Depth and Regularity of Monomial Ideals via Polarization and Combinatorial Optimization

Martinez-Bernal, Jose; Morey, Susan; Villarreal, Rafael H.; Vivares, Carlos E.

doi:10.1007/s40306-018-00308-z

Cited by 12 publications

(5 citation statements)

References 58 publications

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“…The projective dimension, regularity, and algebraic and combinatorial properties of edge ideals of weighted oriented graphs have been studied in [5,8,11,15,21,22]. The first major result about I(D) is an explicit combinatorial expression of Pitones, Reyes and Toledo [15,Theorem 25] for the irredundant decomposition of I(D) as a finite intersection of irreducible monomial ideals.…”

Section: Introductionmentioning

confidence: 99%

Unmixed and Cohen--Macaulay weighted oriented König graphs

Pitones¹,

Reyes²,

Villarreal³

2019

Preprint

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Let D be a weighted oriented graph, whose underlying graph is G, and let I(D) be its edge ideal. If G has no 3-, 5-, or 7-cycles, or G is König, we characterize when I(D) is unmixed. If G has no 3-or 5-cycles, or G is König, we characterize when I(D) is Cohen-Macaulay. We prove that I(D) is unmixed if and only if I(D) is Cohen-Macaulay when G has girth greater than 7 or G is König and has no 4-cycles.

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Section: Introductionmentioning

confidence: 99%

Unmixed and Cohen--Macaulay weighted oriented König graphs

Pitones¹,

Reyes²,

Villarreal³

2019

Preprint

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“…Using a result of [24], we answer a question of Aron Simis and a related question of Antonio Campillo by showing that an oriented graph D is Cohen-Macaulay if and only if the oriented graph U , obtained from D by replacing each weight d i > 3 with d i = 2, is Cohen-Macaulay (Corollary 6). Seemingly, this ought to somewhat facilitate the verification of this property.…”

Section: Introductionmentioning

confidence: 99%

Symbolic Powers of Monomial Ideals and Cohen-Macaulay Vertex-Weighted Digraphs

Gimenez

Martinez-Bernal

Simis

et al. 2018

Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics

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In this paper we study irreducible representations and symbolic Rees algebras of monomial ideals. Then we examine edge ideals associated to vertexweighted oriented graphs. These are digraphs having no oriented cycles of length two with weights on the vertices. For a monomial ideal with no embedded primes we classify the normality of its symbolic Rees algebra in terms of its primary components. If the primary components of a monomial ideal are normal, we present a simple procedure to compute its symbolic Rees algebra using Hilbert bases, and give necessary and sufficient conditions for the equality between its ordinary and symbolic powers. We give an effective characterization of the Cohen-Macaulay vertexweighted oriented forests. For edge ideals of transitive weighted oriented graphs we show that Alexander duality holds. It is shown that edge ideals of weighted acyclic tournaments are Cohen-Macaulay and satisfy Alexander duality.

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“…This subsumes the class of whiskered graphs treated in [38,Corollary 5.1(2)]. For related work on some more special cases, we refer to [2,3,9,16,26,27,39,42,43].…”

Section: An Application To Edge Idealsmentioning

confidence: 97%

Homological Invariants of Powers of Fiber Products

Nguyen

2019

Acta Math Vietnam

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Let R and S be polynomial rings of positive dimensions over a field k. Let I ⊆ R, J ⊆ S be non-zero homogeneous ideals none of which contains a linear form. Denote by F the fiber product of I and J in T = R ⊗ k S. We compute homological invariants of the powers of F using the data of I and J. Under the assumption that either char k = 0 or I and J are monomial ideals, we provide explicit formulas for the depth and regularity of powers of F . In particular, we establish for all s ≥ 2 the intriguing formula depth(T /F s ) = 0. If moreover each of the ideals I and J is generated in a single degree, we show that for all s ≥ 1, reg F s = max i∈ [1,s] Finally, we prove that the linearity defect of F is the maximum of the linearity defects of I and J, extending previous work of Conca and Römer. The proofs exploit the so-called Betti splittings of powers of a fiber product. 2010 Mathematics Subject Classification. 13D02, 13C05, 13D05, 13H99. Proof. This follows from the proof of [46, Corollary 3.8]. 2.4. Elementary facts. Lemma 2.6. Let R, S be affine k-algebras. Let I, J be ideals of R, S, respectively. Then in T = R ⊗ k S, there is an equality I ∩ J = IJ. This standard fact follows from [48, Lemma 2.2(i)].Lemma 2.7. Let R, S be standard graded k-algebras, and M, N be finitely generated graded modules over R, S, respectively. Then for T = R ⊗ k S, there are equalitiesThis lemma is folklore. The first part follows from the description of local cohomology of tensor product due to Goto and Watanabe [29, Theorem 2.2.5]. For the remaining assertions, see [48, Lemma 2.3].Recall that a morphism of noetherian local rings (R, m) θ − → (S, n) is an algebra retract if there exists a local homomorphism (S, n) φ − → (R, m) such that φ • θ is the identity map of R. In that case, φ is called the retraction map of θ. Lemma 2.8 ([48, Lemma 2.4]). Let (R, m) θ − → (S, n) be an algebra retract of noetherian local rings with the retraction map S φ − → R. Let I ⊆ m be an ideal of R. Let J ⊆ n be an ideal containing θ(I)S such that φ(J)R = I. Then there are inequalities ld R (R/I) ≤ ld S (S/J), ld R I ≤ ld S J.

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Depth and Regularity of Monomial Ideals via Polarization and Combinatorial Optimization

Cited by 12 publications

References 58 publications

Unmixed and Cohen--Macaulay weighted oriented König graphs

Unmixed and Cohen--Macaulay weighted oriented König graphs

Symbolic Powers of Monomial Ideals and Cohen-Macaulay Vertex-Weighted Digraphs

Homological Invariants of Powers of Fiber Products

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