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Integral Artin motives II: Perverse motives and Artin Vanishing Theorem

Raphaël Ruimy

TL;DR

The paper introduces a perverse motivic framework for Artin étale motives with integral coefficients, proving the existence of a perverse homotopy $t$-structure on Artin motives over base schemes of dimension at most 2 and showing nonexistence in dimension 4. It builds this via Artin truncation functors, locality/gluing, and a robust six-functor formalism, establishing an Artin Vanishing-type property for integral coefficients. A central achievement is the construction and analysis of the abelian category of perverse Artin motives, including the intermediate extension functor and a Beilinson-style description of simple objects, along with a detailed study of $v$-adic realizations and $t$-exactness. The results connect motivic and étale realizations, revealing when a perverse motivic $t$-structure can exist and how the heart mirrors classical perverse sheaf theory in the Artin setting. This provides a concrete, dimension-sensitive bridge between motivic theory and étale cohomology, with explicit consequences for functoriality and vanishing results.

Abstract

In this text, we are mainly interested in the existence of the perverse motivic t-structures on the category of Artin étale motives with integral coefficients. We construct the perverse homotopy t-structure which is the best possible approximation to a perverse t-structure on Artin motives with rational coefficients. The heart of this t-structure has properties similar to those of the category of perverse sheaves and contains the Ayoub-Zucker motive. With integral coefficients, we construct the perverse motivic t-structure on Artin motives when the base scheme is of dimension at most $2$ and show that it cannot exist in dimension $4$. This construction relies notably on a an analogue for Artin motives of the Artin Vanishing Theorem.

Integral Artin motives II: Perverse motives and Artin Vanishing Theorem

TL;DR

The paper introduces a perverse motivic framework for Artin étale motives with integral coefficients, proving the existence of a perverse homotopy -structure on Artin motives over base schemes of dimension at most 2 and showing nonexistence in dimension 4. It builds this via Artin truncation functors, locality/gluing, and a robust six-functor formalism, establishing an Artin Vanishing-type property for integral coefficients. A central achievement is the construction and analysis of the abelian category of perverse Artin motives, including the intermediate extension functor and a Beilinson-style description of simple objects, along with a detailed study of -adic realizations and -exactness. The results connect motivic and étale realizations, revealing when a perverse motivic -structure can exist and how the heart mirrors classical perverse sheaf theory in the Artin setting. This provides a concrete, dimension-sensitive bridge between motivic theory and étale cohomology, with explicit consequences for functoriality and vanishing results.

Abstract

In this text, we are mainly interested in the existence of the perverse motivic t-structures on the category of Artin étale motives with integral coefficients. We construct the perverse homotopy t-structure which is the best possible approximation to a perverse t-structure on Artin motives with rational coefficients. The heart of this t-structure has properties similar to those of the category of perverse sheaves and contains the Ayoub-Zucker motive. With integral coefficients, we construct the perverse motivic t-structure on Artin motives when the base scheme is of dimension at most and show that it cannot exist in dimension . This construction relies notably on a an analogue for Artin motives of the Artin Vanishing Theorem.

Paper Structure

This paper contains 32 sections, 81 theorems, 313 equations.

Table of Contents

  1. Preliminaries
  2. Categorical complements
  3. Notions of Abelian and Stable Subcategories Generated by a Set of Objects
  4. Induced t-structure
  5. t-structure Generated by a Set of Objects
  6. Idempotent Complete $\mathfrak{p}$-localization
  7. Complements on Étale Motives
  8. Small Étale Site
  9. Change of Coefficients
  10. Strengthening of the Continuity Property
  11. The Perverse t-structure on Torsion Motives
  12. The Perverse t-structure on étale sheaves
  13. The perverse t-structure on torsion motives
  14. Artin Motives
  15. Subcategories of Dimensional Motives
  16. ...and 17 more sections

Key Result

Theorem 1

(Artin Vanishing Theorem) Let $k$ be a field and let $f\colon X\rightarrow Y$ be an affine morphism between $k$-schemes of finite type. Then, the functor is perverse right t-exact (meaning that is preserves negative objects of the perverse t-structure). By duality, the functor $f_!$ is perverse left t-exact.

Theorems & Definitions (171)

  • Theorem
  • Theorem
  • Theorem
  • Theorem
  • Theorem
  • Proposition
  • Theorem
  • Theorem
  • Proposition
  • Definition 1.1.1
  • ...and 161 more