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Mathematics > K-Theory and Homology

arXiv:1810.08028 (math)
[Submitted on 18 Oct 2018 (v1), last revised 11 Jan 2022 (this version, v4)]

Title:Rigidity in etale motivic stable homotopy theory

Authors:Tom Bachmann
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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.
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
Subjects: K-Theory and Homology (math.KT); Algebraic Geometry (math.AG)
Cite as: arXiv:1810.08028 [math.KT]
  (or arXiv:1810.08028v4 [math.KT] for this version)
  https://doi.org/10.48550/arXiv.1810.08028
Journal reference: Algebr. Geom. Topol. 21 (2021) 173-209
Related DOI: https://doi.org/10.2140/agt.2021.21.173

Submission history

From: Tom Bachmann [view email]
[v1] Thu, 18 Oct 2018 13:04:48 UTC (31 KB)
[v2] Thu, 26 Mar 2020 16:47:21 UTC (33 KB)
[v3] Mon, 15 Mar 2021 07:53:07 UTC (33 KB)
[v4] Tue, 11 Jan 2022 10:59:46 UTC (33 KB)
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