Algebraic Cobordism and \'Etale Cohomology

TL;DR

This paper extends Thomason's étale descent theorem to algebraic cobordism and related theories, providing new integral versions and applications in motivic homotopy theory.

Contribution

It generalizes étale descent results to algebraic cobordism modules over Noetherian schemes and develops integral versions with broad applications.

Findings

Étale descent holds for algebraic cobordism and K-theory over general schemes.

Constructs an étale descent spectral sequence for motivic Landweber theories.

Shows étale versions of spectra are cellular and satisfy the six functor formalism.

Abstract

Thomason's \'{e}tale 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 \'etale motivic cohomology is the universal way to impose \'etale descent for these theories. As applications, we describe the \'etale local objects in modules over these spectra and show that they satisfy the full six functor formalism, construct an \'etale descent spectral sequence converging to Bott-inverted motivic Landweber exact theories, and prove cellularity and effectivity of the \'{e}tale versions of these motivic spectra.