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Integral Artin motives I: Smooth objects and the ordinary t-structure

Raphaël Ruimy

TL;DR

The paper develops an integral framework for smooth Artin motives by linking them to étale local systems and Artin representations, and constructs the ordinary motivic t-structure on Artin motives with integral coefficients. It proves that the ℓ-adic realization is t-exact and extends conservativity results from rational to integral coefficients by incorporating v-adic and λ-adic realizations within the Bhatt–Scholze/pro-étale formalism. A detailed description of smooth Artin motives is given via equivalences with lisse and Ind-lisse étale sheaves, and a stratified approach expresses constructible Artin motives as gluings of representation-theoretic data. The work also establishes that torsion motives are Artin motives and develops both the ordinary and perverse (in the sequel) motivic t-structures, providing a robust bridge between motivic and étale–cohomological frameworks with explicit hearts and conservativity results.

Abstract

We link smooth Artin motives to étale local systems and Artin representations. We then construct the ordinary motivic t-structure on Artin motives with integral coefficients and show that the $\ell$-adic realization functor is t-exact.

Integral Artin motives I: Smooth objects and the ordinary t-structure

TL;DR

The paper develops an integral framework for smooth Artin motives by linking them to étale local systems and Artin representations, and constructs the ordinary motivic t-structure on Artin motives with integral coefficients. It proves that the ℓ-adic realization is t-exact and extends conservativity results from rational to integral coefficients by incorporating v-adic and λ-adic realizations within the Bhatt–Scholze/pro-étale formalism. A detailed description of smooth Artin motives is given via equivalences with lisse and Ind-lisse étale sheaves, and a stratified approach expresses constructible Artin motives as gluings of representation-theoretic data. The work also establishes that torsion motives are Artin motives and develops both the ordinary and perverse (in the sequel) motivic t-structures, providing a robust bridge between motivic and étale–cohomological frameworks with explicit hearts and conservativity results.

Abstract

We link smooth Artin motives to étale local systems and Artin representations. We then construct the ordinary motivic t-structure on Artin motives with integral coefficients and show that the -adic realization functor is t-exact.
Paper Structure (29 sections, 66 theorems, 187 equations)

This paper contains 29 sections, 66 theorems, 187 equations.

Table of Contents

  1. Preliminaries
  2. Categorical Preliminaries
  3. Complements on t-categories
  4. Idempotent Complete $\mathfrak{p}$-localization
  5. Stable Categories of Étale motives
  6. Small Étale Site
  7. Change of Coefficients
  8. The -adic Realization and the Bhatt-Scholze Formalism
  9. The λ-adic realization
  10. The $v$-adic Realization
  11. Artin Motives
  12. Subcategories of Dimensional Motives
  13. Generators and Stability
  14. Torsion Motives are Artin Motives
  15. Torsion Étale Motives and Torsion Étale Sheaves
  16. ...and 14 more sections

Key Result

Theorem 1

(orgoayo11abvbvkplhplh2vaish) Let $S$ be a noetherian excellent finite dimensional scheme allowing resolution of singularities by alterations, let $\ell$ be a prime number invertible on $S$ and let $n=0,1$. Then, there is a t-structure on the stable $\infty$-category $\mathcal{DM}^n_{\operatorname{\ is t-exact when the derived category $\mathcal{D}^b_c(S,\mathbb{Q}_\ell)$ of constructible $\ell$-a

Theorems & Definitions (136)

  • Theorem
  • Proposition
  • Theorem
  • Theorem
  • Remark
  • Definition 1.1.1
  • Definition 1.1.2
  • Proposition 1.1.3
  • Proposition 1.1.4
  • proof
  • ...and 126 more