G_E \triangleleft G_F$ the associated absolute Galois groups. We determine precise conditions for the cohomology group $H^n(E)=H^n(G_E,\Fp)$ to be free or trivial as an $\Fp[{\rm Gal}(E/F)]$-module. We examine when these properties for $H^n(E)$ are inherited by $H^k(E)$, $k \gt n$, and, by analogy with cohomological dimension, we introduce notions of cohomological freeness and cohomological triviality. We give examples of $H^n(E)$ free or trivial for each $n\in \N$ with prescribed cohomological dimension.