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Perfect generation for regular algebraic stacks

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Pith's one-line read A single perfect complex generates the derived category of every regular algebraic stack with quasi-finite diagonal.

desk verdict New single-perfect-generator theorem for regular stacks; the proof has a real but localized gap where it cites Perf = D^b_coh without a finite-dimension hypothesis. read the letter →

arxiv 2601.04053 v4 pith:4GJRAD35 submitted 2026-01-07 math.AG math.ACmath.CT

Pat Lank This is my paper · ORCID
classification math.AGmath.ACmath.CT MSC 14A3014D2314F0818G80
keywords algebraicstacksperfectcomplexesderivedcategoriescompactgenerationregularquasi-finitediagonalThomasonconditionrecollement
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

On any regular algebraic stack with quasi-finite diagonal — a geometric space that may have finite automorphism symmetries at points — the paper proves that the entire derived category of quasi-coherent sheaves is generated by one perfect complex. Perfect complexes are the bounded, locally free-like objects that play the role of 'finite resolutions'; having a single one generate everything means every complex can be built from it by shifts, sums, and cones. The proof works for stacks that are not concentrated, do not have separated diagonal, and may have infinite Krull dimension, all of which were previously restrictive assumptions. When the stack is concentrated, the single generator can be chosen compact, making the derived category singly compactly generated. This matters because a one-object generator turns questions about the whole category into questions about one concrete object.

What carries the argument

The central mechanism is the recollement associated to a quasi-compact open immersion j: U → X with closed complement Z. A recollement is a decomposition of a triangulated category into three categories — the open part D_qc(U), the whole D_qc(X), and the closed-support part D_qc,Z(X) — with three adjoint pairs of functors, so that every object in the middle is built from an object of the closed piece and an object of the open piece. Proposition 3.7 shows that a generator of the open part plus a generator of the closed part gives a single generator of the whole. The theorem uses this to glue along a monomorphic splitting sequence of a Nisnevich covering, a finite chain of open substacks on wh

What would settle it

A concrete test: examine a regular Noetherian scheme of infinite Krull dimension (such a scheme exists) and check whether the cited regularity theorem equating perfect and bounded coherent complexes holds there. If it fails, run the paper's induction on a stack whose monomorphic splitting sequence has an infinite-dimensional regular scheme as an intermediate piece; the induction would produce a bounded coherent complex that is not perfect, contradicting Theorem 1.1's conclusion that the generator is perfect.

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Extended reading notes

Core claim

The paper proves Theorem 1.1: for any quasi-compact quasi-separated regular algebraic stack X with quasi-finite diagonal, there is a perfect complex P on X such that every nonzero object E in D_qc(X) has a nonzero morphism from some shift P[n]. Under the extra concentratedness hypothesis, P can be chosen compact, so D_qc(X) is singly compactly generated. The proof is inductive: it uses a monomorphic splitting sequence of a Nisnevich covering to filter X into locally closed pieces, shows each piece carries a single perfect generator via finite duality and the regularity hypothesis, and then glues these generators using recollement diagrams. The novelty is that no separated-diagonal or finite-

Load-bearing premise

The induction step in the proof of Theorem 1.1 assumes that a cited regularity theorem — perfect complexes coincide with bounded coherent complexes — applies to each intermediate stack in the monomorphic splitting sequence; if that theorem secretly requires finite Krull dimension, the induction could produce only a bounded coherent complex, not a perfect one.

Editorial extensions

If this is right

  • For concentrated regular stacks with quasi-finite diagonal, D_qc(X) is singly compactly generated, and the stack satisfies the 1-Thomason condition (Corollary 1.2 and its proof).
  • The result removes the finite-Krull-dimension hypothesis and the separated-diagonal hypothesis from earlier generation theorems for regular stacks.
  • It applies to smooth, finitely presented, quasi-Deligne–Mumford stacks over a DVR, including mixed-characteristic cases, whenever stabilizers are affine and 'nice'.
  • Even when the generator is not compact, the theorem supplies a single perfect complex that generates the whole derived category — a genuinely new class of examples among regular Deligne–Mumford stacks in arbitrary characteristic.
  • The recollement-gluing proposition gives a reusable recipe for combining generators of open and closed pieces into a generator of a stack.

Reading between the lines

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

  • If the cited regularity theorem (perfect complexes coincide with bounded coherent complexes) can be established for regular stacks without any finite-dimension hypothesis, the same induction would likely upgrade Theorem 1.1 to single compact generation for all quasi-compact quasi-separated regular stacks with quasi-finite diagonal, not just concentrated ones.
  • The recollement gluing lemma is not specific to this setting; it could be applied to other triangulated categories with open-closed decompositions, such as equivariant derived categories or categories of matrix factorizations, whenever each piece is known to admit a single generator.
  • A quantitative strengthening is plausible: the single generator is constructed from finitely many piecewise generators, so its complexity (e.g., the number of terms in a perfect resolution) should be bounded by the length of the monomorphic splitting sequence; making this explicit would give a concrete handle on generation time.
  • The non-compactness phenomenon is likely tied to infinite Krull dimension: on infinite-dimensional regular schemes, bounded coherent complexes need not be perfect, so the induction may produce a perfect but non-compact generator; isolating a stack where this happens would mark the boundary of the concentrated case.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper claims that for every quasi-compact quasi-separated regular algebraic stack X with quasi-finite diagonal, the derived category D_qc(X) of complexes with quasi-coherent cohomology is generated by a single perfect complex (Theorem 1.1). In the concentrated case, it further claims that D_qc(X) is singly compactly generated (Corollary 1.2). The proof uses Hall–Rydh's Nisnevich presentations with monomorphic splitting sequences, establishes a 1-Thomason property on the strata, and glues perfect generators along recollements via Proposition 3.7. The main local ingredients are a finite-duality lemma and the assertion that Perf = D^b_coh for regular stacks, used to lift bounded coherent generators to perfect ones.

Significance. If the main theorem is correct, it is a substantial strengthening of known results: it removes separated diagonal, concentratedness, and finite Krull dimension assumptions for generation by a single perfect complex, and in the concentrated case upgrades compact generation to single-object compact generation. The recollement-glueing proposition is clean and potentially reusable. The paper is concise and builds on standard, mostly cited machinery; however, the central proof depends on an unqualified equality Perf = D^b_coh that needs careful verification.

major comments (2)
  1. [Proof of Theorem 1.1 and Corollary 4.7] The induction step invokes [DLMP25, Theorem 3.7] to assert Perf(X_c)=D^b_coh(X_c), and Corollary 4.7 uses the same equality for X itself. The manuscript states no hypotheses for this cited theorem. This is load-bearing: the localization step produces a bounded coherent lift of the generator, and only the Perf=D^b_coh identification makes that lift perfect. As written, the equality is used with no finite Krull dimension or concentratedness assumption. This is not harmless: for a regular Noetherian scheme of infinite Krull dimension (Nagata's example), the inclusion Perf ⊂ D^b_coh is strict, since there exist bounded coherent sheaves of infinite projective dimension; such a scheme is within the scope of Theorem 1.1. The authors must either quote the precise hypotheses of [DLMP25, Theorem 3.7] and verify them, supply a proof, or amend the main theorem's hypotheses. Without this, the inducti
  2. [Proposition 4.3, final paragraph] The conclusion that some B∈B has support equal to Z does not follow as written for arbitrary β. The proof argues that if the support of every Rf_*(B⊗P) were properly contained in Z, then the support of every object of D_qc,Z(X) would be properly contained in Z. This is false when B has more than one element: a finite or infinite coproduct of generators can have support equal to the union of their supports, which may be all of Z even if each individual support is proper. The argument is valid when β=1, because the generating collection is a single object, and this is the only case used in the proof of Theorem 1.1; however, the proposition as stated is not proved. Please restrict the statement to β=1 or supply a correct argument for the general case.
minor comments (4)
  1. [Lemma 4.1] In the statement, 'B⊆D^b_coh(X)' should presumably be 'B⊆D^b_coh(Y)'. As written, the notation is inconsistent with the proof and with the intended use in the proof of Theorem 1.1.
  2. [Lemma 4.4] The proof uses t^{-1}(Z')=Z for a closed subset Z, where Z' is the closure of t(Z), and justifies this by injectivity of t on underlying topological spaces. Injectivity alone is not sufficient for arbitrary quasi-affine morphisms; the equality holds for monomorphisms. Since the application in Theorem 1.1 uses a monomorphism, the hypothesis should be strengthened to 't is a quasi-affine monomorphism'.
  3. [Proof of Theorem 1.1] The application of [HLLP25, Proposition B.1] should state the hypotheses needed for the Verdier localization sequence on bounded coherent categories. The paper currently invokes it without indicating what conditions on X_c are required, making the argument difficult to verify independently.
  4. [Throughout] There are several typographical errors, e.g., 'Specfically' in the proof of Theorem 1.1, 'T ying' in the proof of Proposition 3.1, and 'containes' in Section 4. These do not affect the mathematics but should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found; the main proof is an inductive gluing argument, with the only notable dependency being an overlapping-authorship citation for Perf=D^b_coh, which raises correctness risk rather than circularity.

full rationale

The paper's main theorem is proved by an inductive argument on a monomorphic splitting sequence for a Nisnevich covering, using recollements (Proposition 3.1 and Proposition 3.7) to glue generators. There is no fitted parameter, no quantity defined in terms of the claimed conclusion, and no prediction equal by construction to an input. The only step that calls for scrutiny is the invocation "However, [DLMP25, Theorem 3.7] tells us that Perf=D^b_coh in each case" in the proof of Theorem 1.1 and again in Corollary 4.7. This citation has overlapping authors and is load-bearing insofar as it converts a bounded-coherent lift into a perfect complex. If the cited theorem does not hold under the paper's hypotheses (compare Nagata's infinite-dimensional regular schemes, where Perf is strictly contained in D^b_coh), the proof would be incomplete. That is an external-dependency/correctness concern, not circularity: the cited equality is not derived from Theorem 1.1, is not equivalent to it by construction, and the paper does not redefine regularity, perfectness, or generation in terms of it. No equation in the paper reduces to its own input, and no fitted quantity is renamed as a prediction. Therefore no circular step is exhibited, and the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The proof introduces no new mathematical entities, parameters, or axioms beyond tools taken from cited literature. The central argument rests on standard derived-category technology for stacks, plus a sequence of substantial cited theorems. The only fragile dependency is the equality Perf = D^b_coh, which is used without stating its precise hypotheses and may fail without finite-dimensionality.

assumptions (6)
  • standard math The six-functor formalism and derived categories of quasi-coherent sheaves on algebraic stacks behave as in [HR17a, §1], [Ols07], [LO08a, LO08b].
    All proofs use derived pullback/pushforward adjunctions, RHom, and support formalism for stacks; these are the standard foundations cited at the start of Section 2.
  • domain assumption [HR17a, Theorem 4.14] provides the right adjoints f^×, their conservativity properties, and the formula Rf_* f^× E ≅ RHom(Rf_* O_Y, E).
    Used in Lemma 4.1 and Proposition 4.3 to prove that Rf_* of a generator generates the desired subcategory, and that f^× is conservative.
  • domain assumption [HR18, Theorem 4.1 and Proposition 3.1] provide a finite flat Nisnevich covering V → Y → X and a monomorphic splitting sequence for stacks with quasi-finite diagonal.
    This is the structural starting point of the proof of Theorem 1.1; the induction on the length of the splitting sequence is the backbone of the paper.
  • domain assumption [HLLP25, Proposition B.1] gives a Verdier localization sequence D^b_coh,Z_c(X_c) → D^b_coh(X_c) → D^b_coh(X_{c-1}).
    Used in the induction step of Theorem 1.1 to lift a generator from X_{c-1} to X_c in the bounded coherent/perfect world.
  • domain assumption [DLMP25, Theorem 3.7] says Perf(X) = D^b_coh(X) for the relevant regular algebraic stacks.
    Invoked to identify D^b_coh with Perf in the induction step. This is the flagged assumption: it is not verified in the paper whether [DLMP25] requires finite Krull dimension, and the equality is false for infinite-dimensional regular schemes.
  • standard math For affine schemes, D_qc,Z(X) is compactly generated by perfect complexes supported on Z ([Rou08, Thm 6.8]); objects are homotopy colimits of iterated extensions of such generators ([Sta26, Tag 09SN]).
    Used in Lemma 2.1 and Lemma 2.2 to reduce support statements to checking on perfect generators.

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Cite this review

Pith. "Pith review of Perfect generation for regular algebraic stacks." pith.science (2026). https://pith.science/paper/4GJRAD35

@misc{pith2026260104053,
  author       = {Pith},
  title        = {Pith review of: Perfect generation for regular algebraic stacks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4GJRAD35}},
  note         = {Machine review of arXiv:2601.04053}
}
read the original abstract

We show that the derived category of complexes with quasi-coherent cohomology on a regular Noetherian algebraic stack with quasi-finite diagonal is generated by a single perfect complex. In the concentrated case, the category is singly compactly generated. Key ingredients in the proofs include gluing generators along recollement and the use of suitable filtrations and presentations of the algebraic stack.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Remarks on diagonal dimension for algebraic stacks

    math.AG 2026-05 unverdicted novelty 6.0 of 10

    For smooth, separated, quasi-DM stacks over a regular affine scheme, the diagonal dimension is bounded by a formula in dim R, dim U, and cd(Y); for varieties with mild singularities it is at most 2·dim X.

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