On the Integral Part of A-Motivic Cohomology

Technical Summary: On the Integral Part of A-Motivic Cohomology

Problem Statement
The paper addresses the definition and properties of the "integral part" of motivic cohomology in the context of global function fields of positive characteristic. In the classical setting of number fields, the integral part of motivic cohomology is conjecturally captured by two equivalent approaches: one via the K-theory of regular integral models of varieties, and the other via the $\ell$-adic realization of extensions having good reduction. It is expected that these two definitions coincide (Conjecture C4) and that the resulting space is finite-dimensional over $\mathbb{Q}$ (Conjecture C5).

The author seeks to establish an analogous theory for Anderson $A$-motives, which serve as the function field counterparts to classical mixed motives. The central problem is to define the integral part of $A$-motivic cohomology using maximal integral models (analogous to regular models) and the "good" $\ell$-adic part (extensions with good reduction), and to investigate whether these two definitions coincide and whether the resulting module is finitely generated.

Methodology
The paper employs the framework of Anderson $A$-motives over global and local function fields. The methodology proceeds through several key steps:

1. Foundations of $A$-motives: The author reviews the category of Anderson $A$-motives ($\mathcal{M}_R$), establishing it as an exact category (though not abelian). The extension modules $\text{Ext}^i_{\mathcal{M}_R}(1, M)$ are computed explicitly using the complex $M \xrightarrow{\text{id} - \tau_M} M[j^{-1}]$, where $j$ is the ideal generated by $a \otimes 1 - 1 \otimes \kappa(a)$.
2. Mixedness and Weights: The concept of mixed Anderson $A$-motives is developed using the slope filtration of associated isocrystals at the infinite place $\infty$. The author defines weights and establishes that the category of mixed motives behaves similarly to the number field case regarding extension vanishing for $i > 1$, though the subcategory of pure motives is not semi-simple.
3. Maximal Integral Models: Inspired by Gardeyn's work on $\tau$-sheaves, the paper introduces the notion of maximal integral models for $A$-motives over local and global function fields. Unlike previous approaches, the author does not assume a priori that these models are locally free, proving this property as a consequence of maximality.
   • For local fields, maximal models are shown to exist and be unique.
   • For global fields, the maximal integral model is constructed as the intersection of local maximal models.
4. Comparison of Extensions: The author defines the "integral part" of cohomology ($\text{Ext}^1_{\mathcal{O}_p}(1, M)$) as the image of the maximal integral model under the extension map $\iota$. This is compared against the "good $\ell$-adic part" ($\text{Ext}^1_{\text{good}}(1, M)_\ell$), defined as the kernel of the $\ell$-adic realization map restricted to the inertia subgroup.
5. Regulated Extensions: To address discrepancies between the integral and good parts, the paper introduces "regulated extensions," defined by the additivity of Hodge polygons (analogous to Hodge additivity in Pink's theory).

Key Contributions and Results

• Existence and Uniqueness of Maximal Models: The paper proves that maximal integral models for $A$-motives exist and are unique over both local and global function fields. Furthermore, it is shown that these maximal models are automatically projective (locally free) over the relevant tensor rings, simplifying the technical requirements found in Gardeyn's earlier work on $\tau$-sheaves.
• Inclusion of Integral in Good: The primary result (Theorem D) establishes that the integral part of the $A$-motivic cohomology is contained within the good $\ell$-adic part:
  $$\text{Ext}^1_{\mathcal{O}_p}(1, M) \subseteq \text{Ext}^1_{\text{good}}(1, M)_\ell$$
• Failure of Equality: Contrary to the expectations in the number field setting (Conjecture C4), the paper demonstrates that the inclusion above is generally strict. In Subsection 5.1, an explicit counterexample is constructed for the neutral motive $\mathbf{1}$, showing an extension that has good reduction with respect to a maximal ideal $\ell$ but does not arise from the maximal integral model.
• Finite Generation Issue: The paper notes that the module of integral extensions $\text{Ext}^1_{\mathcal{O}_F}(1, M)$ is typically not finitely generated over $A$, violating the analogue of Conjecture C5. This suggests that the category of all $A$-motives is too large to support a well-behaved motivic cohomology theory without further restrictions.
• Regulated Extensions: The author proposes "regulated extensions" as a refined subcategory where the Hodge polygon of the extension matches the sum of the Hodge polygons of the constituents.
  • It is proven that the map $\iota$ induces an isomorphism between a specific quotient of the motive and the module of regulated extensions (Corollary 5.5).
  • The paper conjectures that for regulated extensions, the integral and good parts coincide (Conjecture 5.8).
  • This conjecture is verified in specific cases: when the motive arises from a Frobenius space with good reduction (Theorem 5.9) and for positive Carlitz twists $A(n)$ (Theorem 5.13).

Significance and Claims
The paper claims to initiate the systematic study of $A$-motivic cohomology for global function fields. Its significance lies in:

1. Establishing the Framework: Providing rigorous definitions for the integral part of $A$-motivic cohomology using maximal models, which serve as the function field analogue of regular integral models.
2. Highlighting Divergence: Demonstrating that the function field picture differs fundamentally from the number field picture, specifically that the integral and good $\ell$-adic definitions do not coincide in general. This challenges the direct transfer of conjectures from number fields to function fields.
3. Proposing a Correction: Introducing the notion of regulated extensions as a necessary condition to restore the expected equality between integral and good parts and to potentially recover finite generation properties.

The author modestly concludes that while the general case of the equality for regulated extensions remains open, the work provides a concrete path forward by identifying the specific structural obstructions (lack of Hodge additivity) and solving the problem for important classes of motives. The results rely heavily on the exact category formalism and the interplay between $A$-motives, isocrystals, and $\ell$-adic representations.