Galois Module Structure and the γ-Filtration

Abstract

We describe a general method for calculating equivariant Euler characteristics. The method exploits the fact that the γ-filtration on the Grothendieck group of vector bundles on a Noetherian quasi-projective scheme has finite length; it allows us to capture torsion information which is usually ignored by equivariant Riemann–Roch theorems. As applications, we study the G-module structure of the coherent cohomology of schemes with a free action by a finite group G and, under certain assumptions, we give an explicit formula for the equivariant Euler characteristic \(\chi (\mathcal{O}_X ) = {\text{H}}^{\text{0}} (X,\mathcal{O}_X ) - {\text{H}}^{\text{1}} (X,\mathcal{O}_X )\) in the Grothendieck group of finitely generated Z[G]-modules, when X is a curve over Z and G has prime order.