Cohomological and categorical concentration

Abstract

Given a torus action on a compact space X, a fundamental result of Borel and Atiyah-Segal asserts that the equivariant cohomology of X is concentrated in the fixed locus X^T, up to inverting enough Chern classes. We prove an analogue for algebraic varieties over an arbitrary field. In fact, we deduce this from a categorification at the level of equivariant derived categories and even equivariant stable motivic homotopy categories, which also gives concentration at the level of Voevodsky motives and for homotopy K-theory.