Equivariant generalized cohomology via stacks

TL;DR

This paper develops a unified approach to equivariant cohomology using stacks, extending classical theorems and providing new computational tools for generalized cohomology theories and equivariant sheaves.

Contribution

It proves a general Borel construction formula for quotient stacks, extends to motivic cohomology and algebraic cobordism, and introduces a new gluing description of equivariant sheaves.

Findings

Borel-equivariant G-theory and Chow groups are compared via a higher Grothendieck-Riemann-Roch theorem.

A general Borel construction formula for quotient stacks is established.

A Bernstein-Lunts-type gluing description of equivariant sheaves is provided.

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.