COHOMOLOGICAL DIMENSION AND SCHREIER’S FORMULA IN GALOIS COHOMOLOGY

JOHN LABUTE NICOLE LEMIRE JAN MINAC JOHN SWALLOW

Abstract. Let p be a prime and F a field containing a primitive pth root of unity. If p > 2 assume also that F is perfect. Then for n ∈ ℕ, the cohomological dimension of the maximal pro-p-quotient G of the absolute Galois group of F is n if and only if the corestriction maps H^{n}(H, 𝔽_{p}) → H^{n}(G, 𝔽_{p}) are surjective for all open subgroups H of index p. Using this result we derive a surprising generalization to dim_{𝔽_{p}} H^{n}(H, 𝔽_{p}) of Schreier’s formula for dim_{𝔽_{p}} H^{1}(H, 𝔽_{p}).