Motivic Atiyah-Segal completion theorem

TL;DR

This paper proves a motivic Atiyah-Segal completion theorem for algebraic and topological K-theories, extending classical results and providing new algebraic and homotopical insights into equivariant cohomology and K-theory.

Contribution

It introduces a motivic version of the Atiyah-Segal completion theorem, extending it to algebraic and semi-topological K-theories and quotient stacks, with improved conditions and new applications.

Findings

Derived completion of E([X/T]) matches the Borel construction.

Alternative proofs of classical completion theorems with relaxed conditions.

New completion theorems in l-adic étale K-theory, semi-topological K-theory, and cyclic homology.

Abstract

Let T be a torus, X a smooth quasi-compact separated scheme equipped with a T-action, and [X/T] the associated quotient stack. Given any localizing A1-homotopy invariant of dg categories E, we prove that the derived completion of E([X/T]) at the augmentation ideal I of the representation ring R(T) of T agrees with the Borel construction associated to the T-action on X. Moreover, for certain localizing A1-homotopy invariants, we extend this result to the case of a linearly reductive group scheme G. As a first application, we obtain an alternative proof of Krishna's completion theorem in algebraic K-theory, of Thomason's completion theorem in \'etale K-theory with coefficients, and also of Atiyah-Segal's completion theorem in topological K-theory. These alternative proofs lead to a spectral enrichment of the corresponding completion theorems and also to the following improvements: in the case of Thomason's completion theorem the base field no longer needs to be separably closed, and in the case of Atiyah-Segal's completion theorem the topological spaces no longer needs to be compact and the equivariant topological K-theory groups no longer need to be finitely generated over the representation ring. As a second application, we obtain new completion theorems in l-adic \'etale K-theory, in (real) semi-topological K-theory and also in periodic cyclic homology. As a third application, we obtain a purely algebraic description of the different equivariant cohomology groups in the literature (motivic, l-adic, (real) morphic, Betti, de Rham, etc). Finally, in two appendixes of independent interest, we extend a result of Weibel on homotopy K-theory from the realm of schemes to the broad setting of quotient stacks and establish some useful properties of (real) semi-topological K-theory.