Integral Artin motives II: Perverse motives and Artin Vanishing Theorem

Technical Summary: Integral Artin Motives II: Perverse Motives and Artin Vanishing Theorem

Problem Statement
The paper addresses the construction of the perverse motivic $t$-structure on the category of Artin étale motives, denoted $\text{DM}^A_{\text{ét}}(S, R)$, where $S$ is a scheme and $R$ is a coefficient ring. While the ordinary motivic $t$-structure is well-understood on subcategories like Artin motives and 1-motives (via works such as [Org04, Ayo11, Vai19]), the existence of a perverse motivic $t$-structure remains a central open problem in the theory of motives. The goal is to define a $t$-structure whose heart corresponds to an abelian category of "mixed motivic sheaves" and for which realization functors (e.g., $\ell$-adic) are $t$-exact. A critical component of this theory is the validity of Artin's Vanishing Theorem, which asserts that the functor $f_!$ is right $t$-exact for affine morphisms $f$.

The specific challenge lies in the distinction between rational coefficients ($R=\mathbb{Q}$) and integral coefficients ($R=\mathbb{Z}$). While the perverse $t$-structure on étale sheaves is well-established, its motivic counterpart is obstructed by the behavior of the Artin truncation functor $\omega_0$ and the constructibility of objects in higher dimensions.

Methodology
The author employs a combination of $\infty$-categorical formalism, gluing techniques, and explicit computations of the Artin truncation functor $\omega_0$.

1. Definition of the Perverse Homotopy $t$-Structure:
   The paper introduces the perverse homotopy $t$-structure ($t_{hp}$) on the category of Artin motives. This structure is generated by Borel-Moore motives $M^{BM}_S(X)[\delta(X)]$ where $X$ is quasi-finite over $S$ and $\delta$ is a dimension function. The definition is motivated by the desire to recover the standard perverse $t$-structure on $\ell$-adic sheaves via realization functors.

2. Pointwise Description:
   A key technical tool is a pointwise characterization of $t_{hp}$. An object $M$ is $t_{hp}$-non-negative if and only if it is bounded below in the ordinary homotopy $t$-structure and, for every point $x \in S$, the Artin truncation of the exceptional pullback, $\omega_0 i^!_x M$, satisfies specific degree bounds relative to $\delta(x)$.

3. Analysis of the Artin Truncation Functor ($\omega_0$):
   The paper rigorously investigates the functor $\omega_0: \text{DM}^{\text{coh}}_{\text{ét}} \to \text{DM}^A_{\text{ét}}$, which is right adjoint to the inclusion of Artin motives into cohomological motives.

   • Rational Case: It is shown that $\omega_0$ preserves constructible objects when $R=\mathbb{Q}$ (extending results from [AZ12, Pep19b]).
   • Integral Case: The paper demonstrates that $\omega_0$ fails to preserve constructibility integrally when the morphism is not quasi-finite. Specifically, for a proper morphism $f: X \to \text{Spec}(k)$ with $X$ regular, $\omega_0 f_* \mathbf{1}_X$ may contain non-constructible components (e.g., copies of $\mathbb{Q}/\mathbb{Z}$), which obstructs the existence of a $t$-structure on constructible motives in higher dimensions.

4. Gluing and Stratification:
   The construction relies on gluing $t$-structures along stratifications of the base scheme $S$. The author utilizes the fact that constructible Artin motives become smooth (and thus identifiable with lisse étale sheaves) on a stratification.

Key Contributions and Results

1. Existence with Rational Coefficients:
   For any excellent scheme $S$, the perverse homotopy $t$-structure exists on $\text{DM}^A_{\text{ét}, c}(S, \mathbb{Q})$. Furthermore, the $\ell$-adic realization functor is $t$-exact. This establishes a robust theory of perverse Artin motives over $\mathbb{Q}$.

2. Artin Vanishing Theorems (Rational Case):
   The paper proves three analogs of Artin's Vanishing Theorem for Artin motives with rational coefficients:

   • Affine Quasi-finite: If $f: X \to S$ is a quasi-finite affine morphism and $X$ is nil-regular, then $f_!$ is $t_{hp}$-exact (Theorem 2.2.1). This holds for $\dim(S) \leq 2$ or if $S$ is of finite type over a finite field.
   • Proper Morphisms: If $f: X \to S$ is proper and $S$ allows resolution of singularities by alterations, then $\omega_0 f_*$ is right $t_{hp}$-exact (Theorem 2.3.1).
   • Open Immersions: If $j: U \to S$ is an open immersion with $U$ nil-regular, $\omega_0 j_*$ is $t_{hp}$-exact (Theorem 2.4.2).

3. Structure of the Heart (Rational Case):
   The heart of the perverse homotopy $t$-structure, denoted $\mathcal{M}^A_{\text{perv}}(S, \mathbb{Q})$, is shown to be an Artinian and Noetherian abelian category. Simple objects are classified as intermediate extensions of simple lisse sheaves (twisted by the dimension function), mirroring the structure of perverse sheaves.

4. Integral Coefficients and Dimensional Constraints:

   • Dimension $\leq 2$: For excellent schemes of dimension at most 2, the perverse homotopy $t$-structure induces a $t$-structure on the subcategory of constructible Artin motives $\text{DM}^A_{\text{ét}, c}(S, \mathbb{Z})$. The $\ell$-adic realization is $t$-exact in this setting (Theorem 3.3.6, Theorem 3.4.3).
   • Dimension $\geq 4$: The paper provides a counterexample showing that for $S = \mathbb{A}^4_k$ (affine 4-space), the perverse homotopy $t$-structure does not induce a $t$-structure on constructible motives with integral coefficients. The obstruction arises because the cohomology of the Artin truncation of the constant motive on a resolution of singularities involves non-constructible modules (specifically related to $H^1$ of the exceptional divisor).
   • Dimension 3: The case of 3-dimensional schemes remains open.

Significance and Claims
The paper claims to resolve the existence of the perverse motivic $t$-structure on Artin motives in low dimensions and with rational coefficients, providing the first unconditional construction of such a structure in the integral setting for $\dim(S) \leq 2$.

The primary significance lies in the reversal of the traditional philosophical approach: rather than assuming the existence of the $t$-structure to prove vanishing theorems, the author first establishes the vanishing theorems (specifically for $f_!$) as a tool to prove the existence of the $t$-structure.

The work highlights a fundamental divergence between rational and integral coefficients in the motivic setting. While rational coefficients allow for a "clean" theory where $\omega_0$ preserves constructibility, integral coefficients introduce pathological behaviors (non-constructible torsion) that prevent the existence of the $t$-structure in dimensions 4 and above. The paper concludes that the perverse homotopy $t$-structure is the "best possible approximation" to a perverse $t$-structure, coinciding with the true perverse motivic $t$-structure whenever the latter is defined (i.e., when realization functors are $t$-exact).

The results are presented as optimal: the existence in dimension 2 is proven, and the non-existence in dimension 4 is demonstrated, leaving the 3-dimensional case as the remaining frontier.