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Frobenius generation for algebraic stacks

This paper introduces the concept of $F$ -finiteness for algebraic stacks in positive characteristic and proves that sufficiently many Frobenius pushforwards generate the bounded derived categories of coherent sheaves on Noetherian concentrated $F$ -finite stacks with quasi-finite and separated diagonal, thereby generalizing and independently recovering a recent result by Ballard et al.

Original authors: Pat Lank, Fei Peng

Published 2026-08-04

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Pat Lank, Fei Peng

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 trying to understand the shape of a complex, multi-layered object, like a giant, twisting sculpture made of invisible threads. In the world of mathematics, specifically a branch called algebraic geometry, these "sculptures" are called algebraic stacks. They are the ultimate generalization of shapes like curves, surfaces, and higher-dimensional spaces, but they can have weird, hidden symmetries and singularities that make them incredibly hard to study. To understand these shapes, mathematicians use a powerful tool called a derived category. Think of this category as a massive library containing every possible "view" or "snapshot" of the shape, organized in a way that reveals its deep structural secrets.

The big question mathematicians have been asking is: Can we find a single, special book (or a small collection of books) in this library that allows us to reconstruct every other book in the collection? If we have such a "master key," we can understand the entire shape just by studying that one key. This is called finding a generator. For simple shapes, we know these keys exist. But for the most complex, twisted stacks, finding an explicit key has been a mystery. The paper you are about to read tackles this mystery in a specific setting: shapes built in a mathematical universe where numbers behave differently, known as positive characteristic (think of a world where counting wraps around, like a clock, but with a prime number of hours).

The Magic of the Frobenius Mirror

In this strange mathematical world, there is a special operation called the Frobenius morphism. You can imagine this as a magical mirror that reflects the entire shape onto itself, but it does so in a way that twists and stretches the underlying fabric of the shape. When you look at the shape through this mirror, you get a new version of it, called the Frobenius pushforward.

The authors of this paper, Pat Lank and Fei Peng, discovered something amazing: if you keep looking at the shape through this magic mirror over and over again (iterating the process), the reflections eventually become so rich and detailed that they can generate the entire library of snapshots. In other words, if you take a sufficiently complex "seed" object and apply this Frobenius mirror enough times, the resulting collection of reflections acts as a classical generator. This means that with enough iterations, you can build any other object in the derived category using only these reflections, along with some standard mathematical operations like adding them together or taking pieces of them.

The New Rulebook: F-Finiteness

To make this work, the authors had to first define what it means for a complex stack to be "well-behaved" enough for this magic to happen. They introduced a new rule called F-finiteness.

Think of F-finiteness as a guarantee that the shape isn't too wild or infinite in its complexity. Just as a finite library has a limited number of books, an F-finite stack has a manageable structure where the Frobenius mirror doesn't create infinite chaos. However, this rule is quite picky. The authors show that while F-finiteness holds for many important types of shapes—such as Deligne–Mumford stacks (which are like shapes with a few extra symmetries attached) and certain classifying stacks representing specific groups—it fails for others.

For instance, they prove that the classifying stack for the additive group ( $B\mathbb{G}_a$ ) is not F-finite because its underlying symmetry group is too "loose" (not linearly reductive). In fact, for a classifying stack to be F-finite, the symmetry group must have a specific property: its "Frobenius kernel" must be linearly reductive. This means the magic trick only works for stacks where the symmetries are "rigid" enough, not for every possible group symmetry.

The Three-Step Construction

The paper doesn't just claim this works; it builds the proof like a master architect constructing a bridge, step by step:

The Local Foundation: First, they look at the tiniest possible pieces of the shape (local rings). They use a clever trick involving Koszul complexes (which are like mathematical scaffolding) to show that even at this microscopic level, the Frobenius reflections eventually contain the "residue field" (the most basic building block of the shape).
The Affine Expansion: Next, they expand from these tiny pieces to slightly larger, flat shapes called affine schemes. They use a result from commutative algebra to glue these local findings together, showing that the generator property holds for these larger, flat patches.
The Global Glue: Finally, they tackle the full, complex algebraic stacks. They use a technique called étale dévissage, which is like taking a complex 3D object, slicing it into manageable flat pieces, solving the problem on each piece, and then gluing the solutions back together. They prove that if the pieces work, the whole stack works, provided the stack is "concentrated" (a technical condition ensuring the shape isn't too spread out) and has a "separated quasi-finite diagonal" (a condition ensuring the symmetries don't get too tangled).

The "Codepth" and How Many Mirrors?

A natural question arises: How many times do you need to look in the mirror before you get a generator? The authors introduce a number called codepth (or a related invariant $\gamma$ ) to estimate this. They show that if you iterate the Frobenius map more than $\lfloor \log_p(N) \rfloor + 1$ times (where $N$ is related to how many sections are needed to describe the shape), you are guaranteed to have a generator. This gives a concrete, calculable limit rather than just saying "it happens eventually."

What This Means for Regularity

One of the coolest side effects of their discovery is a new way to test if a shape is "smooth" or "regular." In the past, mathematicians knew that if a shape is perfectly smooth, the Frobenius mirror behaves in a specific, nice way. The authors show the reverse is also true for these stacks: if the Frobenius pushforwards of a generator stay within the "perfect" category (a specific, well-behaved subset of the library) for many iterations, then the shape must be smooth in that region. This provides a purely categorical test for smoothness, meaning you can tell if a shape is smooth just by looking at how its library of snapshots behaves under the Frobenius mirror, without needing to measure the shape directly.

The Limits of the Magic

The authors are careful to point out where their magic doesn't work. They explicitly show that if a stack is not concentrated (meaning it's too "spread out" or has infinite complexity in its structure), the Frobenius pushforward might not even preserve the "boundedness" of the objects. In their counterexample, they show that for a stack like $B\mathbb{G}_a$ (which is not concentrated), the Frobenius pushforward of a bounded complex can become unbounded, breaking the whole system. This confirms that their assumptions are not just technicalities but essential requirements for the result to hold.

In summary, this paper provides a powerful new toolkit for understanding the geometry of complex algebraic stacks in positive characteristic. It proves that the Frobenius morphism, when iterated enough times, acts as a universal generator for the derived category of coherent sheaves, provided the stack satisfies the F-finiteness and concentrated conditions. It bridges the gap between abstract category theory and concrete geometric properties, offering a way to detect smoothness and generate complex structures from simple, repeated reflections.

Technical Summary: Frobenius Generation for Algebraic Stacks

Problem Statement
The paper addresses the explicit construction of generators for the bounded derived category of coherent sheaves, $D^b_{\text{coh}}(\mathcal{X})$ , on Noetherian algebraic stacks $\mathcal{X}$ in positive characteristic. While the existence of classical and strong generators for such categories is well-established (e.g., [BVdB03, HP26, DLM25]), explicit constructions identifying specific generators remain scarce. In the context of schemes, Ballard–Iyengar–Lank–Mukhopadhyay–Pollitz ([BIL+26]) recently demonstrated that sufficiently many iterates of the Frobenius pushforward of a compact generator yield a classical generator. This work seeks to extend this result to the broader setting of algebraic stacks, a task complicated by the lack of a defined notion of $F$ -finiteness for stacks and the geometric complexities inherent to stacky structures (e.g., stabilizers and non-representable morphisms).

Methodology
The authors employ a geometric approach distinct from the local-to-global tensor action principles used in [BIL+26]. Instead, they leverage Hall–Rydh's étale dévissage ([HR18]) to construct generators geometrically. The argument proceeds in three distinct stages:

Local Rings: The foundation relies on a result by Ballard–Bhatt–Schlank–Stellmacher ([BBST26]), which establishes that the residue field lies in the thick subcategory generated by sufficiently many Frobenius pushforwards of Koszul complexes. This allows the authors to bound the number of iterations required based on numerical invariants of the local ring.
Affine Schemes: The result for local rings is extended to $F$ -finite affine schemes using a commutative algebra result by Letz ([Let21]), which facilitates the gluing of local generation properties.
Algebraic Stacks: The authors utilize Hall–Rydh's étale dévissage to glue these local generators. This involves verifying that the generation property is preserved under open immersions, finite flat surjective morphisms from affine schemes, and étale neighborhoods. This process requires the stack to satisfy specific finiteness conditions, notably being "concentrated" (quasi-compact, quasi-separated, with finite cohomological dimension for derived pushforwards) and having a quasi-finite, separated diagonal.

A key technical innovation is the introduction of a new invariant, $\gamma(F_*\mathcal{O}_X)$ , defined via the differences between the minimal number of generators of Frobenius pushforwards and the depth of local rings. This invariant serves as a tractable substitute for the "codepth" used in the scheme case, allowing for control over the number of Frobenius iterates needed in the étale topology.

Key Contributions and Definitions

Definition of $F$ -finiteness for Stacks: The paper proposes a definition for an algebraic stack $\mathcal{X}$ over $\mathbb{F}_p$ to be $F$ -finite if its absolute Frobenius morphism $F_{\mathcal{X}}$ is concentrated and locally of finite type. This generalizes the scheme-theoretic definition. The authors characterize this property for classifying stacks $BG$ of group schemes, showing $BG$ is $F$ -finite if and only if the Frobenius kernel of $G$ is linearly reductive.
Frobenius Generation Theorem: The main result (Theorem 1.1 / Theorem 6.13) states that for a Noetherian, concentrated, $F$ -finite algebraic stack $\mathcal{X}$ with a separated quasi-finite diagonal, and for any closed subset $Z \subseteq |\mathcal{X}|$ , there exists a perfect complex $G \in \text{Perf}_Z(\mathcal{X})$ such that the sufficiently iterated Frobenius pushforward $R F^e_* G$ is a classical generator for $D^b_{\text{coh}, Z}(\mathcal{X})$ .
Bounding Iterates: The paper provides a bound for the number of iterates $e$ required. For Deligne–Mumford stacks, $e$ can be taken as $\lfloor \log_p(\gamma(F_*\mathcal{O}_U)) \rfloor + 1$ , where $U \to \mathcal{X}$ is an étale presentation from an affine scheme.

Results and Applications

Regularity Criterion: As a consequence of the main theorem, the authors derive a categorical criterion for regularity (Corollary 1.3). A closed subset $Z$ is contained in the regular locus of $\mathcal{X}$ if and only if $R F^e_* G$ remains a perfect complex for all sufficiently large $e$ and any classical generator $G$ of $\text{Perf}_Z(\mathcal{X})$ . This recovers and generalizes Kunz's regularity criterion to the stack setting.
Strong Generators: The paper establishes that for concentrated $F$ -finite Deligne–Mumford stacks, the Frobenius pushforward of a classical generator becomes a strong generator for $D^b_{\text{coh}}(\mathcal{X})$ (Corollary 6.16).
Stacky Curves: The authors partially address when $F^e_* \mathcal{O}_{\mathcal{X}}$ is a strong generator for tame stacky curves. They show that if $R F^e_* \mathcal{O}_{\mathcal{X}}$ is a strong generator, the coarse moduli space of the curve must have genus zero (Proposition 6.20).

Significance and Limitations
The paper claims to independently recover and generalize the main result of [BIL+26] to algebraic stacks. The significance lies in providing explicit generators for derived categories of stacks, a class of objects central to modern algebraic geometry but previously lacking explicit generator constructions in positive characteristic.

The authors are modest regarding the scope of their results, noting specific limitations:

The concentrated condition is necessary for their method, as it ensures compact objects coincide with perfect complexes. They do not know if the result holds for non-concentrated $F$ -finite stacks.
The $F$ -finiteness assumption is necessary even in the affine case; counterexamples exist for stacks locally of finite type over $\mathbb{F}_p$ that are not $F$ -finite (e.g., certain classifying stacks of non-linearly reductive groups).
The paper does not claim to resolve the question of when $F^e_* \mathcal{O}_{\mathcal{X}}$ is a strong generator for general algebraic stacks, noting that this property is restrictive (e.g., failing for smooth projective curves of positive genus).

The work suggests that the techniques of étale dévissage and the invariant $\gamma$ may be useful for establishing other local-global generation results for algebraic stacks beyond the immediate scope of Frobenius generation.