WHEN IS GALOIS COHOMOLOGY FREE OR TRIVIAL?

NICOLE LEMIRE JAN MINAC AND JOHN SWALLOW

Abstract. Let p be a prime and F a field, perfect if p > 2, 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}, 𝔽_{p}) to be free or trivial as an 𝔽_{p}[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 ∈ ℕ with prescribed cohomological dimension.