dorsal/arxiv
View SchemaPro-\'etale motives and solid rigidity
| Authors | Raphaël Ruimy, Swann Tubach, Sebastian Wolf |
|---|---|
| Categories | |
| ArXiv ID | 2601.07358vv1 |
| URL | https://arxiv.org/abs/2601.07358 |
| License | http://creativecommons.org/licenses/by/4.0/ |
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.
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"abstract": "We introduce coefficient systems of pro-\\\u0027etale motives and pro-\\\u0027etale motivic spectra with coefficients in any condensed ring spectrum and show that they afford the six operations. Over locally \\\u0027etale bounded schemes, \\\u0027etale motivic spectra embed into pro-\\\u0027etale 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-\\\u0027etale 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.",
"arxiv_id": "2601.07358",
"authors": [
"Rapha\u00ebl Ruimy",
"Swann Tubach",
"Sebastian Wolf"
],
"categories": [
"math.AG"
],
"license": "http://creativecommons.org/licenses/by/4.0/",
"title": "Pro-\\\u0027etale motives and solid rigidity",
"url": "https://arxiv.org/abs/2601.07358",
"version": "v1"
},
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"id": "arXiv Dataset",
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"variant": "snapshot-2026-01-17",
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