Thomason's completion for K-theory and cyclic homology of quotient stacks

TL;DR

This paper proves new completion theorems for equivariant K-theory and cyclic homology of schemes with group actions, solving Thomason's completion problem and providing explicit descriptions for actions with finite stabilizers.

Contribution

It establishes the derived completion of equivariant K'-theory at the augmentation ideal coincides with the K'-theory of the bar construction, addressing Thomason's problem.

Findings

Derived completion of equivariant K'-theory matches the K'-theory of the bar construction.

Equivariant K-theory and cyclic homology simplify for actions with finite stabilizers.

Explicit descriptions of equivariant Hochschild and other homology groups are provided.

Abstract

We prove several completion theorems for equivariant K-theory and cyclic homology of schemes with group action over a field. One of these shows that for an algebraic space over a field acted upon by a linear algebraic group, the derived completion of equivariant K'-theory at the augmentation ideal of the representation ring of the group coincides with the ordinary K'-theory of the bar construction associated to the group action. This provides a solution to Thomason's completion problem. For action with finite stabilizers, we show that the equivariant K-theory and cyclic homology have non-equivariant descriptions even without passing to their completions. As an application, we describe all equivariant Hochschild and other homology groups for such actions.