$\eta$-periodic motivic stable homotopy theory over fields
Abstract
Over any field of characteristic not 2, we establish a 2-term resolution of the -periodic, 2-local motivic sphere spectrum by shifts of the connective 2-local Witt K-theory spectrum. This is curiously similar to the resolution of the K(1)-local sphere in classical stable homotopy theory. As applications we determine the -periodized motivic stable stems and the -periodized algebraic symplectic and SL-cobordism groups. Along the way we construct Adams operations on the motivic spectrum representing Hermitian K-theory and establish new completeness results for certain motivic spectra over fields of finite virtual 2-cohomological dimension. In an appendix, we supply a new proof of the homotopy fixed point theorem for the Hermitian K-theory of fields.
Keywords
Cite
@article{arxiv.2005.06778,
title = {$\eta$-periodic motivic stable homotopy theory over fields},
author = {Tom Bachmann and Michael J. Hopkins},
journal= {arXiv preprint arXiv:2005.06778},
year = {2021}
}
Comments
60 pages v2: compare Adams operations with those of Fasel--Haution, add corollary about cellularity, mention some further prior work, add missing flatness hypothesis in Lemma 4.8 and expand on its proof, fix some typos v3: minor corrections and extensions v4: another minor revision