On equivariant Euler-Poincaré characteristic in sheaf cohomology

Abstract

Let X be a topological Hausdorff space together with a continuous action of a finite group G. Let R be the ring of integers of a number field F. Let E be a G-sheaf of flat R-modules over X and let $\Phi$ be a G-stable paracompactifying family of supports on X. We show that under some natural cohomological finiteness conditions the Lefschetz number of the action of g in G on the cohomology $H_{\Phi }(X,E){\otimes }_{R}F$ equals the Lefschetz number of the g-action on $H_{\Phi }(X^g,E_{|X^g}){\otimes }_{R}F$, where $X^g$ is the set of fixed points of g in X. More generally, the class ${\sum }_j(-1{)}^j[H_{\Phi }^j(X,E){\otimes }_{R}F]$ in the character group equals a sum of representations induced from irreducible F-rational representations $V_{\lambda }$ of $H$ where $H$ runs in the set of G-conjugacy classes of subgroups of G. The integral coefficients $m_{\lambda }$ in this sum are explicitly determined.

arXiv comment7 pages, no figures