Categorical characterizations of regularity for algebraic stacks

Fei Peng; Kabeer Manali Rahul; Pat Lank; Timothy De Deyn

arxiv: 2504.02813 · v2 · submitted 2025-04-03 · 🧮 math.AG · math.AC

Categorical characterizations of regularity for algebraic stacks

Timothy De Deyn , Pat Lank , Kabeer Manali Rahul , Fei Peng This is my paper

Pith reviewed 2026-05-22 21:25 UTC · model grok-4.3

classification 🧮 math.AG math.AC

keywords algebraic stacksregularityperfect complexesbounded derived categorycoherent sheavesclassical generatorsstrong generatorst-structures

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The pith

For Noetherian algebraic stacks, regularity holds exactly when the perfect complexes equal the bounded coherent derived category.

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

The paper shows that a Noetherian algebraic stack X is regular if and only if its category of perfect complexes coincides with the bounded derived category of coherent sheaves. This equivalence serves as the bridge to extend Neeman's characterizations of regularity, originally for schemes, to stacks via the existence of strong generators and bounded t-structures on the perfect complexes. The result also supplies a criterion for when the bounded derived category of coherent sheaves on such a stack admits a classical generator. A sympathetic reader would care because these statements turn a geometric property into purely categorical data that can be checked inside the derived category without direct reference to local rings at points.

Core claim

For a Noetherian algebraic stack X, X is regular if and only if Perf(X) equals D_coh^b(X). This identity is used to obtain variants of Neeman's results on strong generators and bounded t-structures, and it yields a criterion for the existence of classical generators in D_coh^b(X) for algebraic stacks.

What carries the argument

The equality Perf(X) = D_coh^b(X), which functions as a categorical detector of regularity for the stack.

If this is right

Regularity of the stack can be detected by the existence of a strong generator in Perf(X).
The presence of a bounded t-structure on Perf(X) is equivalent to regularity of the stack.
D_coh^b(X) admits a classical generator precisely when the stack satisfies the stated regularity and generation conditions.
The same categorical tests apply to large classes of Noetherian algebraic stacks beyond schemes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same equivalence might serve as a test for regularity in broader classes of stacks once the Noetherian hypothesis is relaxed in future work.
Categorical regularity criteria could be compared directly with other stack invariants such as the dimension of the inertia stack.
The generator existence result opens a route to compute global dimension bounds uniformly across families of stacks.

Load-bearing premise

The algebraic stack must be Noetherian.

What would settle it

A concrete Noetherian algebraic stack that is singular at some point yet still satisfies Perf(X) = D_coh^b(X), or a regular Noetherian stack where the two categories differ.

read the original abstract

For a Noetherian scheme $X$ of finite Krull dimension, Neeman recently established two characterizations of the regularity of $X$ using strong generators and bounded $t$-structures on $\operatorname{Perf}(X)$. In this note, we obtain variants of Neeman's results for large classes of Noetherian algebraic stacks. An important intermediate step is the fact that $X$ is regular if and only if $\operatorname{Perf}(X)=D_{\operatorname{coh}}^b(X)$, which we establish for Noetherian algebraic stacks. Our approach also yields a criterion for the existence of classical generators for the bounded derived categories of coherent sheaves on algebraic stacks, generalizing previous results for commutative rings and schemes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Neeman's Perf = D_coh^b equivalence and generator criteria extend to Noetherian algebraic stacks, but the finite Krull dimension gap from the scheme case needs checking.

read the letter

The main thing to know is that for a Noetherian algebraic stack X, regularity holds exactly when Perf(X) equals D_coh^b(X). They use this equivalence to obtain variants of Neeman's strong generator and bounded t-structure characterizations, plus a criterion for classical generators on the bounded coherent derived category. Both the equivalence and the generator statements are new for stacks. The abstract positions the equivalence as the key intermediate step they establish, and the rest follows as variants for large classes of such stacks. This is a direct generalization that fills a gap, since stacks appear routinely in moduli work and the scheme results do not automatically carry over. The approach rests on Neeman without visible circularity or parameter fitting. The citation pattern looks standard for this kind of extension. The soft spot is the dimension hypothesis. Neeman's original theorem for schemes requires finite Krull dimension to make the equivalence detect regularity. The claim here drops to Noetherian stacks without mentioning dimension. If there exists a Noetherian stack of infinite dimension where Perf equals D_coh^b but regularity fails, the iff statement collapses. The proofs must either derive finite dimension from Noetherianity for stacks or supply an independent argument; neither is visible in the abstract, so that detail requires verification in the body. This note is for people already working with derived categories of stacks or moduli stacks who know Neeman's paper. A reader in that niche will get immediate value from the statements. It deserves a serious referee because the claims are concrete, the extension is plausible, and the potential dimension issue is checkable rather than fatal.

Referee Report

1 major / 1 minor

Summary. The manuscript extends Neeman's recent characterizations of regularity for Noetherian schemes of finite Krull dimension to Noetherian algebraic stacks. The central result is that a Noetherian algebraic stack X is regular if and only if Perf(X) equals D_coh^b(X). It also obtains variants of the strong-generator and bounded t-structure characterizations, and gives a criterion for the existence of classical generators in D_coh^b(X) on stacks.

Significance. If the results hold, the work provides a useful generalization of Neeman's theorems to algebraic stacks, an important setting in modern algebraic geometry. The intermediate equivalence Perf(X) = D_coh^b(X) detecting regularity is a key step that could facilitate further study of derived categories on stacks.

major comments (1)

[Introduction] Introduction and main theorem (the equivalence Perf(X)=D_coh^b(X) for Noetherian algebraic stacks): the statement omits any finite Krull dimension hypothesis, unlike the scheme case of Neeman cited in the abstract. Noetherianity alone does not imply finite dimension for stacks, and it is unclear whether the proof derives the dimension bound or supplies an independent argument; without this, the central iff claim risks failure on stacks of infinite dimension.

minor comments (1)

[Abstract] The abstract refers to 'variants of Neeman's results' without naming them; a brief enumeration in the introduction would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting this important point about the statement of the main theorem. We address the comment below.

read point-by-point responses

Referee: [Introduction] Introduction and main theorem (the equivalence Perf(X)=D_coh^b(X) for Noetherian algebraic stacks): the statement omits any finite Krull dimension hypothesis, unlike the scheme case of Neeman cited in the abstract. Noetherianity alone does not imply finite dimension for stacks, and it is unclear whether the proof derives the dimension bound or supplies an independent argument; without this, the central iff claim risks failure on stacks of infinite dimension.

Authors: We agree that the finite Krull dimension hypothesis is required, as in Neeman's original result for schemes. Our proof of the equivalence Perf(X) = D_coh^b(X) does not derive a finite dimension bound from the equality itself, nor does it supply an independent argument that would allow the result to hold without this hypothesis. We will therefore revise the abstract, introduction, and main theorem statement to include the assumption that the Noetherian algebraic stack has finite Krull dimension, bringing the result into line with the scheme case. revision: yes

Circularity Check

0 steps flagged

No circularity; extends external Neeman result to stacks

full rationale

The paper cites Neeman's theorem for schemes (external, non-overlapping authors) as the starting point and derives an analogous Perf(X)=D_coh^b(X) equivalence for Noetherian algebraic stacks as a new intermediate step. No self-citations appear load-bearing, no parameters are fitted then renamed as predictions, and no ansatz or uniqueness claim reduces to prior author work by construction. The central claims remain independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard domain assumptions about derived categories of Noetherian stacks; no free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Standard properties of perfect complexes and bounded t-structures on Perf(X) for Noetherian algebraic stacks
Invoked to adapt Neeman's characterizations and to prove the Perf = D_coh^b equivalence.

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discussion (0)

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Remarks on diagonal dimension for algebraic stacks
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Diagonal dimension of a variety with mild singularities is at most twice its Krull dimension; explicit upper bounds are given for smooth morphisms to regular targets.