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Rigidity in étale motivic stable homotopy theory. (English) Zbl 1483.14035

For a scheme \(X\), one can associate the stable motivic homotopy category \(\mathcal{SH}(X)\), the étale motivic stable homotopy category \(\mathcal{SH}_{\text{ét}}(X)\) by localizing the former at the étale hypercovers, whose \(p\)-completion we denote by \(\mathcal{SH}_{\text{ét}}(X)^{\wedge}_p\); there is also the \(\infty\)-category \(\mathcal{SH}(X_{\text{ét}}^{\wedge})\), obtained as the stabilization of the hypercompletion \(\mathcal{Shv}(X_{\text{ét}}^{\wedge})=\mathcal{Shv}(X_{\text{ét}})^{\wedge}\) of its étale \(\infty\)-topos \(\mathcal{Shv}(X_{\text{ét}})\). We have the canonical functor \(e: \mathcal{SH}(X_{\text{ét}}^{\wedge})\to\mathcal{SH}_{\text{ét}}(X)\) induced by the site morphism \(X_{\text{ét}}\to\text{Sm}_{X,\text{ét}}\); its \(p\)-completion we denote by \(e_p^{\wedge}: \mathcal{SH}(X_{\text{ét}}^{\wedge})e_p^{\wedge}\to\mathcal{SH}_{\text{ét}}(X)e_p^{\wedge}\).
One main result in this paper under review is the so-called rigidity in étale motivic stable homotopy theory (Theorem 6.6), which says that under suitable finiteness assumption (\(X\) being locally \(p\)-étale finite and \(p\) is invertible in \(X\)), then \(e_p^{\wedge}: \mathcal{SH}(X_{\text{ét}}^{\wedge})_p^{\wedge}\to\mathcal{SH}_{\text{ét}}(X)_p^{\wedge}\) is an equivalence of \(\infty\)-categories.
A central ingredient in the proof is to construct an object \(\hat{\mathbf{1}}_p(1)[1]\), the so-called twisting spectrum, which is sent to \(\mathbb{G}_m\) by \(e_p^{\wedge}\). This is achieved with the help of the pro-étale topology introduced in [B. Bhatt and P. Scholze, Astérisque 369, 99–201 (2015; Zbl 1351.19001)]. A priori, the object constructed lies in the homotopy category over the pro-étale site, but the author was able to show (in Theorem 3.6) that it comes (uniquely) from the correct category \(\mathcal{SH}(X_{ét}^{\wedge})_p^{\wedge}\).
The rigidity Theorem 6.6 yields a symmetric monoidal étale realization functor \(\mathcal{SH}_{\text{ét}}(X)\to\mathcal{SH}(X_{\text{ét}}^{\wedge})_p^{\wedge}\) (Theorem 7.1). In the end, the endomorphisms of the monoidal unit in \(\mathcal{SH}_{\text{ét}}(X)[1/S]\) (i.e., away from the set \(S\) of primes noninvertible in \(X\)) is determined – it is identified with étale hypercohomology with coefficients in the \(S\)-local classical sphere spectrum (Corollary 7.3), as conjectured by Morel.

MSC:

14F20 Étale and other Grothendieck topologies and (co)homologies
14F42 Motivic cohomology; motivic homotopy theory

Citations:

Zbl 1351.19001
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References:

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