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Motivic Atiyah-Segal completion theorem
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Tabuada, Gonçalo and Van den Bergh, Michel (2024) Motivic Atiyah-Segal completion theorem. International Mathematics Research Notices, 2024 (4). pp. 3497-3550. doi:10.1093/imrn/rnad246 ISSN 1073-7928.
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Official URL: https://doi.org/10.1093/imrn/rnad246
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.
Item Type: | Journal Article | ||||||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||||||
Library of Congress Subject Headings (LCSH): | Geometry, Algebraic, K-theory, Homology theory | ||||||||||||
Journal or Publication Title: | International Mathematics Research Notices | ||||||||||||
Publisher: | Oxford University Press | ||||||||||||
ISSN: | 1073-7928 | ||||||||||||
Official Date: | February 2024 | ||||||||||||
Dates: |
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Volume: | 2024 | ||||||||||||
Number: | 4 | ||||||||||||
Page Range: | pp. 3497-3550 | ||||||||||||
DOI: | 10.1093/imrn/rnad246 | ||||||||||||
Status: | Peer Reviewed | ||||||||||||
Publication Status: | Published | ||||||||||||
Re-use Statement: | This is a pre-copyedited, author-produced version of an article accepted for publication in International Mathematics Research Notices following peer review. The version of record Gonçalo Tabuada, Michel Van den Bergh, Motivic Atiyah-Segal Completion Theorem, International Mathematics Research Notices, 2023;, rnad246, is available online at: https://doi.org/10.1093/imrn/rnad246 | ||||||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||||||
Date of first compliant deposit: | 28 September 2023 | ||||||||||||
RIOXX Funder/Project Grant: |
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