Algebraic cobordism and etale cohomology

Abstract

Thomason's etale 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 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 etale versions of these motivic spectra.