Rigidity in etale motivic stable homotopy theory
Abstract
For a scheme X, denote by SH(X_et^hyp) the stabilization of the hypercompletion of its etale infty-topos, and by SH_et(X) the localization of the stable motivic homotopy category SH(X) at the (desuspensions of) etale hypercovers. For a stable infty-category C, write C_p^comp for the p-completion of C. We prove that under suitable finiteness hypotheses, and assuming that p is invertible on X, the canonical functor e_p^comp: SH(X_et^hyp)_p^comp -> SH_et(X)_p^comp is an equivalence of infty-categories. The primary novelty of our argument is that we use the pro-etale topology to construct directly an invertible object Sptw[1] in SH(X_et^hyp)_p^comp with the property that e_p^comp(Sptw[1]) = Sigma^infty Gm.
Keywords
Cite
@article{arxiv.1810.08028,
title = {Rigidity in etale motivic stable homotopy theory},
author = {Tom Bachmann},
journal= {arXiv preprint arXiv:1810.08028},
year = {2022}
}
Comments
v4: typo in proof of 2.13 v3: remove erroneous remark v2: include results for schemes of finite p-etale dimension at one prime only