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Pro-étale motives and solid rigidity

Raphaël Ruimy, Swann Tubach, Sebastian Wolf

Abstract

We introduce coefficient systems of pro-étale motives and pro-étale motivic spectra with coefficients in any condensed ring spectrum and show that they afford the six operations. Over locally étale bounded schemes, étale motivic spectra embed into pro-étale 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-étale 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.

Pro-étale motives and solid rigidity

Abstract

We introduce coefficient systems of pro-étale motives and pro-étale motivic spectra with coefficients in any condensed ring spectrum and show that they afford the six operations. Over locally étale bounded schemes, étale motivic spectra embed into pro-étale motivic spectra. We then use the framework of condensed category theory to define a solidification process for any -linear condensed category. Pro-étale 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 -adic realization functor. The solid realization functor is compatible with change of coefficients, which allows one to recover the -adic realization functor while remaining in a setting of presentable categories.
Paper Structure (16 sections, 74 theorems, 219 equations)

This paper contains 16 sections, 74 theorems, 219 equations.

Table of Contents

  1. Solid sheaves on schemes.
  2. Recollections on the pro-étale topos
  3. Abelian categories.
  4. Derived categories
  5. Solid sheaves through condensed category theory
  6. Pro-étale motives.
  7. Basic definitions and properties
  8. Functoriality
  9. Localization
  10. Ambidexterity
  11. The six functors for pro-étale motives
  12. The embedding theorem from étale motives
  13. The rigidity theorem.
  14. Solid motives.
  15. Embedding solid sheaves into solid motives
  16. ...and 1 more sections

Key Result

Theorem 1

(thm:solid, prop:derivedSolid_Lambda, prop:cons_are_solid and prop:cons_are_solid_rational) Let $X$ be a qcqs scheme.

Theorems & Definitions (183)

  • Theorem 1
  • Theorem 2
  • Proposition 3
  • Theorem 4
  • Theorem 5
  • Proposition 6
  • Proposition 1.1.2
  • proof
  • Remark 1.1.3: Size issues
  • Lemma 1.1.4
  • ...and 173 more