Galois module structure of cohomology groups for tamely ramified coverings of algebraic varieties
type: Item
instance of: scholarly article
title: Galois module structure of cohomology groups for tamely ramified coverings of algebraic varieties
zbMATH Open document ID: 0602.14017
DOI: 10.1016/0022-314X(86)90032-6
published in: Journal of Number Theory
publication date: 1986-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 14C35
Mathematics Subject Classification ID: 14E20
Mathematics Subject Classification ID: 14H30
zbMATH DE Number: 3970902
zbMATH Keywords: Galois module structure of cohomology groups
zbMATH Keywords: Brauer character
author: Shoichi Nakajima
reviewed by: Andy R. Magid
MaRDI profile type: Publication
cites work: Q3922829
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description: scientific article; zbMATH DE number 3970902