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The derived $\infty$ -category of Frobenius modules

This paper establishes a t-exact equivalence between the derived $\infty$ -category of Frobenius modules and the $\infty$ -category of Frobenius modules in the derived category for any quasi-compact $\mathbb{F}_p$ -scheme with affine diagonal, thereby generalizing previous results from regular Noetherian schemes and proving Zariski descent for these categories.

Original authors: Klaus Mattis, Timo Weiß

Published 2026-07-30

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CC BY 4.0

Original authors: Klaus Mattis, Timo Weiß

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are an architect trying to understand the shape of a city. In the world of mathematics, specifically a branch called algebraic geometry, cities are built from "schemes"—abstract structures that act like maps for solving equations. In these cities, there is a special kind of magic spell called the "Frobenius endomorphism." Think of this spell as a cosmic photocopier that takes a building (a mathematical object) and stamps it with a specific pattern based on a prime number $p$ . When you apply this spell to a building, you get a new version of it, and studying how these buildings change under the spell reveals deep secrets about the city's structure.

For a long time, mathematicians have been trying to build a perfect "translation guide" between two different ways of looking at these spell-stamped buildings. One way looks at the buildings as they are right now (the "ordinary" view), and the other way looks at them as a collection of all possible variations and histories (the "derived" view). The goal is to prove that these two views are actually just different lenses on the exact same reality. This is crucial because the "derived" view is much more powerful for solving hard problems, but it's only been proven to work perfectly when the city is built on very smooth, regular ground. The big question was: Does this perfect translation still work if the city is messy, has sharp corners, or isn't perfectly smooth?

This paper, written by Klaus Mattis and Timo Weiß, answers that question with a resounding "yes," but with a specific condition. The authors prove that for a wide class of these mathematical cities—specifically those that are "quasi-compact" (meaning they aren't infinitely sprawling) and have an "affine diagonal" (a technical way of saying the city's layout is reasonably well-behaved, like a city where every neighborhood connects neatly)—the translation guide works perfectly. They show that the "derived" view of Frobenius modules (the spell-stamped buildings) is exactly equivalent to the "Frobenius modules of the derived view." In simpler terms, you can take the messy, complex history of these buildings, apply the magic spell, and get the same result as if you applied the spell first and then looked at the history.

The authors had to overcome a major hurdle. In their previous work, they could only prove this for cities that were "regular" and "Noetherian" (math-speak for cities that are perfectly smooth and follow strict, finite rules). In those perfect cities, the magic spell was "flat," meaning it didn't distort the buildings at all. But in the messier, more general cities the authors are interested in, the spell does distort things; it's not flat. This distortion usually breaks the translation guide. To fix this, the authors didn't try to force the old rules to work. Instead, they built a new framework using "infinity-categories," which are like super-tools that can handle infinite layers of complexity and distortion without breaking.

They proved that even when the spell twists the buildings, the relationship between the "before" and "after" views remains a perfect match, as long as the city isn't too chaotic. They did this by showing that both sides of the equation behave like "Zariski sheaves." Imagine a sheaf as a puzzle where if you know the pieces for every small neighborhood, you can perfectly reconstruct the whole picture. The authors showed that you can build the solution for a whole city just by solving it for its small, affine (simple) neighborhoods and then stitching them together.

To make this work, they relied on a powerful theorem by Schwede and Shipley, which is like a master key. This key says that if a mathematical structure has a special "generator" (a single building block that can create everything else in the structure), then the whole structure is equivalent to a category of modules over a specific ring (a set of rules for combining numbers). The authors found that on these geometric schemes, both sides of their equation have these special generators, and the rules for combining them are identical. This proved that the two sides are not just similar, but mathematically identical.

The paper also discovered a helpful side effect: the "derived" category of these Frobenius modules follows the rules of "Zariski descent." This means that if you have a local rule for a neighborhood, and you have a consistent way to glue those rules together across the whole city, you can trust that the global rule exists and is unique. This is a fundamental property that makes the mathematics much more robust and easier to use for future discoveries.

In short, Mattis and Weiß have expanded the territory where we know the "derived" view of Frobenius modules works perfectly. They removed the strict requirement that the mathematical city must be perfectly smooth. Now, we know that as long as the city is reasonably well-connected and not infinitely sprawling, the deep structural relationship between these spell-stamped objects holds true, even in the presence of distortion. This opens the door for applying these powerful mathematical tools to a much wider range of geometric problems, allowing mathematicians to explore more complex and "messy" structures with the confidence of having a perfect translation guide.

Technical Summary: The Derived $\infty$ -Category of Frobenius Modules

Problem Statement

In algebraic geometry over a field of characteristic $p > 0$ , the study of modules equipped with an action of the absolute Frobenius endomorphism $F: X \to X$ is central. A Frobenius module is defined as a pair $(M, \kappa_M)$ where $M$ is a quasi-coherent $\mathcal{O}_X$ -module and $\kappa_M: M \to F_*M$ is an $\mathcal{O}_X$ -linear morphism.

Previous work by the authors in [MW24] established a canonical $t$ -exact equivalence between the derived $\infty$ -category of Frobenius modules and the category of Frobenius modules within the derived $\infty$ -category:
$D(\text{Frob}(\text{QCoh}(X), F_*)) \xrightarrow{\sim} \text{Frob}(D(\text{QCoh}(X)), D(F_*))$
However, this result was restricted to regular Noetherian $F_p$ -schemes. The restriction was necessary because, in the regular Noetherian case, Kunz's theorem guarantees that the Frobenius morphism is flat, making the pushforward functor $F_*$ exact. For arbitrary schemes, $F_*$ is not exact, and the proof strategy employed in [MW24] fails.

The central problem addressed in this paper is to extend the aforementioned equivalence to a broader class of schemes, specifically removing the regularity and Noetherian hypotheses, while working within the framework of stable $\infty$ -categories.

Methodology

The authors employ a combination of $\infty$ -categorical descent, module theory over ring spectra, and structural properties of derived categories. The proof strategy proceeds in three main stages:

1. Reduction to Affine Schemes via Zariski Descent

The authors first establish that the relevant derived $\infty$ -categories satisfy Zariski descent. They define a "geometric" scheme as a quasi-compact scheme with an affine diagonal (also known as semi-separated).

They prove that the presheaf $D^+(\text{Frob}(\text{QCoh}(-), F_*))$ defines a Zariski sheaf on the small Zariski site of a geometric scheme.
This relies on showing that the construction of Frobenius modules is functorial and limit-preserving, and that the derived functor $D^+$ preserves limits under specific base change conditions (flat base change).
Consequently, the global equivalence problem reduces to the case where $X = \text{Spec}(R)$ is an affine scheme.

2. Identification with Module Spectra (Schwede–Shipley Theorem)

For the affine case $X = \text{Spec}(R)$ , the authors identify both sides of the proposed equivalence with categories of module spectra over specific $E_1$ -ring spectra.

Source Side: They analyze $D(\text{Frob}(\text{Mod}^\heartsuit_R, F_*))$ , the derived $\infty$ -category of Frobenius modules over the abelian category of $R$ -modules. They construct a compact projective generator $A$ for this category.
Target Side: They analyze $\text{Frob}(D(\text{Mod}^\heartsuit_R), D(F_*))$ . They construct a corresponding compact projective generator $B$ for this category.
Schwede–Shipley Theorem: Using the $\infty$ -categorical version of Gabriel's theorem (Schwede–Shipley), they show that a presentable stable $\infty$ -category with a compact projective generator is equivalent to the category of module spectra over the endomorphism ring spectrum of that generator.
The authors prove that the endomorphism ring spectra $\text{End}(A)$ and $\text{End}(B)$ are equivalent. Specifically, they are shown to be equivalent to the discrete (non-commutative) ring $R[F]^{\text{op}}$ (or $R[\tau]^{\text{op}}$ in the language of $\tau$ -sheaves), viewed as an $E_1$ -ring.

3. Extension to Left-Completeness

Having established the equivalence for bounded-above objects ( $D^+$ ), the authors extend the result to the full derived $\infty$ -categories ( $D$ ).

They prove that for geometric schemes, the derived $\infty$ -categories of both Frobenius modules and quasi-coherent sheaves are left-complete. This relies on verifying a weaker version of Grothendieck's axiom AB4 $^*_n(\omega)$ for the category of Frobenius modules.
Using a lemma stating that a $t$ -exact functor between left-complete stable $\infty$ -categories is an equivalence if its restriction to bounded-above objects is an equivalence, they lift the affine result to the full derived categories.

Key Contributions and Results

Main Theorem (Theorem A / Theorem 5.7)

For any geometric $F_p$ -scheme $X$ (quasi-compact with affine diagonal), there exists a canonical $t$ -exact equivalence of presentable stable $\infty$ -categories:
$D(\text{Frob}(\text{QCoh}(X), F_*)) \xrightarrow{\sim} \text{Frob}(D(\text{QCoh}(X)), D(F_*))$
This result generalizes the previous work [MW24] by removing the assumptions that $X$ be regular and Noetherian.

Descent Results

Theorem C (Proposition 4.13): The presheaf $D^+(\text{Frob}(\text{QCoh}(-), F_*))$ defines a Zariski sheaf on the small Zariski site of a geometric scheme.
Theorem D: The full derived $\infty$ -category $D(\text{Frob}(\text{QCoh}(-), F_*))$ also defines a Zariski sheaf with values in presentable stable $\infty$ -categories.
Corollary E: An analogous descent result holds for the derived $\infty$ -category of Cartier modules.

Structural Properties

Left-Completeness: The paper proves that for geometric schemes, the derived $\infty$ -categories of Frobenius modules are left-complete (Proposition 5.2). This is a non-trivial structural result derived from the properties of the underlying abelian categories.
Functoriality: The construction of Frobenius modules is shown to be functorial in the underlying $\infty$ -category and its endofunctor, preserving limits (Proposition 2.5).

Significance and Claims

The paper claims that the derived $\infty$ -category framework is essential for these results. The authors note that if one were to replace the derived $\infty$ -category with the ordinary derived category (the homotopy category), the target categories would fail to admit a triangulated structure, rendering the equivalence impossible.

The significance of the work lies in:

Generalization: It extends the structural understanding of Frobenius modules to a much wider class of schemes (geometric schemes), which includes all quasi-compact separated schemes, without requiring regularity.
Methodological Robustness: It demonstrates that the $\infty$ -categorical approach, specifically utilizing the Schwede–Shipley recognition theorem and descent theory, provides a robust framework for handling non-exact functors (like $F_*$ on non-regular schemes) where classical derived category methods fail.
Descent: It establishes that the derived theory of Frobenius and Cartier modules satisfies Zariski descent, allowing local-to-global arguments in positive characteristic geometry.

The authors do not claim to solve open problems regarding the specific cohomological computations of these modules but rather provide the correct categorical framework in which such computations can be performed for general geometric schemes. The work is presented as a foundational step, generalizing the results of [MW24] and [BB11, BB13, BP09] to a non-Noetherian, non-regular setting.

Technical Summary: The Derived ∞\infty∞-Category of Frobenius Modules