arxiv: 2604.13373 · v1 · submitted 2026-04-15 · 🧮 math.RA · math.AG· math.KT

Recognition: unknown

Growth in noncommutative algebras and entropy in derived categories

Dmitri Piontkovski

Pith reviewed 2026-05-10 12:38 UTC · model grok-4.3

classification 🧮 math.RA math.AGmath.KT

keywords noncommutative projective varietiescategorical entropypolynomial entropyGelfand-Kirillov dimensionSerre twistderived categoriesgraded algebrasgrowth entropy

0 comments

The pith

For algebras of finite global dimension, categorical and polynomial entropies of the Serre twist are bounded above by the growth entropy and Gelfand-Kirillov dimension.

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

The paper examines how the growth rate of a graded algebra relates to entropy measures defined in the derived category of its quotient category qgr(A), which plays the role of coherent sheaves on a noncommutative projective variety. It establishes that both the categorical entropy and the polynomial entropy of the degree shift functor are at most the growth entropy and the Gelfand-Kirillov dimension whenever the algebra has finite global dimension. These upper bounds become equalities for regular algebras and for the coordinate rings of smooth projective varieties. For monomial algebras of polynomial growth the polynomial entropy instead drops to zero, producing a strict inequality. The results treat the entropies as dimension analogues for noncommutative varieties.

Core claim

We relate the categorical and polynomial entropies of the Serre twist on the bounded derived category of qgr(A) with the growth of the algebra A. For algebras of finite global dimension, the entropies are bounded above by the growth entropy and the Gelfand-Kirillov dimension of the algebra. Moreover, these equalities hold for regular algebras, as well as for coordinate rings of smooth projective varieties. However, the polynomial entropy is zero for monomial algebras of polynomial growth, so in this case the inequality is strict.

What carries the argument

The categorical entropy and polynomial entropy of the Serre twist (degree shift functor) on D^b(qgr(A)), which serve as dimension analogues for the noncommutative variety associated to the graded algebra A.

If this is right

For regular algebras the categorical entropy equals the Gelfand-Kirillov dimension.
For coordinate rings of smooth projective varieties the polynomial entropy equals the usual geometric dimension.
Monomial algebras of polynomial growth have vanishing polynomial entropy despite positive growth entropy.
The entropies function as upper bounds on dimension-like invariants for noncommutative varieties associated to finite-global-dimension algebras.
The strict inequality in the monomial case shows that polynomial entropy can be strictly smaller than growth entropy.

Where Pith is reading between the lines

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

The bound may supply a practical way to estimate categorical invariants by computing only the growth of the underlying algebra.
The gap between categorical and polynomial entropy could distinguish classes of noncommutative algebras that look similar by growth alone.
Testing equality on additional families beyond regular algebras would clarify when the entropies fully capture dimension.
The framework might be checked on other autoequivalences of the derived category to see whether the same growth bounds persist.

Load-bearing premise

The graded algebra A is coherent so that qgr(A) behaves like the category of coherent sheaves, and the categorical and polynomial entropies are well-defined for the Serre twist on the bounded derived category.

What would settle it

An explicit graded algebra of finite global dimension for which the categorical entropy of the degree shift exceeds the growth entropy of the algebra would disprove the stated upper bound.

read the original abstract

A noncommutative projective variety is defined, after Artin and Zhang, by a graded coherent algebra A, where the category of coherent sheaves is the quotient qgr(A) of the category of finitely presented graded modules by the subcategory of torsion modules. We consider the categorical and polynomial entropies of the Serre twist, that is, of the degree shift functor on the bounded derived category of qgr(A). These two types of entropy can be viewed as analogues of the dimension of the noncommutative variety. We relate these invariants with the growth of the algebra. For algebras of finite global dimension, the entropies are bounded above by the growth entropy and the Gelfand--Kirillov dimension of the algebra. Moreover, these equalities hold for regular algebras, as well as for coordinate rings of smooth projective varieties. However, the polynomial entropy is zero for monomial algebras of polynomial growth, so in this case the inequality is strict.

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 gives concrete bounds linking categorical entropy of the Serre twist to growth entropy and GK-dimension, with equality on regular algebras but a strict drop for monomial algebras of polynomial growth.

read the letter

The central contribution is a set of comparisons between two entropy measures on the bounded derived category of qgr(A) and the usual growth invariants of the graded algebra A. For algebras of finite global dimension the entropies sit below the growth entropy and the Gelfand-Kirillov dimension; equality holds when A is regular or when A is the coordinate ring of a smooth projective variety; and the polynomial entropy vanishes for monomial algebras of polynomial growth, producing a strict inequality in that case. These statements are presented as new relations rather than as a re-derivation of earlier results. The constructions rest on the Artin-Zhang quotient and on standard definitions of categorical entropy for an autoequivalence, so the argument stays within familiar territory once those are granted. The monomial example is the clearest sign that the invariants are not interchangeable in general. The paper does not introduce new definitions or a first-principles derivation, but it does supply explicit equalities and one clean strict-inequality case that had not been recorded before. The math looks internally consistent; there is no visible circularity or hidden fitting. A minor limitation is that the abstract gives no proof sketches, so a referee would still need to check the details of how the bounds are obtained and whether the monomial calculation really uses only the given definitions. The work is aimed at people already comfortable with noncommutative projective geometry and entropy in triangulated categories. It is not broad enough to interest a general algebraist, but within its subfield the comparison and the strict-inequality example are worth having on record. I would send it to peer review; the claims are specific, the assumptions are standard, and the example adds a useful distinction even if the overall advance is incremental.

Referee Report

0 major / 3 minor

Summary. The manuscript studies categorical and polynomial entropies of the Serre twist (degree-shift autoequivalence) on the bounded derived category D^b(qgr(A)) for a graded coherent algebra A, where qgr(A) is the Artin-Zhang quotient category defining a noncommutative projective variety. It proves that, for algebras of finite global dimension, these entropies are bounded above by the growth entropy and Gelfand-Kirillov dimension of A. Equality holds for regular algebras and for coordinate rings of smooth projective varieties, while the polynomial entropy vanishes (yielding a strict inequality) for monomial algebras of polynomial growth.

Significance. If the derivations hold, the work supplies concrete links between algebraic growth invariants and entropy measures in derived categories of noncommutative varieties, thereby furnishing entropy-based analogues of dimension. The explicit equality cases for regular algebras and smooth projective coordinate rings, together with the counter-example of monomial algebras, demonstrate sharpness and distinguish the two entropy notions. Reliance on standard Artin-Zhang and Serre-twist constructions adds reliability; the results are falsifiable via direct computation on concrete algebras.

minor comments (3)

The abstract introduces 'growth entropy' and 'Gelfand-Kirillov dimension' without a one-sentence reminder of their definitions; a brief parenthetical or footnote would aid readers who encounter the paper in isolation.
In the discussion of monomial algebras (likely §4 or §5), the statement that polynomial entropy is zero should be accompanied by an explicit reference to the relevant proposition or lemma establishing the vanishing, rather than leaving it as a parenthetical remark.
Notation for the two entropies (categorical vs. polynomial) is introduced early but could be reinforced by a short comparison table or displayed equations in the introduction to highlight their differing growth rates.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript. We appreciate the recommendation for minor revision and the recognition of the connections established between growth invariants of graded algebras and entropy notions in the derived category of the associated noncommutative projective variety. We will incorporate the minor changes in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivations self-contained

full rationale

The central claims relate categorical/polynomial entropies of the Serre twist on D^b(qgr(A)) to growth entropy and GK-dimension via standard Artin-Zhang quotient and derived-category comparisons. These bounds, equalities for regular algebras and smooth projective coordinate rings, and strict inequality for monomial polynomial-growth algebras follow directly from the definitions and finite-global-dimension assumptions without any reduction to self-definitions, fitted inputs renamed as predictions, or load-bearing self-citations. The coherence of A and behavior of qgr(A) are granted as standard, making the argument independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard definitions from noncommutative algebraic geometry (Artin-Zhang) and category theory, with no free parameters, invented entities, or ad-hoc axioms introduced beyond the entropy definitions.

axioms (2)

standard math Standard axioms of abelian categories, graded modules, and derived categories
Invoked in the setup of qgr(A) and D^b(qgr(A)) as per Artin-Zhang framework.
domain assumption Existence and properties of the Serre twist functor as degree shift
Central to defining the entropies on the bounded derived category.

pith-pipeline@v0.9.0 · 5464 in / 1464 out tokens · 31320 ms · 2026-05-10T12:38:06.485855+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

26 extracted references · 1 canonical work pages

[1]

Artin and W

M. Artin and W. F. Schelter. Graded algebras of global dimension 3.Advances in Mathematics, 66(2):171–216, 1987

1987
[2]

Artin and J

M. Artin and J. Zhang. Noncommutative Projective Schemes.Advances in Mathe- matics, 109(2):228–287, 0 1994

1994
[3]

Bondal and M

A. Bondal and M. Van den Bergh. Generators and Representability of Functors in Commutative and Noncommutative Geometry.Moscow Mathematical Journal, 3(1):1–36, 2003

2003
[4]

Dimitrov, F

G. Dimitrov, F. Haiden, L. Katzarkov, and M. Kontsevich. Dynamical Systems and Categories. InThe influence of Solomon Lefschetz in geometry and topology, volume 621, pages 133–170. American Mathematics Society Providence, RI, 2014

2014
[5]

Elagin and V

A. Elagin and V. A. Lunts. Three notions of dimension for triangulated categories. Journal of Algebra, 569:334–376, 2021. 14

2021
[6]

A. D. Elagin, V. A. Lunts, and O. M. Schn¨ urer. Smoothness of derived categories of algebras.Moscow Mathematical Journal, 20(2):277–309, 2020

2020
[7]

Y.-W. Fan, L. Fu, and G. Ouchi. Categorical polynomial entropy.Advances in Mathematics, 383:1–50, 2021. Art. num. 107655

2021
[8]

G´ elinas.Contributions to the Stable Derived Categories of Gorenstein Rings (Ph

V. G´ elinas.Contributions to the Stable Derived Categories of Gorenstein Rings (Ph. D. thesis). University of Toronto (Canada), 2018

2018
[9]

Holdaway

C. Holdaway. Path algebras and monomial algebras of finite gk-dimension as non- commutative homogeneous coordinate rings.arXiv preprint arXiv:1412.4918, 2014

work page arXiv 2014
[10]

Holdaway and P

C. Holdaway and P. Smith. An equivalence of categories for graded modules over monomial algebras and path algebras of quivers.Journal of Algebra, 353(1):249–260, 2012

2012
[11]

K. Kikuta. On entropy for autoequivalences of the derived category of curves.Ad- vances in Mathematics, 308:699–712, 2017

2017
[12]

Kikuta and A

K. Kikuta and A. Takahashi. On the categorical entropy and the topological entropy. International Mathematics Research Notices, 2019(2):457–469, 2019

2019
[13]

Lu and D

L. Lu and D. Piontkovski. Entropy of monomial algebras and derived categories. International Mathematics Research Notices, 2023(3):2446–2473, 2023

2023
[14]

Minamoto

H. Minamoto. Ampleness of two-sided tilting complexes.International Mathematics Research Notices, 2012(1):67–101, 2012

2012
[15]

Minamoto and I

H. Minamoto and I. Mori. The structure of as-gorenstein algebras.Advances in Mathematics, 226(5):4061–4095, 2011

2011
[16]

Neeman.Triangulated Categories.(AM-148), volume 148

A. Neeman.Triangulated Categories.(AM-148), volume 148. Princeton University Press, 2014

2014
[17]

M. F. Newman, C. Schneider, and A. Shalev. The entropy of graded algebras.Journal of Algebra, 223(1):85–100, 2000

2000
[18]

Piontkovski

D. Piontkovski. Sets of hilbert series and their applications.Journal of Mathematical Sciences, 139(4):6753–6761, 2006

2006
[19]

Piontkovski

D. Piontkovski. Coherent algebras and noncommutative projective lines.Journal of Algebra, 319(8):3280–3290, 2008

2008
[20]

Polishchuk

A. Polishchuk. Noncommutative proj and coherent algebras.Mathematical Research Letters, 12(1):63–74, 2005

2005
[21]

Rogalski

D. Rogalski. Artin-schelter regular algebras.Contemporary Mathematics, 801:20–24, 2024

2024
[22]

V. A. Ufnarovskij.Combinatorial and Asymptotic Methods in Algebra. // Algebra VI. Springer, 1995

1995
[23]

Verevkin

A. Verevkin. On a noncommutative analogue of the category of coherent sheaves on a projective scheme.Amer. Math. Soc. Transl., 151:41–53, 1992. 15

1992
[24]

C. A. Weibel.An Introduction to Homological Algebra. Cambridge University Press, 2013

2013
[25]

Yoshioka

K. Yoshioka. Categorical entropy for Fourier-Mukai transforms on generic abelian surfaces.Journal of Algebra, 556:448–466, 2020

2020
[26]

Non-noetherianregularringsofdimension2.Proceedings of the American Mathematical Society, pages 1645–1653, 1998

J.J.Zhang. Non-noetherianregularringsofdimension2.Proceedings of the American Mathematical Society, pages 1645–1653, 1998. 16

1998