Equivariant localization and completion in cyclic homology and derived loop spaces

TL;DR

This paper establishes an equivariant localization theorem for derived loop spaces and Hochschild homology of quotient stacks, linking their completions to fixed point stacks and derived de Rham cohomology, extending classical theorems to a derived setting.

Contribution

It proves an equivariant localization theorem and an Atiyah-Segal type completion theorem for cyclic homology in the context of derived algebraic geometry, connecting these to fixed points and derived de Rham cohomology.

Findings

Derived loop spaces of quotient stacks are equivalent after completion along semisimple parameters.

Completed periodic cyclic homology corresponds to 2-periodic derived de Rham cohomology.

The results unify fixed point formulas with derived algebraic geometry techniques.

Abstract

We prove an equivariant localization theorem over an algebraically closed field of characteristic zero for smooth quotient stacks by reductive groups $X/G$ in the setting of derived loop spaces as well as Hochschild homology and its cyclic variants. We show that the derived loop spaces of the stack $X/G$ and its classical $z$-fixed point stack $\pi_0(X^z)/G^z$ become equivalent after completion along a semisimple parameter $[z] \in G//G$, implying the analogous statement for Hochschild and cyclic homology of the dg category of perfect complexes $\text{Perf}(X/G)$. We then prove an analogue of the Atiyah-Segal completion theorem in the setting of periodic cyclic homology, where the completion of the periodic cyclic homology of $\text{Perf}(X/G)$ at the identity $[e] \in G//G$ is identified with a 2-periodic version of the derived de Rham cohomology of $X/G$. Together, these results identify the completed periodic cyclic homology of a stack $X/G$ over a parameter $[z] \in G//G$ with the 2-periodic derived de Rham cohomology of its $z$-fixed points.