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Algebraic cobordism and étale cohomology

Elden Elmanto, Marc Levine, Markus Spitzweck and Paul Arne Østvær

Geometry & Topology 26 (2022) 477–586

Thomason’s étale descent theorem for Bott periodic algebraic K–theory is generalized to any MGL 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 K–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.

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motivic homotopy theory, étale motives, slice spectral sequence, algebraic cobordism, algebraic K-theory, motivic cohomology, étale cohomology, descent, slice filtration
Mathematical Subject Classification 2010
Primary: 14F20, 14F42, 19E15, 19D55, 55P42, 55P43
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
Elden Elmanto
Department of Mathematics
Harvard University
Cambridge, MA
United States
Marc Levine
Fakultät für Mathematik
Universität Duisburg-Essen
Markus Spitzweck
Fakultät für Mathematik
Universität Osnabrück
Paul Arne Østvær
Department of Mathematics “Federigo Enriques”
University of Milan
Department of Mathematics
University of Oslo