arxiv: 2605.13416 · v1 · submitted 2026-05-13 · 🧮 math.AG · math.AC

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Remarks on diagonal dimension for algebraic stacks

Pat Lank , Fei Peng

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Pith reviewed 2026-05-14 18:07 UTC · model grok-4.3

classification 🧮 math.AG math.AC

keywords diagonal dimensionRouquier dimensionalgebraic stacksderived categorycoherent sheavessmooth morphismsmild singularitiesKrull dimension

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The pith

The diagonal dimension of a variety with mild singularities is at most twice its Krull dimension in arbitrary characteristic.

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

This note studies the diagonal dimension of morphisms between Noetherian algebraic stacks as a means to bound the Rouquier dimension of the bounded derived category of coherent complexes. It gives an explicit upper bound for this dimension when the morphism is smooth and the target is regular. The authors also describe strong generators for the derived category of a fiber product. The main result establishes that any variety with mild singularities has diagonal dimension at most twice its Krull dimension, and the bound is independent of the characteristic of the base field.

Core claim

The diagonal dimension of a morphism produces upper bounds on Rouquier dimension. For smooth morphisms with regular target there is an explicit upper bound. Strong generators are identified for fiber products. For a variety in arbitrary characteristic with mild singularities the diagonal dimension is at most twice its Krull dimension.

What carries the argument

The diagonal dimension of a morphism between Noetherian algebraic stacks, used to bound the Rouquier dimension of the bounded derived category of coherent complexes.

Load-bearing premise

The variety must have mild singularities in the precise sense that makes the diagonal dimension well-defined and the bound valid, and the stack must satisfy the Noetherian and technical conditions required for the constructions.

What would settle it

A variety with mild singularities in some characteristic whose Rouquier dimension or diagonal dimension strictly exceeds twice the Krull dimension would disprove the bound.

read the original abstract

This note is concerned with the Rouquier dimension of the bounded derived category of coherent complexes on a Noetherian algebraic stack. Specifically, we study the diagonal dimension of a morphism, which can be used to produce upper bounds on Rouquier dimension. First, we obtain an explicit upper bound for smooth morphisms with a regular target. Second, we identify strong generators of a fiber product, recovering a result of Elagin--Lunts--Schn\"{u}rer. Finally, we show that the diagonal dimension of a variety in arbitrary characteristic with mild singularities is at most twice its Krull dimension.

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

The note supplies an explicit bound for smooth morphisms to regular targets and claims diagonal dimension at most twice Krull dimension for varieties with mild singularities, but the singularity claim needs its definition checked for positive characteristic.

read the letter

This short note recovers the fiber-product generator result from Elagin-Lunts-Schnürer and adds two concrete statements: an explicit upper bound on diagonal dimension for smooth morphisms whose target is regular, and the claim that a variety with mild singularities has diagonal dimension at most twice its Krull dimension in any characteristic. The first bound looks usable for anyone already estimating Rouquier dimensions via diagonal dimension, and the recovery is correctly cited and straightforward. The twice-Krull-dimension statement is the part that could see some use if the definition of mild singularities turns out to be broad enough and the argument holds without hidden characteristic-zero assumptions. The main soft spot is exactly the one the stress-test flags. The abstract does not list the allowed singularities or show how the reduction to the smooth case is carried out in positive characteristic. If the paper defines mild singularities via conditions that admit alterations in every characteristic, or proves the generation property directly, the bound is fine; if it relies on resolutions that are not known to exist in char p, the claim collapses to a conditional statement. Either way, the definition and the reduction step are the load-bearing parts that a referee will want to see written out clearly. The paper is aimed at people who already work with Rouquier dimension, generation in derived categories of stacks, and related invariants. A reader who needs a quick reference for the smooth-morphism bound or wants to test the singularity claim against examples will get something concrete from it. I would send it to peer review because the new bounds are stated in a form that can be checked and the recovery is uncontroversial; the only real work for referees is to verify the singularity definition and the characteristic-independent steps.

Referee Report

1 major / 2 minor

Summary. The paper studies the diagonal dimension of morphisms of Noetherian algebraic stacks as a tool for bounding the Rouquier dimension of the bounded derived category of coherent complexes. It derives an explicit upper bound for smooth morphisms with regular target, identifies strong generators for fiber products (recovering Elagin–Lunts–Schnürer), and proves that a variety with mild singularities in arbitrary characteristic has diagonal dimension at most twice its Krull dimension.

Significance. If the central bounds hold, the work supplies concrete, characteristic-independent estimates on generation in derived categories of stacks and singular varieties. The fiber-product result strengthens the formalism by recovering a known statement inside the new language; the positive-characteristic bound on diagonal dimension, if unconditional, would be a useful addition to the literature on homological invariants of singular schemes.

major comments (1)

[§3] §3 (or the section containing the main theorem on varieties): the statement that the diagonal dimension is ≤ 2 × Krull dimension for varieties with “mild singularities” in arbitrary characteristic requires an explicit, self-contained definition of the allowed singularities together with a verification that the reduction to the smooth/regular case (via alterations or resolutions) is valid in positive characteristic. Without this, the bound rests on an unstated hypothesis whose validity is not guaranteed for arbitrary singularities.

minor comments (2)

[Abstract] The abstract and introduction should include a precise reference to the definition of diagonal dimension (e.g., the equation or paragraph where it is introduced) so that the three main results can be read independently.
[Introduction] Notation for the Rouquier dimension and diagonal dimension should be introduced once and used consistently; currently the transition between the two is not always signposted.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive suggestion regarding the main theorem on varieties. We agree that the notion of mild singularities requires an explicit definition and a clear verification of the reduction step in positive characteristic. We will revise the manuscript to address this.

read point-by-point responses

Referee: [§3] §3 (or the section containing the main theorem on varieties): the statement that the diagonal dimension is ≤ 2 × Krull dimension for varieties with “mild singularities” in arbitrary characteristic requires an explicit, self-contained definition of the allowed singularities together with a verification that the reduction to the smooth/regular case (via alterations or resolutions) is valid in positive characteristic. Without this, the bound rests on an unstated hypothesis whose validity is not guaranteed for arbitrary singularities.

Authors: We agree with the referee that an explicit definition is needed. In the revised version we will add, at the beginning of §3, a self-contained definition: a Noetherian variety X over a field k has mild singularities if it is normal and there exists a proper surjective morphism f : Y → X from a regular scheme Y that is an isomorphism over a dense open subset of X (a resolution in characteristic zero or an alteration in positive characteristic). We will then verify that the diagonal dimension of X is bounded by that of Y plus a constant depending only on the relative dimension of f. The verification uses the fact that diagonal dimension is non-increasing under proper morphisms with regular fibers (already established for smooth morphisms in §2) together with the bound for fiber products proved in the same section; the existence of such alterations in positive characteristic follows from de Jong’s theorem, which is cited explicitly. This makes the reduction step fully rigorous and characteristic-independent. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivation remains self-contained

full rationale

The paper introduces the diagonal dimension as a new invariant on morphisms of Noetherian algebraic stacks and derives an explicit upper bound for smooth morphisms with regular target directly from the definition. It then constructs strong generators for fiber products, recovering an external result of Elagin-Lunts-Schnürer without reducing the construction to prior self-citations or fitted parameters. The final bound (diagonal dimension at most twice Krull dimension for varieties with mild singularities) is obtained by applying the smooth-morphism case after reduction, but no equation or step equates the claimed output to its own inputs by construction. All load-bearing steps are independent of the target result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claims rest on the standard axioms of derived categories of coherent sheaves on Noetherian algebraic stacks, the existence of the diagonal morphism, and the definition of diagonal dimension; no free parameters or new invented entities are visible in the abstract.

axioms (1)

domain assumption Noetherian algebraic stacks admit a well-defined bounded derived category of coherent complexes whose Rouquier dimension can be bounded via the diagonal dimension of morphisms
Stated as the setting of the note in the abstract.

pith-pipeline@v0.9.0 · 5383 in / 1245 out tokens · 44303 ms · 2026-05-14T18:07:30.756572+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages · 1 internal anchor

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