Rigidity in etale motivic stable homotopy theory

TL;DR

This paper establishes an equivalence between certain stabilized etale homotopy categories under finiteness and invertibility conditions, using pro-etale topology to construct key invertible objects.

Contribution

It introduces a novel approach using pro-etale topology to prove an equivalence of stabilized etale motivic homotopy categories under specific conditions.

Findings

Proves the equivalence of SH(X_et^hyp)_p^comp and SH_et(X)_p^comp under certain hypotheses.

Constructs an invertible object Sptw[1] in SH(X_et^hyp)_p^comp using pro-etale topology.

Demonstrates the functor e_p^comp is an equivalence in the given setting.

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.