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Rigidity in étale motivic stable homotopy theory

Tom Bachmann

Algebraic & Geometric Topology 21 (2021) 173–209

For a scheme X, denote by 𝒮(Xét) the stabilization of the hypercompletion of its étale –topos, and by 𝒮ét(X) the localization of the stable motivic homotopy category 𝒮(X) at the (desuspensions of) étale hypercovers. For a stable –category 𝒞, write 𝒞p for the p–completion of 𝒞.

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

ep: 𝒮(X ét) p𝒮 ét(X)p

is an equivalence of –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 𝒮ét(X) are given by étale hypercohomology with coefficients in the S–local classical sphere spectrum:

[1[1S],1[1S]]𝒮ét(X) ét0(X,1[1S]).

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 1̂p(1)[1] 𝒮(Xét)p with the property that ep(1̂p(1)[1]) Σ𝔾m 𝒮ét(X)p.

motivic homotopy theory, étale cohomology
Mathematical Subject Classification 2010
Primary: 14F20, 14F42
Received: 29 May 2019
Revised: 24 March 2020
Accepted: 11 May 2020
Published: 25 February 2021
Tom Bachmann
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA
United States
Mathematisches Institut
LMU Munich