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

Amalendu Krishna, Ritankar Nath

TL;DR

The paper advances algebraic Atiyah–Segal type completion theorems for schemes with linear group actions by proving that the derived I_G-completion of equivariant K-theory and Hochschild-type homology can be computed from ordinary theories of bar and Borel constructions. It develops a robust twisting/decomposition framework and leverages Lurie’s derived completion, motivic homotopy theory, and inertia-stack techniques to obtain completions at augmentation and at all maximal ideals of R(G). It further extends these results to actions with finite stabilizers, Deligne–Mumford stacks, and various homology theories (HH, HC^{-}, HC, HP), giving explicit descriptions in many cases and establishing descent results in étale and cdh topologies. The approach yields both integral and rational statements, with special groups offering integral equivalences, and clarifies when Morita-type reductions can be employed. Collectively, these results enable practical computation of equivariant K-theory and Hochschild-type homology via ordinary (non-equivariant) theories of quotient, Borel, or inertia constructions, significantly broadening computational access to equivariant invariants in algebraic geometry.

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.

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

TL;DR

The paper advances algebraic Atiyah–Segal type completion theorems for schemes with linear group actions by proving that the derived I_G-completion of equivariant K-theory and Hochschild-type homology can be computed from ordinary theories of bar and Borel constructions. It develops a robust twisting/decomposition framework and leverages Lurie’s derived completion, motivic homotopy theory, and inertia-stack techniques to obtain completions at augmentation and at all maximal ideals of R(G). It further extends these results to actions with finite stabilizers, Deligne–Mumford stacks, and various homology theories (HH, HC^{-}, HC, HP), giving explicit descriptions in many cases and establishing descent results in étale and cdh topologies. The approach yields both integral and rational statements, with special groups offering integral equivalences, and clarifies when Morita-type reductions can be employed. Collectively, these results enable practical computation of equivariant K-theory and Hochschild-type homology via ordinary (non-equivariant) theories of quotient, Borel, or inertia constructions, significantly broadening computational access to equivariant invariants in algebraic geometry.

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.

Paper Structure

This paper contains 60 sections, 108 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. Equivariant $K$-theory via ordinary $K$-theory of bar construction
  3. Equivariant $K$-theory via ordinary $K$-theory of Borel construction
  4. Completion of equivariant $KH$-theory
  5. Completions at other ideals of $R(G)$
  6. Thomason's completion of equivariant homology theories
  7. Results for action with finite stabilizers
  8. Future direction
  9. Layout of the paper
  10. Notations
  11. Geometry of representation ring
  12. Finite generation
  13. Relation with class functions
  14. Results of Edidin-Graham
  15. Equivariant $K$-theory and Hochschild homology
  16. ...and 45 more sections

Key Result

Theorem 1.1

Let $k$ be a field and $G$ a linear algebraic group over $k$ acting on an algebraic space $X$. Let $m \in k^\times$ be an integer. We now have the following.

Theorems & Definitions (206)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Remark 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 196 more