Thomason’s étale descent theorem for Bott periodic algebraic
–theory is
generalized to any
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
–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.
A correction was submitted on 4 March 2024 and posted onlineon 14 March 2024.
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