Algebraic Cobordism and Étale Cohomology

Abstract

Thomason's étale descent theorem for Bott periodic algebraic $K$-theory \cite{aktec} is generalized to any $MGL$ module over a regular Noetherian scheme of finite dimension. Over arbitrary Noetherian schemes of finite dimension, this generalizes the analog 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.

published version with revised acknowledgements, 70 pages, to appear in Geometry & Topology

References in corpus (8)

• Higher Topos Theory
• Motivic and Real Etale Stable Homotopy Theory
• Hyperdescent and étale K-theory
• Cdh descent in equivariant homotopy K-theory
• Bivariant theories in motivic stable homotopy
• On Modules Over Motivic Ring Spectra
• The eta-inverted sphere over the rationals
• A primer for unstable motivic homotopy theory

Cited by in corpus (1)

• Topological models for stable motivic invariants of regular number rings