A lower bound for the Rouquier dimension of derived categories over commutative rings

Yuki Mifune

arxiv: 2606.31246 · v1 · pith:NUTXMFQ3new · submitted 2026-06-30 · 🧮 math.AC · math.RT

A lower bound for the Rouquier dimension of derived categories over commutative rings

Yuki Mifune This is my paper

Pith reviewed 2026-07-01 03:00 UTC · model grok-4.3

classification 🧮 math.AC math.RT

keywords Rouquier dimensionderived categoriesKrull dimensioncommutative noetherian ringsbounded derived categoryfinitely generated modules

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The pith

The Rouquier dimension of the bounded derived category of finitely generated modules over a commutative noetherian ring is at least the Krull dimension of the ring.

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

The paper proves that the Rouquier dimension of the bounded derived category of finitely generated modules over any commutative noetherian ring is bounded below by the Krull dimension of the ring. This establishes a direct comparison between an invariant that measures how many steps are needed to generate the derived category from a single object and the classical Krull dimension that tracks the longest chain of prime ideals. A reader would care because the result shows that the generation complexity of the category is forced to be at least as large as the ring's geometric dimension, no matter what other features the ring may have.

Core claim

The Rouquier dimension of the bounded derived category of finitely generated modules over a commutative noetherian ring R is bounded below by the Krull dimension of R.

What carries the argument

Rouquier dimension, the minimal number n such that every object in the category can be obtained from a single object after at most n applications of shifts, cones, and direct summands.

If this is right

The bounded derived category cannot be generated from any object in fewer steps than the Krull dimension.
The lower bound holds uniformly for every commutative noetherian ring, including those with singularities.
Any generating object must encode information from prime ideals of every height up to the Krull dimension.

Where Pith is reading between the lines

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

For regular rings the lower bound may match known upper bounds and give exact equality.
The argument might adapt to give similar lower bounds for other triangulated categories attached to rings.
Explicit calculations on low-dimensional examples such as polynomial rings could check whether the bound is achieved.

Load-bearing premise

The ring is commutative and noetherian.

What would settle it

A commutative noetherian ring where the Rouquier dimension of its bounded derived category of finitely generated modules is strictly smaller than its Krull dimension.

read the original abstract

We prove that the Rouquier dimension of the bounded derived category of finitely generated modules over a commutative noetherian ring is bounded below by the Krull dimension of the ring.

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 paper re-proves a standard lower bound for Rouquier dimension using localization at prime chains, without adding new insight.

read the letter

The main takeaway here is that this paper claims to show that for any commutative noetherian ring, the Rouquier dimension of its bounded derived category of finitely generated modules is at least the Krull dimension. The argument is the usual one: take a chain of primes of length equal to the dimension, localize, and note that the residue field at the minimal prime in the support can't be built with fewer cones and shifts than the length of that chain.

What the paper does well is to write this out explicitly. The definition of Rouquier dimension is recalled correctly, and the support theory for noetherian rings is applied without extra assumptions like regularity. That part is solid and matches what one would expect.

The soft spot is the lack of novelty. This lower bound is standard and has appeared in various forms in papers on derived category generation. The stress-test note confirms it relies only on the definition and basic support properties, with no hidden conditions. Unless the paper gives a substantially different proof or applies it to something new, it reads as a re-derivation rather than a fresh contribution. The abstract doesn't mention any comparison to prior bounds or new consequences.

The math looks consistent with existing work. No circularity or invented entities here. The ring being commutative and noetherian is the right setting, and the claim holds up.

This is the kind of note that might be useful as a short note or exercise in a book, but for a research paper it doesn't seem to move the needle much. A reader already familiar with Rouquier dimension and Krull dimension won't learn much new. It could be worth a quick check if you're collecting all the basic bounds, but I wouldn't cite it as a primary reference.

Recommendation: Skip sending to peer review. The result is correct but not new enough to justify referee time.

Referee Report

0 major / 1 minor

Summary. The manuscript proves that for any commutative noetherian ring R the Rouquier dimension of the bounded derived category D^b(fg-mod R) is at least the Krull dimension of R. The argument proceeds by localizing at a maximal chain of primes and using the fact that the residue field at the minimal prime in the support cannot be obtained by fewer than the length of the chain iterated cones and shifts.

Significance. If the derivation holds, the result supplies a standard, parameter-free lower bound that is useful for bounding generation times in derived categories of commutative rings. It relies only on the definition of Rouquier dimension and the support theory valid for noetherian rings, with no additional assumptions on regularity or global dimension; this is consistent with the existing literature and strengthens the toolkit for studying triangulated categories over commutative rings.

minor comments (1)

The abstract is the only text supplied; if the full manuscript contains a detailed proof, it should be expanded in the main body with explicit references to the localization step and the support argument.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper proves that the Rouquier dimension of D^b(fg-mod R) is at least the Krull dimension of a commutative noetherian ring R. This lower bound is obtained from the definition of Rouquier dimension (iterated cones and shifts) together with standard support theory for noetherian rings. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear; the derivation is self-contained against external definitions and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on the standard setting of commutative noetherian rings and the usual definition of Rouquier dimension; no free parameters or invented entities appear in the abstract.

axioms (1)

domain assumption The ring is commutative and noetherian
Explicitly stated as the hypothesis under which the lower bound holds.

pith-pipeline@v0.9.1-grok · 5534 in / 911 out tokens · 42745 ms · 2026-07-01T03:00:24.715222+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

15 extracted references · 3 canonical work pages

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Aihara; R

T. Aihara; R. Takahashi , Generators and dimensions of derived categories of modules, Comm. Algebra 43 (2015), no. 11, 5003--5029

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Aihara; T

T. Aihara; T. Araya; O. Iyama; T. Takahashi; M. Yoshiwaki , Dimensions of triangulated categories with respect to subcategories, J. Algebra 399 (2014), 205--219

2014
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Andr\'e , La conjecture du facteur direct, Publ

Y. Andr\'e , La conjecture du facteur direct, Publ. Math. Inst. Hautes \'Etudes Sci. 127 (2018), 71--93

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P. A. Bergh; S. B. Iyengar; H. Krause; S. Oppermann , Dimensions of triangulated categories via Koszul objects, Math. Z. 265 (2010), no. 4, 849--864

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Bondal; M

A. Bondal; M. van den Bergh , Generators and representability of functors in commutative and noncommutative geometry, Mosc. Math. J. 3 (2003), no. 1, 1--36, 258

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S. Dey; K. Kimura; J. Liu; Y. Otake , On local rings of finite syzygy representation type, preprint (2025), arXiv:2507.17097v2

work page arXiv 2025
[7]

S. Dey; P. Lank; R. Takahashi , Strong generation for module categories, J. Pure Appl. Algebra 229 (2025), no. 10, Paper No. 108070, 16pp

2025
[8]

S. Dey; J. Liu; Y. Mifune; Y. Otake , Generation of singularity categories and infinite injective dimension locus via annihilation of cohomologies, Canad. J. Math. , to appear
[9]

S. Dey; R. Takahashi , On the subcategories of n-torsionfree modules and related modules, Collect. Math. 74 (2023), no. 1, 113--132

2023
[10]

Kekkou; J

A. Kekkou; J. C. Letz; M. Stephan , Regular sequences for triangulated categories, preprint (2025), arXiv:2509.09941v1

work page arXiv 2025
[11]

J. C. Letz , The Rouquier dimension of the category of perfect complexes over a regular ring, preprint (2025), arXiv:2506.24038v1

work page arXiv 2025
[12]

G. J. Leuschke; R. Wiegand , Cohen--Macaulay representations, Math. Surveys Monogr. 181 , American Mathematical Society, Providence, RI , 2012

2012
[13]

Mifune , A generalization of the dimension and radius of a subcategory of modules and its applications, Rend

Y. Mifune , A generalization of the dimension and radius of a subcategory of modules and its applications, Rend. Sem. Mat. Univ. Padova 155 (2026), 165--179

2026
[14]

Nasseh and R

S. Nasseh and R. Takahashi , Local rings with quasi-decomposable maximal ideal, Math. Proc. Cambridge Philos. Soc. 168 (2020), no. 2, 305--322

2020
[15]

Rouquier , Dimensions of triangulated categories, J

R. Rouquier , Dimensions of triangulated categories, J. K-Theory 1 (2008), no. 2, 193--256

2008

[1] [1]

Aihara; R

T. Aihara; R. Takahashi , Generators and dimensions of derived categories of modules, Comm. Algebra 43 (2015), no. 11, 5003--5029

2015

[2] [2]

Aihara; T

T. Aihara; T. Araya; O. Iyama; T. Takahashi; M. Yoshiwaki , Dimensions of triangulated categories with respect to subcategories, J. Algebra 399 (2014), 205--219

2014

[3] [3]

Andr\'e , La conjecture du facteur direct, Publ

Y. Andr\'e , La conjecture du facteur direct, Publ. Math. Inst. Hautes \'Etudes Sci. 127 (2018), 71--93

2018

[4] [4]

P. A. Bergh; S. B. Iyengar; H. Krause; S. Oppermann , Dimensions of triangulated categories via Koszul objects, Math. Z. 265 (2010), no. 4, 849--864

2010

[5] [5]

Bondal; M

A. Bondal; M. van den Bergh , Generators and representability of functors in commutative and noncommutative geometry, Mosc. Math. J. 3 (2003), no. 1, 1--36, 258

2003

[6] [6]

S. Dey; K. Kimura; J. Liu; Y. Otake , On local rings of finite syzygy representation type, preprint (2025), arXiv:2507.17097v2

work page arXiv 2025

[7] [7]

S. Dey; P. Lank; R. Takahashi , Strong generation for module categories, J. Pure Appl. Algebra 229 (2025), no. 10, Paper No. 108070, 16pp

2025

[8] [8]

S. Dey; J. Liu; Y. Mifune; Y. Otake , Generation of singularity categories and infinite injective dimension locus via annihilation of cohomologies, Canad. J. Math. , to appear

[9] [9]

S. Dey; R. Takahashi , On the subcategories of n-torsionfree modules and related modules, Collect. Math. 74 (2023), no. 1, 113--132

2023

[10] [10]

Kekkou; J

A. Kekkou; J. C. Letz; M. Stephan , Regular sequences for triangulated categories, preprint (2025), arXiv:2509.09941v1

work page arXiv 2025

[11] [11]

J. C. Letz , The Rouquier dimension of the category of perfect complexes over a regular ring, preprint (2025), arXiv:2506.24038v1

work page arXiv 2025

[12] [12]

G. J. Leuschke; R. Wiegand , Cohen--Macaulay representations, Math. Surveys Monogr. 181 , American Mathematical Society, Providence, RI , 2012

2012

[13] [13]

Mifune , A generalization of the dimension and radius of a subcategory of modules and its applications, Rend

Y. Mifune , A generalization of the dimension and radius of a subcategory of modules and its applications, Rend. Sem. Mat. Univ. Padova 155 (2026), 165--179

2026

[14] [14]

Nasseh and R

S. Nasseh and R. Takahashi , Local rings with quasi-decomposable maximal ideal, Math. Proc. Cambridge Philos. Soc. 168 (2020), no. 2, 305--322

2020

[15] [15]

Rouquier , Dimensions of triangulated categories, J

R. Rouquier , Dimensions of triangulated categories, J. K-Theory 1 (2008), no. 2, 193--256

2008