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Algebraic cobordism and
étale cohomology
Elden Elmanto, Marc Levine, Markus Spitzweck and Paul Arne Østvær |
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Geometry & Topology 26 (2022) 477–586
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Abstract |
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Thomason’s étale descent theorem for Bott periodic algebraic –theory is generalized to any module over a regular Noetherian scheme of finite dimension. Over arbitrary Noetherian schemes of finite dimension, this generalizes the analogue of Thomason’s theorem for Weibel’s homotopy –theory. This is achieved by amplifying the effects from the case of motivic cohomology, using the slice spectral sequence in the case of the universal example of algebraic cobordism. We also obtain integral versions of these statements: Bousfield localization at étale motivic cohomology is the universal way to impose étale descent for these theories. As applications, we describe the étale local objects in modules over these spectra and show that they satisfy the full six functor formalism, construct an étale descent spectral sequence converging to Bott-inverted motivic Landweber exact theories, and prove cellularity and effectivity of the étale versions of these motivic spectra. A correction was submitted on 4 March 2024 and posted online on 14 March 2024. |
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Keywords
motivic homotopy theory, étale motives, slice spectral
sequence, algebraic cobordism, algebraic K-theory, motivic
cohomology, étale cohomology, descent, slice filtration
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Mathematical Subject Classification 2010
Primary: 14F20, 14F42, 19E15, 19D55, 55P42, 55P43
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Supplementary material |
References |
Publication
Received: 7 February 2019
Revised: 1 March 2021
Accepted: 31 March 2021
Published: 15 June 2022
Proposed: Mark Behrens
Seconded: Stefan Schwede, Haynes R Miller
Correction posted: 14 March 2024 |
Authors |
| Elden Elmanto | |
| Department of Mathematics Harvard University Cambridge, MA United States |
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| Marc Levine | |
| Fakultät für Mathematik Universität Duisburg-Essen Essen Germany |
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| Markus Spitzweck | |
| Fakultät für Mathematik Universität Osnabrück Osnabrück Germany |
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| Paul Arne Østvær | |
| Department of Mathematics “Federigo
Enriques” University of Milan Milan Italy |
Department of Mathematics University of Oslo Oslo Norway |
| © 2022 MSP (Mathematical Sciences Publishers). |
