Profinite Borel completeness and smooth Artin motives

Abstract

The purpose of this paper is twofold. In the first part, we revisit the description of the $\infty$-category of Borel complete equivariant spectra for a finite group given by Mathew-Naumann-Noel, introduce a version with coefficients, and then consider Borel equivariance for profinite groups. Here we identify two generally differing notions: levelwise Borel completeness and the hypercompletion thereof. In the second part, we study variants of smooth Artin motives, which are subcategories of the $\infty$-categories of effective Nisnevich and étale Voevodsky motives over a base scheme $S$ that are controlled by the étale fundamental group $\ddot{I}{€}_1^{\tilde{\mathrm{A}}\text{c◯}\mathrm{t}}(S)$. In the Nisnevich case, we extend a theorem of Voevodsky and identify smooth Artin motives with modules over the Bredon cohomology spectrum for the profinite group $\ddot{I}{€}_1^{\tilde{\mathrm{A}}\text{c◯}\mathrm{t}}(S)$. In the étale case, we show that the difference between our two notions of profinite Borel completeness is precisely the difference between étale sheaves and hypersheaves on finite étale schemes.

56 pages; v2: changes to section 5, fixed what is now 5.32

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