Galois module structure of cohomology groups for tamely ramified coverings of algebraic varieties

AbstractLet f:X → Y be a tamely ramified finite Galois covering of projective varieties over a field k, and let F be a coherent G-sheaf (G = Gal(XY)) on X. Concerning the k[G]-module structure of the cohomology groups Hi(X, F) (i ≥ 0), we first give a result (Theorem 1) for X and Y of arbitrary dimension. It shows, in particular, that Hn(X, F) is k[G]-projective when Hi(X, F) = 0 for i ≠ n. Next, we further assume that X and Y are non-singular curves and F is locally free. In this case an exact “relation” is given between the k[G]-modules H0(X, F) and H1(X, F) (Theorem 2; we have Hi(X, F) = 0 for i ≥ 2). Our interest lies mainly in the case char k > 0