Nicole Lemire, Jan Minac, John Swallow: When is Galois cohomology free or trivial?

nlemire@uwo.ca, minac@uwo.ca, joswallow@davidson.edu

Submission: 2004, Oct 30

Let $p$ be a prime and $F$ a field, perfect if $p>2$, 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 $\Fp[\Gal(E/F)]$-module. We examine when these properties for $H^n(E)$ are inherited by $H^k(E)$, $k>n$, 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.

2000 Mathematics Subject Classification: 11S25, 16D70

Keywords and Phrases: Galois cohomology, Milnor K-theory, norm map, Hilbert 90, cyclic extension, Galois module

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