Frobenius generation for algebraic stacks

Fei Peng; Pat Lank

REVIEW 4 major objections 4 minor 48 references

Frobenius generation for algebraic stacks

T0 review · 4 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read This paper proves that on a broad class of positive-characteristic algebraic stacks, enough Frobenius pushforwards of a single perfect complex generate the entire bounded derived category of coherent sheaves with prescribed support.

desk verdict A genuine extension of Frobenius generation from schemes to stacks; the proof leans on Hall–Rydh dévissage and the flatness correction is flagged in a footnote, so the stress-test worry is real but not fatal. read the letter →

arxiv 2512.05026 v3 pith:2FN66CT2 submitted 2025-12-04 math.AG

Pat Lank , Fei Peng This is my paper · ORCID

classification math.AG MSC 14A3014A2013A3514F08

keywords FrobeniuspushforwardderivedcategoriesalgebraicstacksF-finitenessclassicalgeneratorsstrongpositivecharacteristicDeligne–Mumford

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes that, on a large class of algebraic stacks in positive characteristic, Frobenius pushforwards of a single perfect complex generate the whole bounded derived category of coherent sheaves supported on any closed substack, provided one pushes forward enough times. To state this, the authors introduce a notion of F-finiteness for stacks based on properness of the Frobenius morphism, since finiteness fails for basic examples such as classifying stacks. The main theorem covers concentrated F-finite stacks with separated quasi-finite diagonal whose Frobenius is representable by algebraic spaces; Deligne–Mumford stacks of finite type over an F-finite scheme are the principal examples. A sympathetic reader should care because this gives an explicit recipe for generators in settings—like good moduli spaces—where no such recipe existed, and it generalizes the known scheme-level statement.

What carries the argument

The load-bearing objects are the absolute Frobenius morphism F:S→S and its iterated pushforward RF^e_* on the derived category. The paper's new notion of F-finiteness—Noetherian plus proper Frobenius—replaces the scheme-level notion of finite Frobenius because on stacks Frobenius need not be representable by schemes; properness still yields an exact enough functor. The proof is carried by an étale dévissage theorem that lets the authors reduce generation from an arbitrary stack to three étale-local situations, and by the relative categories D^b_coh,Z, which record objects supported on a closed substack. For Deligne–Mumford stacks, the number of iterates is controlled by γ(F_*O_U), the suprem

What would settle it

For a concrete test, let S be a genus-one tame stacky curve over an algebraically closed field of characteristic p, e.g. a quotient of an elliptic curve by a finite group. Compute, for e≫0, the minimal n with ⟨RF^e_*Perf(S)⟩_{n+1}=D^b_coh(S). If no finite n exists, the main theorem fails on a stack satisfying its hypotheses; if n is finite, compare it with the bound ceil(log_p N) from Proposition 4.21 to test sharpness.

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Extended reading notes

Core claim

The central claim is Theorem 4.18: if S is a concentrated F-finite algebraic stack with separated quasi-finite diagonal and the absolute Frobenius F:S→S is representable by algebraic spaces, then for every closed subset Z⊆|S| and all e sufficiently large there exists a perfect complex G supported on Z such that the thick closure of the Frobenius pushforward RF^e_*G is the entire bounded derived category D^b_coh,Z(S). Equivalently, the Frobenius pushforwards of perfect complexes classically generate the bounded derived category of coherent sheaves with support in Z. The proof proceeds by an étale dévissage that reduces the statement to three cases—open immersions, finite flat surjective cover

Load-bearing premise

The proof assumes that the étale dévissage theorem applies to the class of morphisms that are representable by algebraic spaces, separated, finitely presented, quasi-finite and flat, and in particular that every stack in sight admits a finite flat surjective cover by an affine scheme; if that dévissage step fails, the induction collapses.

Editorial extensions

If this is right

For every concentrated F-finite Deligne–Mumford stack with separated diagonal, bounded derived categories of coherent sheaves supported on a closed substack admit an explicit classical generator of the form RF^e_*G with G perfect.
The same conclusion holds for the broader class of Artin stacks with Frobenius representable by algebraic spaces; the theorem's hypotheses are satisfied, for instance, by Deligne–Mumford stacks of finite presentation over an F-finite scheme.
On a Deligne–Mumford stack, the number of Frobenius iterates required is at most ceil(log_p(N)), where N is the minimal number of local sections generating F_*O_U over an étale affine cover; N is finite and independent of the chosen closed substack.
A regularity dichotomy holds: on such stacks, a closed substack lies in the regular locus exactly when Frobenius pushforwards of a generator remain perfect complexes.
For separated Deligne–Mumford stacks, a classical generator of the perfect complexes becomes, after sufficiently many Frobenius pushforwards, a strong generator of D^b_coh.
For separated Deligne–Mumford stacks, a classical generator of the perfect complexes becomes, after sufficiently many Frobenius pushforwards, a strong generator of D^b_coh.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The étale dévissage strategy suggests that Frobenius generation is an étale-local property: if the dévissage theorem were unavailable in some broader class, one might still expect generation to descend from étale covers by other means, so the theorem's hypotheses are likely not optimal.
Because BG_m is F-finite but its Frobenius is not representable by algebraic spaces, the theorem sharply separates stacks with representable Frobenius from those without; testing generation on such classifying stacks would show whether the representability hypothesis is genuinely needed.
One can test the bound in Proposition 4.21 on explicit tamely stacky curves: computing γ(F_*O_U) should give the exact minimal e, and comparing it with the actual generation threshold would measure how sharp the log_p bound is.
The genus-zero conclusion for strong generation by F_*O_X on stacky curves suggests that the structure sheaf is a much weaker generator than a well-chosen perfect complex; understanding this gap for higher-genus stacky curves would clarify the role of the generator G in the main theorem.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know that this paper does what it claims: it generalizes Ballard–Iyengar–Lank–Mukhopadhyay–Pollitz from F-finite schemes to concentrated F-finite algebraic stacks with quasi-finite separated diagonal. The new definition of F-finiteness via a proper Frobenius is natural, and Example 3.2 (BG_m) explains convincingly why the naive finite-morphism notion is too restrictive. The main theorem is Theorem 4.18, and the proof strategy via étale dévissage is appropriate for stacks, where local rings are not available.

What is actually new: the F-finiteness notion for stacks, the relative version Proposition 4.8 (plus the Kunz-style Remark 4.19), and the main generation theorem. Corollary 4.20 upgrades to strong generation for separated Deligne–Mumford stacks, and Proposition 4.24 on stacky curves is a nice application. The paper is honest about what it recovers and cites prior work properly, including the authors' own BIL+23.

The soft spot is exactly what the stress-test note flags: the reliance on [HR18, Theorem E] in Theorem 4.18 without stating the theorem or verifying all its hypotheses in the text. But the footnote on p.15 explicitly mentions the typo in (I2) and refers to the corrected flat version in [DLM25, Prop 5.10]. Since the class E in the proof is defined to include flat morphisms, the correction aligns with their setup. So this is a presentational gap, not a load-bearing flaw. Still, a referee should ask them to spell out the exact statement of [HR18, Theorem E] and why each hypothesis is satisfied. Proposition 4.15 also assumes the existence of a finite flat surjective cover from an affine scheme; that comes from the dévissage theorem, so it is justified if one accepts that theorem.

I cannot fully verify every step—several proofs cite lengthy external results (e.g., Lemma 4.14 relies on a splitting result from DLMRP25, and approximation by compacts is imported from HLLP25). But that is normal for this area, and the cited results are independently published. I see no red flag that would make the argument circular or depend on a false statement.

Who should read this: people working on derived categories of algebraic stacks or prime-characteristic geometry. It deserves a serious referee. I would likely cite it if I worked in the area. The referee should focus on the dévissage verification and the flatness correction, but the paper is in good shape.

Referee Report

4 major / 4 minor

Summary. The paper introduces a notion of F-finiteness for algebraic stacks: a Noetherian stack over F_p is F-finite if its absolute Frobenius is proper (Definition 3.5). The main theorem (Theorem 4.18) states that if S is a concentrated F-finite algebraic stack with separated quasi-finite diagonal and whose Frobenius is representable by algebraic spaces, then for every closed Z⊆|S| and e≫0 there is G∈Perf_Z(S) such that D^b_coh,Z(S)=⟨RF^e_*G⟩. This generalizes the Ballard–Iyengar–Lank–Mukhopadhyay–Pollitz theorem from schemes to stacks. The proof reduces the desired property to three cases using Hall–Rydh's étale dévissage theorem [HR18, Theorem E]: open immersions, finite flat surjective affine covers, and étale neighborhoods. The paper also gives an explicit bound on the number of Frobenius iterates for Deligne–Mumford stacks (Proposition 4.21) and discusses a relative Kunz criterion and a stacky-curve application (Proposition 4.24).

Significance. If the main theorem is correct, it is a substantial advance: it identifies explicit generators for bounded derived categories of coherent sheaves on a large class of algebraic stacks, including Deligne–Mumford stacks, and it provides a structural refinement of Kunz's theorem in the stack setting. The paper is clearly structured and makes appropriate use of prior results such as compact approximation [HLLP25] and the Hall–Rydh dévissage framework. The relative formulation around Hypothesis 4.9 is natural and likely to be useful for future work. However, the central induction is presented as a black box: [HR18, Theorem E] is not stated, and footnote 2 acknowledges a correction to item (I2) without giving the corrected statement. Several lemmas in the induction have proofs that are too terse to verify the exact hypotheses. With these details supplied, the paper would be a solid contribution.

major comments (4)

[§4.2, Theorem 4.18 proof] The proof of the main theorem is entirely an application of [HR18, Theorem E], but that theorem is not stated and its hypotheses are not verified beyond three bullet points. Footnote 2 acknowledges a 'minor typo' in (I2) and refers to [DLM25, Prop. 5.10], but does not state the corrected condition. Since the validity of the induction depends on matching the corrected statement (notably the flatness of the finite surjective affine cover), please state the theorem and its correction, and verify explicitly that Propositions 4.12, 4.15 and 4.17 satisfy all closure conditions.
[§3.2, Lemma 3.7] The proof of Lemma 3.7 shows only that F:U→U is locally of finite type, not that it is proper. In the factorization F=f∘F_{U/X}, properness of f does not imply properness of the composition unless F_{U/X} is proper (or finite). The proof should supply the missing argument that F_{U/X} is proper/finite for a smooth morphism from a scheme, or cite a result. This lemma is used in Proposition 3.9(1)=>(3) and Corollary 3.12, and hence underpins the observations in Theorem 4.18.
[§4.2, Lemma 4.14] The proof establishes that E is a direct summand of Rf_*Lf^*E, not that E is isomorphic to an object in the essential image. If 'essentially dense' is intended in the usual triangulated-category sense that every object is a direct summand of an object in the image, this should be stated, and Proposition 4.15 should explain why this suffices for thick generation. Otherwise, a stronger statement is required for the finite flat dévissage step.
[§4.2, Proposition 4.21] In step (2), the sentence 'we see that (s')_*G is a classical generator' is not what is proved: the proof immediately argues for L(s')^*G and uses L(s')^*G throughout. If the pushforward statement is intended, it needs a separate justification; if not, the text should be corrected to say L(s')^*G. This step is needed for the claimed explicit bound in the Deligne–Mumford case.

minor comments (4)

[Footnote 2, §4.2] Typo: 'convience' should be 'convenience'.
[§4.2, Proposition 4.21] Typo: 'Lema 4.8(3)' should be 'Lemma 4.8(3)'. In addition, the proof of Proposition 4.17 contains the garbled expression 'D^b_coh,Z(X)⊆⟨RF^e_*Perf_{Z∩W}(X)⊕RF^e_*Perf_Z(X)⟩' which should be rewritten for clarity.
[References] The titles '[DLMV25]' and '[LMV25]' contain 'Meausuring' instead of 'Measuring'.
[§4.2, Proposition 4.24] The proof appears to use essential density in the wrong direction: if Rπ_* is essentially dense, it does not in general follow that pushing forward a strong generator yields a strong generator. Please clarify or revise this argument if the proposition is to be retained.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: Theorem 4.18 is proved by reducing to Hall–Rydh dévissage and prior published results, not by assuming its own conclusion.

full rationale

The derivation chain for Theorem 4.18 is a genuine induction: Hypothesis 4.9 is a named placeholder, and the theorem is proved by verifying the three dévissage closure conditions in Propositions 4.12, 4.15, and 4.17, then invoking the external Hall–Rydh theorem [HR18, Theorem E]. The proof does not assume the target statement. The affine/scheme base case Proposition 4.8 uses [BIL+23], a prior result of one of the authors, but this is used as input, not derived from Theorem 4.18; the stack-theoretic extension is the new content. The footnote on p. 15 relies on a correction to [HR18, Theorem E] from [DLM25, Proposition 5.10], which is a self-citation, and this is a real self-containedness/correctness concern: the corrected statement is not reproduced and its hypotheses are not verified. However, that is not circularity, because the corrected dévissage theorem is independent of the present paper's conclusion. Similarly, other self-citations such as [HLLP25] supply approximation-by-compacts results but do not encode Hypothesis 4.9. The abstract's phrase 'independently recovers' [BIL+23] is bibliographically overstated, since the proof invokes [BIL+23] directly, but this is an overclaim about provenance rather than a circular derivation. No fitted parameter is renamed as a prediction, and no definition is equivalent by construction to the theorem's conclusion.

Assumptions & free parameters 1 free parameters · 3 assumptions · 1 invented entities

The paper's central claim depends on the Hall–Rydh dévissage theorem as a black box, plus the notion of concentratedness and approximation by compacts for stacks. The new entity is the definition of F-finite stack itself. No parameters are fitted to data; mathematics is not empirical.

free parameters (1)

e (number of Frobenius iterates) = e≫0, and in the DM case e ≥ ⌈log_p(γ(F_* O_U))⌉
Not fitted to data; it is the iteration count. But the bound depends on a local minimal generation number γ(F_* O_U), which is computed from the scheme U and is not explicitly computed for general stacks, only proven finite.

assumptions (3)

domain assumption Hall–Rydh étale dévissage theorem [HR18, Theorem E]
Central induction in Theorem 4.18 and Proposition 4.21; the class E is to be shown equal to D using this theorem. If the theorem has hypotheses not met, the proof fails.
domain assumption Quasi-finite separated diagonal implies approximation by compacts and 1-Thomason condition ([HLLP25, HR17])
Used for Proposition 4.15 and Lemma 4.14 to ensure essential density and compact generation; stated as known.
domain assumption Stacks are concentrated (compact objects are perfects) [HR17, Lemma 4.4]
Used throughout to identify Perf with compact objects and to ensure the perverse operations behave well.

invented entities (1)

F-finite algebraic stack (proper Frobenius)
purpose: To extend the notion of F-finiteness from schemes to stacks, allowing non-representable Frobenius maps
This is a new definition proposed in the paper. It is not experimentally verified; it is a mathematical concept. The paper proves it coincides with classical notion on schemes.

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Cite this review

Pith. "Pith review of Frobenius generation for algebraic stacks." pith.science (2026). https://pith.science/paper/2FN66CT2

@misc{pith2026251205026,
  author       = {Pith},
  title        = {Pith review of: Frobenius generation for algebraic stacks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2FN66CT2}},
  note         = {Machine review of arXiv:2512.05026}
}

abstract

We introduce a notion of $F$-finiteness for algebraic stacks in positive characteristic. Our main result shows that sufficiently many Frobenius pushforwards generate the bounded derived categories of coherent sheaves on Noetherian concentrated $F$-finite algebraic stacks with quasi-finite and separated diagonal. This generalizes, and independently recovers, a result of Ballard--Iyengar--Lank--Mukhopadhyay--Pollitz.

Discussion (0). Continue with ORCID to comment.

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Abel Castorena and Frank Neumann. On actions of Frobenius morphisms for moduli stacks of principal bundles over algebraic curves. Bull. Sci. Math. , 193:26, 2024. Id/No 103450

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Closedness of the singular locus and generation for derived categories

Souvik Dey and Pat Lank. Closedness of the singular locus and generation for derived categories. arXiv:2403.19564 https://arXiv.org/abs/2403.19564, 2024. Accepted for publication at J.\ of Algebra

arXiv 2024