#### Volume 26, issue 2 (2022)

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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
##### Abstract

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.

##### Keywords
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