Depth of powers of squarefree monomial ideals

Huy T\`ai H\`a; Louiza Fouli; Susan Morey

arxiv: 1907.00401 · v2 · pith:CYVFGHQ4new · submitted 2019-06-30 · 🧮 math.AC

Depth of powers of squarefree monomial ideals

Louiza Fouli , Huy T\`ai H\`a , Susan Morey This is my paper

Pith reviewed 2026-05-25 12:18 UTC · model grok-4.3

classification 🧮 math.AC

keywords squarefree monomial idealsdepthpowershyperforestsedgewise domination numberinitially regular sequenceshypergraphs

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The pith

Two bounds on the depths of powers of squarefree monomial ideals are derived when the ideals correspond to hyperforests.

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

The paper establishes two general bounds for the depths of all powers of squarefree monomial ideals that arise from hyperforests. These bounds extend earlier results that controlled only the depth of the ideal itself, not its higher powers. The earlier results expressed depth in terms of the edgewise domination number of the hypergraph and the length of initially regular sequences. A reader would care because depth measures how far an ideal is from being a complete intersection and controls the length of minimal free resolutions. The new bounds therefore give uniform control over the homological behavior of every power without recomputing resolutions each time.

Core claim

We derive two general bounds for the depths of powers of squarefree monomial ideals corresponding to hyperforests. These bounds generalize known bounds for the depths of squarefree monomial ideals, which were given in terms of the edgewise domination number of the corresponding hypergraphs and the lengths of initially regular sequences with respect to the ideals.

What carries the argument

The edgewise domination number of the hypergraph associated to the ideal, together with the lengths of initially regular sequences, which together supply upper and lower bounds on depth(I^k) for every k.

If this is right

The same combinatorial data that bounds depth(I) also bounds depth(I^k) for every positive integer k.
Lower bounds coming from initially regular sequences remain valid after taking powers.
Upper bounds coming from the edgewise domination number remain valid after taking powers.
Depth can be estimated for arbitrary powers without constructing the full minimal free resolution of each power.

Where Pith is reading between the lines

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

The same technique might produce bounds when the hypergraph is only nearly acyclic rather than fully acyclic.
Exact depth formulas could be recovered for special classes of hyperforests where the two bounds coincide.
The results suggest checking whether the edgewise domination number controls other invariants such as regularity of powers.

Load-bearing premise

The squarefree monomial ideals must correspond to hyperforests, that is, the associated hypergraphs must be acyclic.

What would settle it

An explicit hyperforest whose squarefree monomial ideal I satisfies depth(I^2) falling strictly outside the interval predicted by the two derived bounds.

read the original abstract

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.

Referee Report

0 major / 1 minor

Summary. The manuscript derives two general bounds for the depths of powers of squarefree monomial ideals corresponding to hyperforests. These bounds generalize known bounds for the depths of squarefree monomial ideals (the k=1 case), expressed in terms of the edgewise domination number of the corresponding hypergraphs and the lengths of initially regular sequences with respect to the ideals.

Significance. If the stated bounds hold and are derived correctly from the hyperforest hypothesis, the result would extend combinatorial control of depth from the base ideal to its powers, which is relevant for questions about asymptotic depth behavior and associated primes in combinatorial commutative algebra.

minor comments (1)

The abstract alone does not contain the explicit statements of the two bounds, the precise definitions of the combinatorial quantities used, or any indication of the proof strategy, preventing direct verification of the generalization claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for summarizing our work on bounds for depths of powers of squarefree monomial ideals associated to hyperforests and for noting its potential relevance to asymptotic depth and associated primes. The recommendation is listed as uncertain, but the report contains no specific major comments or questions for us to address point by point.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper states that it derives two general bounds on depth(I^k) for squarefree monomial ideals I corresponding to hyperforests, presented explicitly as generalizations of previously known bounds (for the k=1 case) that are expressed in terms of the edgewise domination number and lengths of initially regular sequences. The hyperforest (acyclicity) hypothesis is identified as the structural condition enabling the extension to powers, with no equations or steps shown to reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations. The abstract and claim structure indicate an independent mathematical derivation resting on external prior results rather than internal redefinition or renaming.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, background axioms, or new postulated entities.

pith-pipeline@v0.9.0 · 5570 in / 1034 out tokens · 25649 ms · 2026-05-25T12:18:44.526263+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We derive two general bounds for the depths of powers of squarefree monomial ideals corresponding to hyperforests... depth R/I^s ≥ max{ε(G)−s+1,1}
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The common important underlying idea... if α(G) is an invariant... depth R/I^s ≥ max{α(G)−s+1,1}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages · 1 internal anchor

[1]

Bandari, J

S. Bandari, J. Herzog, and T. Hibi, Monomial ideals whose depth fu nction has any given number of strict local maxima. Ark. Mat. 52 (2014), 11–19

work page 2014
[2]

Beyarslan, H.T

S. Beyarslan, H.T. H` a, and T.N. Trung, Regularity of powers of f orests and cycles. Journal of Algebraic Combinatorics, 42 (2015), no. 4, 1077-1095

work page 2015
[3]

Brodmann, The asymptotic nature of the analytic spread

M. Brodmann, The asymptotic nature of the analytic spread. Ma th. Proc. Cambridge Philos. Soc. 86 (1979), no. 1, 35–39

work page 1979
[4]

Bruns and J

W. Bruns and J. Herzog, Cohen-Macaulay rings. Cambridge Stud ies in Advanced Mathematics, 39. Cambridge University Press, Cambridge, 1993. xii+403 pp

work page 1993
[5]

Burch, Codimension and Analytic Spread

L. Burch, Codimension and Analytic Spread. Proc. Cambridge Philo s. Soc. 72 (1972), 369-373

work page 1972
[6]

Caviglia, H.T

G. Caviglia, H.T. H` a, J. Herzog, M. Kummini, N. Terai, and N.V. Trun g, Depth and regularity modulo a principal ideal. J. Algebraic Combin. 49 (2019), no. 1, 1–20

work page 2019
[7]

Dao and J

H. Dao and J. Schweig, Projective dimension, graph domination pa rameters, and independence complex homology. J. Combin. Theory Ser. A 120 (2013), 453–469

work page 2013
[8]

Dao and J

H. Dao and J. Schweig, Bounding the projective dimension of a squ arefree monomial ideal via domination in clutters. Proc. Amer. Math. Soc. 143 (2015), no. 2, 555–565

work page 2015
[9]

Eisenbud and C

D. Eisenbud and C. Huneke, Cohen-Macaulay Rees Algebras and t heir Specializations. J. Algebra 81 (1983) 202-224

work page 1983
[10]

Faridi, Simplicial trees are sequentially Cohen-Macaulay

S. Faridi, Simplicial trees are sequentially Cohen-Macaulay. J. Pu re Appl. Algebra 190 (2004), no. 1-3, 121–136

work page 2004
[11]

Fouli and S

L. Fouli and S. Morey, A lower bound for depths of powers of ed ge ideals. J. Algebraic Combin. 42 (2015), no. 3, 829–848

work page 2015
[12]

Initially regular sequences and depths of ideals

L. Fouli, H.T. H` a, and S. Morey, Initially regular sequences and d epth of ideals. Preprint (2019), arXiv:1810.01512. 8

work page internal anchor Pith review Pith/arXiv arXiv 2019
[13]

Isidoro, R

G. Isidoro, R. Enrique, and R.H. Villarreal, Blowup algebras of squ are-free monomial ideals and some links to combinatorial optimization problems. Rocky Mountain J. Math . 39 (2009), no. 1, 71–102

work page 2009
[14]

D. R. Grayson and M. E. Stillman, Macaulay2, a software system for research in algebraic geometry, Available at https://faculty.math.illinois.edu/Macaulay2/

work page
[15]

H` a, H.D

H.T. H` a, H.D. Nguyen, N.V. Trung, and T.N. Trung, Depth funct ions of powers of homogeneous ideals. Preprint (2019), arXiv:1904.07587

work page arXiv 2019
[16]

H` a and M

H.T. H` a and M. Sun, Squarefree monomial ideals that fail the pe rsistence property and non-increasing depth. Acta Math. Vietnam. 40 (2015), no. 1, 125–137

work page 2015
[17]

Hang and T.N

N.T. Hang and T.N. Trung, The behavior of depth functions of co ver ideals of unimodular hypergraphs. Ark. Mat. 55 (2017), no. 1, 89–104

work page 2017
[18]

Herzog and T

J. Herzog and T. Hibi, Monomial Ideals. Graduate Texts in Mathe matics 260, Springer, 2011

work page 2011
[19]

Herzog and T

J. Herzog and T. Hibi, The depth of powers of an ideal. J. Algebra 291 (2005), no. 2, 534–550

work page 2005
[20]

Herzog, T

J. Herzog, T. Hibi, N.V. Trung, and X. Zheng, Standard graded vertex cover algebras, cycles and leaves. Trans. Amer. Math. Soc. 360 (2008), no. 12, 6231–6249

work page 2008
[21]

Herzog and M

J. Herzog and M. Vladoiu, Squarefree monomial ideals with const ant depth function. J. Pure Appl. Algebra 217 (2013), no. 9, 1764–1772

work page 2013
[22]

T. Hibi, A. Higashitani, K. Kimura, and A.B. O’Keefe, Depth of initial ideals of normal edge rings. Comm. Algebra 42 (2014), no. 7, 2908–2922

work page 2014
[23]

L.T. Hoa, K. Kimura, N. Terai, and T.N. Trung, Stability of depths of symbolic powers of Stanley- Reisner ideals. J. Algebra 473 (2017), 307–323

work page 2017
[24]

Hoa and T.N

L.T. Hoa and T.N. Trung, Stability of depth and Cohen-Macaulayn ess of integral closures of powers of monomial ideals. Acta Math. Vietnam. 43 (2018), no. 1, 67–81

work page 2018
[25]

Lin and P

K.-N. Lin and P. Mantero, Hypergraphs with high projective dime nsion and 1-dimensional hypergraphs. Internat. J. Algebra Comput. 27 (2017), no. 6, 591–617

work page 2017
[26]

Lin and P

K.-N. Lin and P. Mantero, Projective dimension of string and cyc le hypergraphs. Comm. Algebra 44 (2016), no. 4, 1671–1694

work page 2016
[27]

Matsuda, T

K. Matsuda, T. Suzuki, and A. Tsuchiya, Nonincreasing depth f unction of monomial ideals. Glasg. Math. J. 60 (2018), no. 2, 505–511

work page 2018
[28]

Miranda-Neto, Analytic spread and non-vanishing of asymp totic depth

C.B. Miranda-Neto, Analytic spread and non-vanishing of asymp totic depth. Math. Proc. Cambridge Philos. Soc. 163 (2017), no. 2, 289–299

work page 2017
[29]

Morey, Depths of powers of the edge ideal of a tree

S. Morey, Depths of powers of the edge ideal of a tree. Comm. Algebra 38 (2010), no. 11, 4042–4055

work page 2010
[30]

Morey and R.H

S. Morey and R.H. Villarreal, Edge ideals: Algebraic and combinator ial properties. Progress in Com- mutative Algebra 1 , 85-126, de Gruyter, Berlin, 2012

work page 2012
[31]

Popescu, Graph and depth of a monomial squarefree ideal

D. Popescu, Graph and depth of a monomial squarefree ideal. P roc. Amer. Math. Soc. 140 (2012), no. 11, 3813–3822

work page 2012
[32]

Simis, W.V

A. Simis, W.V. Vasconcelos, and R.H. Villarreal, On the ideal theory of graphs. J. Algebra 167 (1994), 389–416

work page 1994
[33]

Terai and N.V

N. Terai and N.V. Trung, On the associated primes and the dept h of the second power of squarefree monomial ideals. J. Pure Appl. Algebra 218 (2014), no. 6, 1117–112 9. 9 Department of Mathematical Sciences, New Mexico State Univ ersity, P.O. Box 30001, Department 3MB, Las Cruces, NM 88003 E-mail address : lfouli@nmsu.edu URL: http://www.web.nmsu.edu/~lfo...

work page 2014

[1] [1]

Bandari, J

S. Bandari, J. Herzog, and T. Hibi, Monomial ideals whose depth fu nction has any given number of strict local maxima. Ark. Mat. 52 (2014), 11–19

work page 2014

[2] [2]

Beyarslan, H.T

S. Beyarslan, H.T. H` a, and T.N. Trung, Regularity of powers of f orests and cycles. Journal of Algebraic Combinatorics, 42 (2015), no. 4, 1077-1095

work page 2015

[3] [3]

Brodmann, The asymptotic nature of the analytic spread

M. Brodmann, The asymptotic nature of the analytic spread. Ma th. Proc. Cambridge Philos. Soc. 86 (1979), no. 1, 35–39

work page 1979

[4] [4]

Bruns and J

W. Bruns and J. Herzog, Cohen-Macaulay rings. Cambridge Stud ies in Advanced Mathematics, 39. Cambridge University Press, Cambridge, 1993. xii+403 pp

work page 1993

[5] [5]

Burch, Codimension and Analytic Spread

L. Burch, Codimension and Analytic Spread. Proc. Cambridge Philo s. Soc. 72 (1972), 369-373

work page 1972

[6] [6]

Caviglia, H.T

G. Caviglia, H.T. H` a, J. Herzog, M. Kummini, N. Terai, and N.V. Trun g, Depth and regularity modulo a principal ideal. J. Algebraic Combin. 49 (2019), no. 1, 1–20

work page 2019

[7] [7]

Dao and J

H. Dao and J. Schweig, Projective dimension, graph domination pa rameters, and independence complex homology. J. Combin. Theory Ser. A 120 (2013), 453–469

work page 2013

[8] [8]

Dao and J

H. Dao and J. Schweig, Bounding the projective dimension of a squ arefree monomial ideal via domination in clutters. Proc. Amer. Math. Soc. 143 (2015), no. 2, 555–565

work page 2015

[9] [9]

Eisenbud and C

D. Eisenbud and C. Huneke, Cohen-Macaulay Rees Algebras and t heir Specializations. J. Algebra 81 (1983) 202-224

work page 1983

[10] [10]

Faridi, Simplicial trees are sequentially Cohen-Macaulay

S. Faridi, Simplicial trees are sequentially Cohen-Macaulay. J. Pu re Appl. Algebra 190 (2004), no. 1-3, 121–136

work page 2004

[11] [11]

Fouli and S

L. Fouli and S. Morey, A lower bound for depths of powers of ed ge ideals. J. Algebraic Combin. 42 (2015), no. 3, 829–848

work page 2015

[12] [12]

Initially regular sequences and depths of ideals

L. Fouli, H.T. H` a, and S. Morey, Initially regular sequences and d epth of ideals. Preprint (2019), arXiv:1810.01512. 8

work page internal anchor Pith review Pith/arXiv arXiv 2019

[13] [13]

Isidoro, R

G. Isidoro, R. Enrique, and R.H. Villarreal, Blowup algebras of squ are-free monomial ideals and some links to combinatorial optimization problems. Rocky Mountain J. Math . 39 (2009), no. 1, 71–102

work page 2009

[14] [14]

D. R. Grayson and M. E. Stillman, Macaulay2, a software system for research in algebraic geometry, Available at https://faculty.math.illinois.edu/Macaulay2/

work page

[15] [15]

H` a, H.D

H.T. H` a, H.D. Nguyen, N.V. Trung, and T.N. Trung, Depth funct ions of powers of homogeneous ideals. Preprint (2019), arXiv:1904.07587

work page arXiv 2019

[16] [16]

H` a and M

H.T. H` a and M. Sun, Squarefree monomial ideals that fail the pe rsistence property and non-increasing depth. Acta Math. Vietnam. 40 (2015), no. 1, 125–137

work page 2015

[17] [17]

Hang and T.N

N.T. Hang and T.N. Trung, The behavior of depth functions of co ver ideals of unimodular hypergraphs. Ark. Mat. 55 (2017), no. 1, 89–104

work page 2017

[18] [18]

Herzog and T

J. Herzog and T. Hibi, Monomial Ideals. Graduate Texts in Mathe matics 260, Springer, 2011

work page 2011

[19] [19]

Herzog and T

J. Herzog and T. Hibi, The depth of powers of an ideal. J. Algebra 291 (2005), no. 2, 534–550

work page 2005

[20] [20]

Herzog, T

J. Herzog, T. Hibi, N.V. Trung, and X. Zheng, Standard graded vertex cover algebras, cycles and leaves. Trans. Amer. Math. Soc. 360 (2008), no. 12, 6231–6249

work page 2008

[21] [21]

Herzog and M

J. Herzog and M. Vladoiu, Squarefree monomial ideals with const ant depth function. J. Pure Appl. Algebra 217 (2013), no. 9, 1764–1772

work page 2013

[22] [22]

T. Hibi, A. Higashitani, K. Kimura, and A.B. O’Keefe, Depth of initial ideals of normal edge rings. Comm. Algebra 42 (2014), no. 7, 2908–2922

work page 2014

[23] [23]

L.T. Hoa, K. Kimura, N. Terai, and T.N. Trung, Stability of depths of symbolic powers of Stanley- Reisner ideals. J. Algebra 473 (2017), 307–323

work page 2017

[24] [24]

Hoa and T.N

L.T. Hoa and T.N. Trung, Stability of depth and Cohen-Macaulayn ess of integral closures of powers of monomial ideals. Acta Math. Vietnam. 43 (2018), no. 1, 67–81

work page 2018

[25] [25]

Lin and P

K.-N. Lin and P. Mantero, Hypergraphs with high projective dime nsion and 1-dimensional hypergraphs. Internat. J. Algebra Comput. 27 (2017), no. 6, 591–617

work page 2017

[26] [26]

Lin and P

K.-N. Lin and P. Mantero, Projective dimension of string and cyc le hypergraphs. Comm. Algebra 44 (2016), no. 4, 1671–1694

work page 2016

[27] [27]

Matsuda, T

K. Matsuda, T. Suzuki, and A. Tsuchiya, Nonincreasing depth f unction of monomial ideals. Glasg. Math. J. 60 (2018), no. 2, 505–511

work page 2018

[28] [28]

Miranda-Neto, Analytic spread and non-vanishing of asymp totic depth

C.B. Miranda-Neto, Analytic spread and non-vanishing of asymp totic depth. Math. Proc. Cambridge Philos. Soc. 163 (2017), no. 2, 289–299

work page 2017

[29] [29]

Morey, Depths of powers of the edge ideal of a tree

S. Morey, Depths of powers of the edge ideal of a tree. Comm. Algebra 38 (2010), no. 11, 4042–4055

work page 2010

[30] [30]

Morey and R.H

S. Morey and R.H. Villarreal, Edge ideals: Algebraic and combinator ial properties. Progress in Com- mutative Algebra 1 , 85-126, de Gruyter, Berlin, 2012

work page 2012

[31] [31]

Popescu, Graph and depth of a monomial squarefree ideal

D. Popescu, Graph and depth of a monomial squarefree ideal. P roc. Amer. Math. Soc. 140 (2012), no. 11, 3813–3822

work page 2012

[32] [32]

Simis, W.V

A. Simis, W.V. Vasconcelos, and R.H. Villarreal, On the ideal theory of graphs. J. Algebra 167 (1994), 389–416

work page 1994

[33] [33]

Terai and N.V

N. Terai and N.V. Trung, On the associated primes and the dept h of the second power of squarefree monomial ideals. J. Pure Appl. Algebra 218 (2014), no. 6, 1117–112 9. 9 Department of Mathematical Sciences, New Mexico State Univ ersity, P.O. Box 30001, Department 3MB, Las Cruces, NM 88003 E-mail address : lfouli@nmsu.edu URL: http://www.web.nmsu.edu/~lfo...

work page 2014