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When is Galois cohomology free or trivial?

Nicole Lemire·Jan Minac·John Swallow
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Abstract

Let p be a prime and F a field containing a primitive pth root of unity. Let E/F be a cyclic extension of degree p and G_E < 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.

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Nicole Lemire, Jan Minac, John Swallow. 2005-03-01. When is Galois cohomology free or trivial?. https://arxiv.org/abs/math/0410617

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