Cohomology of G-sheaves in positive characteristic

TL;DR

This paper develops a refined equivariant K-theory for schemes over fields of positive characteristic, linking modular representation theory with algebraic geometry, and generalizes a modular Riemann-Roch theorem in the one-dimensional case.

Contribution

It introduces a new refinement of equivariant K-theory that incorporates modular representation theory and extends the modular Riemann-Roch theorem to broader contexts.

Findings

Refined equivariant K-theory incorporating modular representations

Generalization of Nakajima's modular Riemann-Roch theorem

Enhanced understanding of Galois modules and wild ramification

Abstract

Let X be a noetherian scheme defined over an algebraically closed field of positive characteristic p, and G be a finite group, of order divisible by p, acting on X. We introduce a refinement of the equivariant K-theory of X to take into account the information related to modular representation theory. As an application, in the 1-dimensional case, we generalize a modular Riemann-Roch theorem given by S.Nakajima, extending the link between Galois modules and wild ramification.