2025/03/13 by Elmanto, Elden, Kubrak, Dmitry, Sosnilo, Vladimir · 2 citations
#Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics #K-Theory and Homology (math.KT) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2503.09928
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 RdAffK([X/G]) of right Kan extension of the K-theory functor from schemes to stacks with the completion of K-theory of the category Perf([X/G]) at the augmentation ideal of K0(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.