Motivic Atiyah-Segal completion theorem

Abstract

Let T be a torus, X a smooth separated scheme of finite type equipped with a T -action, and [X/T] the associated quotient stack. Given any localizing A1 -homotopy invariant of dg categories E (homotopy K -theory, algebraic K -theory with coefficients, étale K -theory with coefficients, l -adic algebraic K -theory, l -adic étale K -theory, semi-topological K -theory, topological K -theory, periodic cyclic homology, etc), 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 classical Borel construction associated to the T -action o 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 étale K -theory with coefficients, and also of Atiyah-Segal’s completion theorem in topological K -theory (for those topological M -spaces Xan arising from analytification; M is a(ny) maximal compact Lie subgroup of Gan⁠). 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 k no longer needs to be separably closed, and in the case of Atiyah-Segal’s completion theorem the topological M -space Xan no longer needs to be compact and the M -equivariant topological K -theory groups of Xan no longer need to be finitely generated over the representation ring R(M) ⁠. As a second application, we obtain new completion theorems in l -adic étale K -theory, in semi-topological K -theory and also in periodic cyclic homology. As a third application, we obtain a description of the different equivariant cohomology groups in the literature (motivic, l-adic, morphic, Betti, de Rham, etc) in terms of derived completion. 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 semi-topological K -theory