Rigidity in étale motivic stable homotopy theory

Bachmann, Tom

doi:10.2140/agt.2021.21.173

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Rigidity in étale motivic stable homotopy theory

Tom Bachmann

Algebraic & Geometric Topology 21 (2021) 173–209

DOI: 10.2140/agt.2021.21.173

Abstract

For a scheme $X$ , denote by $𝒮 ℋ (X_{ét}^{\land})$ the stabilization of the hypercompletion of its étale $\infty$ –topos, and by ${𝒮 ℋ}_{ét} (X)$ the localization of the stable motivic homotopy category $𝒮 ℋ (X)$ at the (desuspensions of) étale hypercovers. For a stable $\infty$ –category $𝒞$ , write $𝒞_{p}^{\land}$ 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}^{\land} : 𝒮 ℋ {(X_{ét}^{\land})}_{p}^{\land} \to {𝒮 ℋ}_{ét} {(X)}_{p}^{\land}

is an equivalence of $\infty$ –categories. This generalizes the rigidity theorems of Suslin and Voevodsky (Invent. Math. 123 (1996) 61–94), Ayoub (Ann. Sci. École Norm. Sup. 47 (2014) 1–145) and Cisinski and Déglise (Compos. Math. 152 (2016) 556–666) to the setting of spectra. We deduce that under further regularity hypotheses on $X$ , if $S$ is the set of primes not invertible on $X$ , then the endomorphisms of the $S$ –local sphere in ${𝒮 ℋ}_{ét} (X)$ are given by étale hypercohomology with coefficients in the $S$ –local classical sphere spectrum:

{[1 [1 ∕ S], 1 [1 ∕ S]]}_{{𝒮 ℋ}_{ét} (X)} ≃ ℍ_{ét}^{0} (X, 1 [1 ∕ S]) .

This confirms a conjecture of Morel.

The primary novelty of our argument is that we use the pro-étale topology of Bhatt and Scholze (Astérisque 369 (2015) 99–201) to construct directly an invertible object ${\hat{1}}_{p} (1) [1] \in 𝒮 ℋ {(X_{ét}^{\land})}_{p}^{\land}$ with the property that $e_{p}^{\land} ({\hat{1}}_{p} (1) [1]) ≃ Σ^{\infty} 𝔾_{m} \in {𝒮 ℋ}_{ét} {(X)}_{p}^{\land}$ .

Keywords

motivic homotopy theory, étale cohomology

Mathematical Subject Classification 2010

Primary: 14F20, 14F42

References

Publication

Received: 29 May 2019

Revised: 24 March 2020

Accepted: 11 May 2020

Published: 25 February 2021

Authors

Tom Bachmann
Department of Mathematics Massachusetts Institute of Technology Cambridge, MA United States	Mathematisches Institut LMU Munich München Germany
http://tom-bachmann.com

Volume 21, issue 1 (2021)