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 ${\mathbb {A}}<^>{1}$-homotopy invariant of dg categories $E$ (homotopy $K$-theory, algebraic $K$-theory with coefficients, etale $K$-theory with coefficients, $l$-adic algebraic $K$-theory, $l$-adic etale $K$-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 on $X$. Moreover, for certain localizing ${\mathbb {A}}<^>{1}$-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 (for those topological $M$-spaces $X<^>{\textrm {an}}$ arising from analytification; $M$ is a(ny) maximal compact Lie subgroup of $G<^>{\textrm {an}}$). 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 $X<^>{\textrm {an}}$ no longer needs to be compact and the $M$-equivariant topological $K$-theory groups of $X<^>{\textrm {an}}$ 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 etale $K$-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.