Uniform bounds on projective dimension and Castelnuovo-Mumford regularity

Alessandro De Stefani; Giulio Caviglia

arxiv: 2409.12787 · v2 · pith:3HXPJ3LFnew · submitted 2024-09-19 · 🧮 math.AC

Uniform bounds on projective dimension and Castelnuovo-Mumford regularity

Giulio Caviglia , Alessandro De Stefani This is my paper

Pith reviewed 2026-05-23 20:25 UTC · model grok-4.3

classification 🧮 math.AC

keywords projective dimensionCastelnuovo-Mumford regularitysyzygieshomogeneous idealspolynomial ringsuniform boundsminimal free resolutions

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The pith

Uniform bounds on projective dimension and Castelnuovo-Mumford regularity of homogeneous ideals depend only on a fraction of their syzygies and remain independent of the number of variables.

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

This paper establishes uniform effective upper bounds for the projective dimension and Castelnuovo-Mumford regularity of homogeneous ideals inside a standard graded polynomial ring over a field. The bounds depend on partial data from either the beginning or the end of the minimal free resolution rather than the full set of syzygies. They extend an earlier result that bounded regularity from half the syzygies to bounds on both invariants from a smaller fraction of them. A reader would care because the independence from the number of variables suggests that certain homological properties stabilize across rings of arbitrary size. This could reduce the computational cost of determining full resolutions in practice.

Core claim

For any homogeneous ideal in a standard graded polynomial ring over a field, effective uniform upper bounds exist on both the projective dimension and the Castelnuovo-Mumford regularity; these bounds are independent of the number of variables and can be read off from data extracted from only a fraction of the syzygies at the start or the end of the resolution.

What carries the argument

Partial syzygy data from the minimal free resolution, used to control the length and degrees of the entire resolution.

If this is right

Knowing the initial segment of the syzygies determines an explicit upper bound on the projective dimension that holds for any number of variables.
The Castelnuovo-Mumford regularity is likewise bounded using only a fraction of the syzygies from either end of the resolution.
The same partial data yields simultaneous bounds on both invariants.
All such bounds remain valid when the base field is arbitrary.

Where Pith is reading between the lines

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

Algorithms could stop computing the resolution once the relevant fraction of syzygies is obtained and still certify the remaining invariants.
The technique might apply to other graded invariants that are controlled by the resolution, such as Betti numbers in middle degrees.
Testing the bounds on explicit families of ideals with known large resolutions could reveal how tight the fraction needs to be.

Load-bearing premise

Data from only a fraction of the syzygies is sufficient to produce explicit uniform bounds on the full projective dimension and regularity in the standard graded setting over any field.

What would settle it

An explicit homogeneous ideal in a polynomial ring over a field whose projective dimension or Castelnuovo-Mumford regularity exceeds every candidate bound derived from any fixed positive fraction of its syzygies.

read the original abstract

In this article we obtain uniform effective upper bounds for the projective dimension and the Castelnuovo-Mumford regularity of homogeneous ideals inside a standard graded polynomial ring $S$ over a field. Such bounds are independent of the number of variables of $S$, in the spirit of Stillman's conjecture and of the Ananyan-Hochster's theorem, and depend on partial data extracted from the beginning or the end of the resolution. In this direction, we extend a result of McCullough from 2012 regarding a bound on regularity in terms of half the syzygies to a bound on the projective dimension and the regularity of an ideal in terms of a fraction of the syzygies.

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

Extends McCullough 2012 to bound both pd and regularity from a fraction of the syzygies with uniform effective constants independent of variables.

read the letter

This paper extends McCullough's 2012 regularity bound to simultaneous control of projective dimension and Castelnuovo-Mumford regularity, using data from only a fraction of the syzygies rather than half, and keeps the bounds uniform and independent of the number of variables in the polynomial ring over any field. The abstract states the bounds are effective and depend on partial information from the start or end of the resolution, in the direction of Stillman's conjecture. That is the main new content. The work does well by making the extension to projective dimension explicit and by keeping the result effective rather than existential. It also handles any field, which broadens the prior result. The soft spots are limited. The abstract gives no explicit fraction or form of the bounds, so it is not yet clear how much smaller the fraction is or whether the constants remain reasonable in practice. If the proofs follow similar lines to McCullough, the lift to projective dimension is the real step. No circularity or hidden dependence on the number of variables shows up in the stated claim. This is for people working on effective bounds in commutative algebra and on Stillman's conjecture. A reader who needs concrete dependence on partial syzygy data would get something usable from it. The paper deserves a serious referee because the direction is central and the claims are stated sharply enough to be verified or refuted.

Referee Report

0 major / 2 minor

Summary. The manuscript establishes uniform effective upper bounds on the projective dimension and Castelnuovo-Mumford regularity of homogeneous ideals in a standard graded polynomial ring over a field. These bounds are independent of the number of variables and are determined by data extracted from a fixed fraction of the syzygies at either the beginning or the end of the minimal free resolution. The work extends McCullough's 2012 result, which bounded regularity in terms of half the syzygies, to simultaneous bounds on both projective dimension and regularity using a smaller fraction of the syzygies.

Significance. If the claimed bounds hold, the paper advances the program initiated by Stillman's conjecture and the Ananyan-Hochster theorem by supplying explicit, effective constants that depend only on partial resolution data rather than the full resolution or the number of variables. This partial-data approach is a concrete strengthening of McCullough's earlier regularity bound and could facilitate both theoretical comparisons and computational applications in commutative algebra.

minor comments (2)

The abstract and introduction would benefit from an explicit statement of the fraction of syzygies required (e.g., one-third or one-quarter) rather than the generic phrase 'a fraction of the syzygies.'
Notation for the partial resolution data (beginning versus end) should be introduced with a short diagram or table in §2 to improve readability for readers unfamiliar with the McCullough 2012 setup.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading, positive summary, and recommendation to accept the manuscript. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper extends an independent 2012 result by McCullough on regularity bounds from half the syzygies to uniform bounds on projective dimension and Castelnuovo-Mumford regularity from a fraction of the syzygies (beginning or end of the resolution). The abstract and description position the work as building on external prior results in the spirit of Stillman and Ananyan-Hochster, with no self-definitional steps, fitted parameters renamed as predictions, or load-bearing self-citations. The central claims rely on standard graded polynomial ring setup over any field and partial resolution data, remaining self-contained without reduction to inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only; no free parameters, invented entities, or nonstandard axioms are mentioned. The work operates inside the standard framework of graded modules over polynomial rings.

axioms (1)

standard math Standard setup of homogeneous ideals in a standard graded polynomial ring over a field
The paper states its results inside this conventional commutative algebra setting.

pith-pipeline@v0.9.0 · 5638 in / 1186 out tokens · 25759 ms · 2026-05-23T20:25:06.702227+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

Ananyan and M

T. Ananyan and M. Hochster. Small subalgebras of polynomial rin gs and Stillman’s Conjecture. J. Amer. Math. Soc. , 33(1):291–309, 2020

work page 2020
[2]

Bayer and M

D. Bayer and M. Stillman. A theorem on reﬁning division orders by th e reverse lexicographic order. Duke Math. J. , 55(2):321–328, 1987

work page 1987
[3]

Caviglia, M

G. Caviglia, M. Chardin, J. McCullough, I. Peeva, and M. Varbaro. Regularity of prime ideals. Math. Z., 291(1-2):421–435, 2019

work page 2019
[4]

Caviglia and A

G. Caviglia and A. De Stefani. Bounds on the number of generator s of prime ideals. arXiv: 2108.05683 , 2021

work page arXiv 2021
[5]

Caviglia and A

G. Caviglia and A. De Stefani. Linearly presented modules and boun ds on the Castelnuovo-Mumford regularity of ideals. Proc. Amer. Math. Soc. , 150(4):1397–1404, 2022

work page 2022
[6]

Caviglia and E

G. Caviglia and E. Sbarra. Characteristic-free bounds for the C astelnuovo-Mumford regularity. Compos. Math., 141(6):1365–1373, 2005

work page 2005
[7]

M. Chardin. Some results and questions on Castelnuovo-Mumfor d regularity. In Syzygies and Hilbert functions, volume 254 of Lect. Notes Pure Appl. Math. , pages 1–40. Chapman & Hall/CRC, Boca Raton, FL, 2007

work page 2007
[8]

Eisenbud

D. Eisenbud. Commutative algebra , volume 150 of Graduate Texts in Mathematics . Springer-Verlag, New York, 1995. With a view toward algebraic geometry

work page 1995
[9]

Eisenbud and S

D. Eisenbud and S. Goto. Linear free resolutions and minimal multip licity. J. Algebra , 88(1):89–133, 1984

work page 1984
[10]

A. Galligo. Th´ eor` eme de division et stabilit´ e en g´ eom´ etrie analytique locale. Ann. Inst. Fourier (Greno- ble), 29(2):vii, 107–184, 1979

work page 1979
[11]

McCullough

J. McCullough. A polynomial bound on the regularity of an ideal in t erms of half of the syzygies. Math. Res. Lett., 19(3):555–565, 2012

work page 2012
[12]

McCullough and I

J. McCullough and I. Peeva. Counterexamples to the Eisenbud- Goto regularity conjecture. J. Amer. Math. Soc. , 31(2):473–496, 2018

work page 2018
[13]

I. Peeva. Graded syzygies, volume 14 of Algebra and Applications. Springer-Verlag London, Ltd., London, 2011. Department of Mathematics, Purdue University, 150 N. Unive rsity Street, West Lafayette, IN 47907-2067, USA Email address : gcavigli@purdue.edu Dipartimento di Matematica, Universit `a di Genova, Via Dodecaneso 35, 16146 Genova, Italy Email addres...

work page 2011

[1] [1]

Ananyan and M

T. Ananyan and M. Hochster. Small subalgebras of polynomial rin gs and Stillman’s Conjecture. J. Amer. Math. Soc. , 33(1):291–309, 2020

work page 2020

[2] [2]

Bayer and M

D. Bayer and M. Stillman. A theorem on reﬁning division orders by th e reverse lexicographic order. Duke Math. J. , 55(2):321–328, 1987

work page 1987

[3] [3]

Caviglia, M

G. Caviglia, M. Chardin, J. McCullough, I. Peeva, and M. Varbaro. Regularity of prime ideals. Math. Z., 291(1-2):421–435, 2019

work page 2019

[4] [4]

Caviglia and A

G. Caviglia and A. De Stefani. Bounds on the number of generator s of prime ideals. arXiv: 2108.05683 , 2021

work page arXiv 2021

[5] [5]

Caviglia and A

G. Caviglia and A. De Stefani. Linearly presented modules and boun ds on the Castelnuovo-Mumford regularity of ideals. Proc. Amer. Math. Soc. , 150(4):1397–1404, 2022

work page 2022

[6] [6]

Caviglia and E

G. Caviglia and E. Sbarra. Characteristic-free bounds for the C astelnuovo-Mumford regularity. Compos. Math., 141(6):1365–1373, 2005

work page 2005

[7] [7]

M. Chardin. Some results and questions on Castelnuovo-Mumfor d regularity. In Syzygies and Hilbert functions, volume 254 of Lect. Notes Pure Appl. Math. , pages 1–40. Chapman & Hall/CRC, Boca Raton, FL, 2007

work page 2007

[8] [8]

Eisenbud

D. Eisenbud. Commutative algebra , volume 150 of Graduate Texts in Mathematics . Springer-Verlag, New York, 1995. With a view toward algebraic geometry

work page 1995

[9] [9]

Eisenbud and S

D. Eisenbud and S. Goto. Linear free resolutions and minimal multip licity. J. Algebra , 88(1):89–133, 1984

work page 1984

[10] [10]

A. Galligo. Th´ eor` eme de division et stabilit´ e en g´ eom´ etrie analytique locale. Ann. Inst. Fourier (Greno- ble), 29(2):vii, 107–184, 1979

work page 1979

[11] [11]

McCullough

J. McCullough. A polynomial bound on the regularity of an ideal in t erms of half of the syzygies. Math. Res. Lett., 19(3):555–565, 2012

work page 2012

[12] [12]

McCullough and I

J. McCullough and I. Peeva. Counterexamples to the Eisenbud- Goto regularity conjecture. J. Amer. Math. Soc. , 31(2):473–496, 2018

work page 2018

[13] [13]

I. Peeva. Graded syzygies, volume 14 of Algebra and Applications. Springer-Verlag London, Ltd., London, 2011. Department of Mathematics, Purdue University, 150 N. Unive rsity Street, West Lafayette, IN 47907-2067, USA Email address : gcavigli@purdue.edu Dipartimento di Matematica, Universit `a di Genova, Via Dodecaneso 35, 16146 Genova, Italy Email addres...

work page 2011