Pro-\'etale motives and solid rigidity

TL;DR

This paper develops a framework for pro-étale motives using condensed category theory, establishing a solidification process and a solid realization functor that extends classical $ ext{ell}$-adic realizations.

Contribution

It introduces coefficient systems of pro-étale motives, defines a solidification process for condensed categories, and constructs a solid realization functor extending $ ext{ell}$-adic realizations.

Findings

Pro-étale motives can be embedded into pro-étale motivic spectra over certain schemes.

The solidification process closely relates to Fargues-Scholze's solid sheaves, demonstrating a rigidity result.

Solid sheaves on schemes support the six operations and enable a new solid realization functor for motives.

Abstract

We introduce coefficient systems of pro-\'etale motives and pro-\'etale motivic spectra with coefficients in any condensed ring spectrum and show that they afford the six operations. Over locally \'etale bounded schemes, \'etale motivic spectra embed into pro-\'etale motivic spectra. We then use the framework of condensed category theory to define a solidification process for any $\widehat{\mathbb{Z}}$-linear condensed category. Pro-\'etale motives naturally enhance to a condensed category and we show that their solidification is very close to the category of solid sheaves defined by Fargues-Scholze, suitably modified to work on schemes: this is a rigidity result. As a consequence, we obtain that in contrast with the rigid-analytic setting, solid sheaves on schemes afford the six operations, and we obtain a solid realization functor of motives, extending the $\ell$-adic realization functor. The solid realization functor is compatible with change of coefficients, which allows one to recover the $\mathbb{Q}_\ell$-adic realization functor while remaining in a setting of presentable categories.