Rigidity in étale motivic stable homotopy theory

Abstract

For a scheme $X$, denote by $\mathcal{𝒮}\mathcal{\mathscr{H}}\left(X_{ét}^{\wedge }\right)$ the stabilization of the hypercompletion of its étale $\infty$–topos, and by ${\mathcal{𝒮}\mathcal{\mathscr{H}}}_{ét}\left(X\right)$ the localization of the stable motivic homotopy category $\mathcal{𝒮}\mathcal{\mathscr{H}}\left(X\right)$ at the (desuspensions of) étale hypercovers. For a stable $\infty$–category $\mathcal{𝒞}$, write ${\mathcal{𝒞}}_p^{\wedge }$ for the $p$–completion of $\mathcal{𝒞}$.

We prove that under suitable finiteness hypotheses, and assuming that $p$ is invertible on $X$, the canonical functor

$$e_p^{\wedge }:\mathcal{𝒮}\mathcal{\mathscr{H}}{\left(X_{ét}^{\wedge }\right)}_p^{\wedge }\to {\mathcal{𝒮}\mathcal{\mathscr{H}}}_{ét}{\left(X\right)}_p^{\wedge }$$

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 ${\mathcal{𝒮}\mathcal{\mathscr{H}}}_{ét}\left(X\right)$ are given by étale hypercohomology with coefficients in the $S$–local classical sphere spectrum:

$${\left[\mathbf{1}\left[1∕S\right],\mathbf{1}\left[1∕S\right]\right]}_{{\mathcal{𝒮}\mathcal{\mathscr{H}}}_{ét}\left(X\right)}\simeq {ℍ}_{ét}^0\left(X,\mathbf{1}\left[1∕S\right]\right).$$

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 ${\widehat{\mathbf{1}}}_p\left(1\right)\left[1\right]\in \mathcal{𝒮}\mathcal{\mathscr{H}}{\left(X_{ét}^{\wedge }\right)}_p^{\wedge }$ with the property that $e_p^{\wedge }\left({\widehat{\mathbf{1}}}_p\left(1\right)\left[1\right]\right)\simeq {\mathrm{\Sigma }}^{\infty }{\mathbb{𝔾}}_m\in {\mathcal{𝒮}\mathcal{\mathscr{H}}}_{ét}{\left(X\right)}_p^{\wedge }$.