Powers of edge ideals of regularity three bipartite graphs

Alilooee, Ali; Banerjee, Arindam

doi:10.1216/jca-2017-9-4-441

Cited by 46 publications

(43 citation statements)

References 14 publications

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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%

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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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“…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%

“…We have reg I = reg I 2 = 8, reg(mI) = 9. The fiber product F satisfies reg F 2 = reg(mI) + 1 = 10 > max i∈ [1,2] {reg I i + 2 − i, reg J i + 2 − i} = 9.…”

Section: Regularitymentioning

confidence: 99%

Homological Invariants of Powers of Fiber Products

Nguyen

2019

Acta Math Vietnam

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“…If w j = 1 for all j, then I(D) recovers the usual edge ideal of its (undirected) underlying graph. Edge ideals of (undirected) graphs have been investigated extensively in the literature [1,2,3,4,5,6,7,14,19,21,24,25,29]. In general, edge ideals of weighted oriented graphs are different from edge ideals of edge-weighted (undirected) graphs defined by Paulsen and Sather-Wagstaff [27].…”

Section: Introductionmentioning

confidence: 99%

Regularity and projective dimension of powers of edge ideal of the disjoint union of some weighted oriented gap-free bipartite graphs

Zhu

Wang

et al. 2019

J. Algebra Appl.

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In this paper we provide some precise formulas for regularity of powers of edge ideal of the disjoint union of some weighted oriented gap-free bipartite graphs. For the projective dimension of such an edge ideal, we give its exact formula. Meanwhile, we also give the upper and lower bounds of projective dimension of higher power of such edge ideals. Some examples show that these formulas are related to direction selection.

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“…Computing and finding bounds for the regularity of powers of a monomial ideal have been studied by a number of researchers (see for example [1], [2], [3], [4], [5], [7], [11], [14] [15]). This work is motivated by a recent paper of Hang and Trung [12].…”

Section: Introductionmentioning

confidence: 99%