Equivariant generalized cohomology via stacks

Abstract

We prove a general form of the statement that the cohomology of a quotient stack can be computed by the Borel construction. It also applies to the lisse extensions of generalized cohomology theories like motivic cohomology and algebraic cobordism. We use this to prove a (higher) equivariant Grothendieck-Riemann-Roch theorem, comparing Borel-equivariant G-theory and equivariant Chow groups. We also give a Bernstein-Lunts-type gluing description of the infinity-category of equivariant sheaves on a scheme X, in terms of nonequivariant sheaves on X and sheaves on its Borel construction.