On filtered algebraic $K$-theory of stacks I: characteristic zero
Elden Elmanto, Dmitry Kubrak, Vladimir Sosnilo
TL;DR
This work proves an Atiyah–Segal type completion theorem for algebraic K-theory of quotient stacks in characteristic zero, identifying the I_G-adic completion of K([X/G]) with the right Kan extension of K-theory from derived affine schemes when the stack X→BG is representable and ANS. It develops a general framework showing how AS completion extends to A^1-invariant localizing invariants (like KH) and to Hochschild-type invariants (HH, HC^-, HC, HP) via loop-stack formalisms and unipotent-loop completions, with descent arguments grounded in cdh and Nisnevich theories and equivariant resolution of singularities. The paper also provides a counterexample illustrating the necessity of ANS-type hypotheses, and it lays out applications to functorial pushforwards and motivic filtrations on derived stacks, aiming to extend motivic filtrations on schemes to stacks. Overall, the results give intrinsic descriptions of completions of invariants on stacks and offer tools for computing completed K-theory and related invariants in valuable geometric settings, including singular and derived targets. These contributions significantly advance the motivic filtration program for stacks and establish a robust K-theoretic AS completion theory in broad geometric contexts.
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.