Equivariant Euler–Poincaré characteristic in sheaf cohomology

Let X be a Hausdorff space equipped with a continuous action of a finite group G and a G-stable family of supports Φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Phi}$$\end{document}. Fix a number field F with ring of integers R. We study the class χ=∑j(-1)j[HΦj(X,E)⊗RF]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\chi = \sum_j (-1)^j [H^j_\Phi (X, \mathcal{E}) \otimes_R F]}$$\end{document} in the character group of G over F for any flat G-sheaf E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{E}}$$\end{document} of R-modules over X. Under natural cohomological finiteness conditions we give a formula for χ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\chi}$$\end{document} with respect to the basis given by the irreducible characters of G. We discuss applications of our result concerning the cohomology of arithmetic groups.