Yorick Fuhrmann | Searcharxiv

arXiv2026

Profinite Borel completeness and smooth Artin motives

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 \'etale Voevodsky motives over a base scheme $S$ that are controlled by the \'etale fundamental group $\pi_1^{\mathrm{\'et}}(S)$. 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 $\pi_1^{\mathrm{\'et}}(S)$. In the \'etale case, we show that the difference between our two notions of profinite Borel completeness is precisely the difference between \'etale sheaves and hypersheaves on finite \'etale schemes.

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arXiv2025

Modular fixed points in equivariant homotopy theory

We show that the derived $\infty$-category of permutation modules is equivalent to the category of modules over the Eilenberg-MacLane spectrum associated to a constant Mackey functor in the $\infty$-category of equivariant spectra. On such module categories we define a modular fixed point functor using geometric fixed points followed by an extension of scalars and identify it with the modular fixed point functor on derived permutation modules introduced by Balmer-Gallauer. As an application, we show that the Picard group of such a module category for a $p$-group is given by the group of class functions satisfying the Borel-Smith conditions. In the language of representation theory, this result was first obtained by Miller.

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