On equivariant Euler-Poincar\'e characteristic in sheaf cohomology

TL;DR

This paper establishes a formula relating the Lefschetz number of a group action on sheaf cohomology to fixed point data, generalizing classical results in equivariant topology with explicit coefficient calculations.

Contribution

It introduces a new equivariant Euler-Poincaré characteristic formula for sheaf cohomology under finite group actions, including explicit coefficient determination.

Findings

Lefschetz number of group action equals fixed point Lefschetz number under certain conditions

Character group sum of cohomology representations is explicitly described

Provides explicit coefficients for the induced representations in the sum

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.