Cohomology of G-sheaves in positive characteristic

Let G be a finite group, and X a noetherian G-scheme defined on an algebraically closed field k, whose characteristic divides the order of G. We define a refinement of the equivariant K-theory of X devoted to give a better account of the information related to modular representation theory. The construction relies in an essential way on the work of M. Auslander in modular representation theory and the use of sheaves of “rings with several objects”. The main applications of this “modular K-theory” are in dimension one, where we show how it allows to extend the work of S. Nakajima.