On equivariant Euler-Poincar\'e 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 $\sum_j (-1)^j [H^j_\Phi (X,E) \otimes_R F]$ 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.