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.

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

References in corpus (3)

• Higher Topos Theory
• Motivic and Real Etale Stable Homotopy Theory
• A note on stable recollements

Cited by in corpus (3)

• Remarks on étale motivic stable homotopy theory
• Topological models for stable motivic invariants of regular number rings
• Galois reconstruction of Artin-Tate $\mathbb{R}$-motivic spectra