On filtered algebraic $K$-theory of stacks I: characteristic zero

Abstract

Given a compact Lie group $G$ acting on a space $X$, the classical Atiyah-Segal completion theorem identifies topological $K$-theory of the homotopy quotient $X/G$ with an explicit completion of $G$-equivariant topological $K$-theory of $X$. We prove an analog of this result for algebraic $K$-theory over a field of characteristic 0. In our setting $G$ is a reductive group that acts on a derived algebraic space $X$ with the assumption that all stabilizer groups are nice (in the sense of Alper). Our main result identifies the value $R^{\mathrm{dAff}}K([X/G])$ of right Kan extension of the $K$-theory functor from schemes to stacks with the completion of $K$-theory of the category $\mathrm{Perf}([X/G])$ at the augmentation ideal of $K_0(\mathrm{Rep}(G))$. The main novelty of our results is that $X$ is allowed to be singular or even derived. This generality is achieved by employing and improving analogous versions of completion theorem for negative cyclic homology (after Ben-Zvi--Nadler and Chen) and for homotopy $K$-theory (after van den Bergh--Tabuada). We also show that in the singular setting the completion theorem does not necessarily hold without the nice stabilizer assumption. We view our results as a part of the general paradigm of extending the motivic filtration on algebraic $K$-theory of schemes to algebraic $K$-theory of stacks.