Voevodsky motives and motives with modulus in positive characteristic

1. Problem Statement and Context

The paper addresses a fundamental question in the theory of motives: the relationship between Voevodsky's triangulated category of motives ($DM^{eff}(k)$), which is built on smooth varieties and is $\mathbb{A}^1$-invariant, and the triangulated category of motives with modulus ($MDM^{eff}(k)$), built on modulus pairs (varieties equipped with effective Cartier divisors) to capture non-$\mathbb{A}^1$-invariant cohomology theories (e.g., coherent cohomology).

• The Core Question: In characteristic $p > 0$, does the "modulus" (the divisor and its multiplicity) carry independent information in the rational category of motives with modulus?
• The Conjecture: It was previously known (via work by Miyazaki) that higher Chow groups with modulus become independent of the multiplicity of the divisor after inverting $p$. The paper aims to prove the motivic analogue: that after tensoring with $\mathbb{Q}$, the category of motives with modulus is equivalent to Voevodsky's category, rendering the multiplicities of the modulus "invisible."
• The Challenge: Proving this without assuming resolution of singularities, a standard tool in motivic homotopy theory that is not available in positive characteristic.

2. Methodology

The author employs a combination of sheaf-theoretic techniques, topological refinements, and geometric alteration theorems to bypass the need for resolution of singularities.

A. The Interior Generically Finite $h$-Site

A central innovation is the introduction of a new covering notion on the category of modulus pairs ($MSch$): the interior generically finite $h$-covering.

• Definition: A family of morphisms is an interior generically finite $h$-covering if it is built from Zariski open coverings and "proper interior generically finite" coverings.
• Key Property: A proper interior generically finite covering $\{N_i \to M\}$ requires that the map is proper and generically finite, and crucially, that for every point in the interior ($M^\circ = M \setminus |M_\infty|$), there is a neighborhood where the base change is finite and surjective.
• Refinement: The author proves (Proposition 2.2) that any such covering can be refined, up to morphisms in a specific class $\Sigma$ (minimal morphisms inducing isomorphisms on the interior), by a modulus $qfh$ covering (a topology defined by Kelly and Miyazaki).

B. Bhatt–Snowden's Refined Alteration Theorem

To construct the necessary coverings, the paper relies on the Refined Alteration Theorem by Bhatt and Snowden.

• Theorem: For a variety $X$ over a perfect field $k$ and a closed subvariety $Z$, there exists an alteration $f: Y \to X$ such that $Y$ is smooth, $f^{-1}(Z)$ is a strict normal crossings divisor, and $f$ is finite étale near a chosen finite set of points.
• Application: This allows the construction of coverings of smooth modulus pairs by log smooth modulus pairs (where the total space is smooth and the divisor is a strict normal crossings divisor), even without global resolution of singularities.

C. Sheaf Equivalences

The proof leverages existing equivalences established by Kelly and Miyazaki between different sheaf categories:
$$Shv_{Mqfh}(MSch, \mathbb{Q}) \simeq Shv_{Mqfh}(MCor, \mathbb{Q}) \simeq Shv_{MNis}(MCor, \mathbb{Q})$$
This allows the author to translate geometric covering properties into statements about the derived categories of sheaves.

3. Key Contributions and Results

Main Theorem (Theorem 1.1)

Let $k$ be a perfect field of characteristic $p > 0$. The localization functor
$$\omega^{eff}: MDM^{eff}(k, \mathbb{Q}) \to DM^{eff}(k, \mathbb{Q})$$
is an equivalence of triangulated categories.

• Implication: In positive characteristic, after tensoring with $\mathbb{Q}$, the category of motives with modulus collapses to Voevodsky's category. The "modulus" structure (specifically the multiplicities of the divisors) does not generate new rational motives.
• Corollary: If $X$ is a normal proper variety and $D, D'$ are effective Cartier divisors with the same support such that $X \setminus |D|$ is smooth, then:
  $$M(X, D)_\mathbb{Q} \simeq M(X, D')_\mathbb{Q}$$
  in $MDM^{eff}(k, \mathbb{Q})$.

Compactly Generated Structure (Theorem 1.2 & 3.6)

The paper establishes that $MDM^{eff}(k, \mathbb{Q})$ is compactly generated by motives of log smooth modulus pairs ($M \in MSmls$).

• Specifically, any smooth modulus pair $M$ can be "covered" in the derived category by a finite family of log smooth modulus pairs $\{N_i\}$ such that $Q_{tr}(M)$ is a direct summand of $\bigoplus Q_{tr}(N_i)$.
• This result is crucial because log smooth pairs are geometrically well-behaved (smooth total space, normal crossings divisor), allowing the use of the Refined Alteration Theorem.

Comparison Theorem (Theorem 3.7)

The proof of the main equivalence relies on showing that the generators of $MDM^{eff}(k, \mathbb{Q})$ (log smooth proper pairs) lie in the essential image of the functor $\omega^{eff}$.

• Using a previous result (from the author's paper [8]), the author shows that for a log smooth pair $(M, M_\infty)$ with $M_\infty$ reduced, the motive $M(M, M_\infty)_\mathbb{Q}$ is isomorphic to $\omega^{eff}(M(M^\circ)_\mathbb{Q})$.
• Since $\omega^{eff}$ is fully faithful and strongly additive, and the generators are in its image, the functor is an equivalence.

4. Significance

1. Resolution-Free Proof: The most significant technical achievement is proving this equivalence without assuming resolution of singularities. This makes the result robust for arithmetic geometry and fields of positive characteristic where resolution is not guaranteed.
2. Clarification of Modulus Motives: The result clarifies the role of modulus in positive characteristic. While modulus motives are essential for capturing non-$\mathbb{A}^1$-invariant phenomena (like coherent cohomology), the rational theory in positive characteristic does not distinguish between different multiplicities of the modulus. The "modulus" information is essentially torsion in the rational setting.
3. Methodological Advancement: The introduction of the interior generically finite $h$-site and its interplay with the modulus $qfh$ topology provides a new toolkit for studying motives with modulus. It demonstrates how geometric alterations (Bhatt–Snowden) can be adapted to the modulus setting to replace resolution of singularities.
4. Unification: It unifies the theory of motives with modulus with Voevodsky's classical theory in the rational, positive characteristic setting, suggesting that the "new" cohomology theories captured by modulus motives in this context are already encoded in the standard motivic framework once rational coefficients are applied.

In summary, Matsumoto proves that in positive characteristic, the complex structure of motives with modulus simplifies drastically over $\mathbb{Q}$, collapsing to the classical Voevodsky category, and achieves this through a novel synthesis of alteration theorems and refined covering topologies.