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arXiv · 1810.08028

Rigidity in etale motivic stable homotopy theory

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.

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Tom Bachmann. 2018-10-18. Rigidity in etale motivic stable homotopy theory. https://doi.org/10.2140/agt.2021.21.173

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