Galois module structure of cohomology groups for tamely ramified coverings of algebraic varieties
Let \(f: X\to Y\) be a tamely ramified finite Galois covering of projective varieties over the field k and let F be a coherent \(G (=Gal(X/Y))\quad sheaf\) on X. The author proves that there is a finite complex of finitely generated projective k[G]-modules whose cohomology groups are \(H^*(X,F)\) (as k[G]-modules). The proof consists basically of annotations to his earlier proof of a similar result in the unramified case [the author, Invent. Math. 75, 1-8 (1984)]. In case X and Y are curves, the author produces k[G]-modules which connect the two cohomology groups \(H^ 0=H^ 0(X,E)\) and \(H^ 1=H^ 1(X,E)\) for locally free E. These modules realize the Brauer character \(ch_ G(H^ 0)-ch_ G(H^ 1)\) and, with a Schanuel lemma argument, show how each of \(H^ 0, H^ 1\) determines the other as k[G]-module, in a sort of ''equivariant Riemann-Roch'' formula.
Galois module structure of cohomology groups
scientific article; zbMATH DE number 3970902