A generalization of Eagon–Reiner’s theorem and a characterization of bi-CM<sub><i>t</i></sub> bipartite and chordal graphs

Haghighi, Hassan; Fakhari, S. A. Seyed; Yassemi, Siamak; Zaare-Nahandi, Rahim

doi:10.1080/00927872.2018.1427244

Cited by 4 publications

(8 citation statements)

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“…Yanagawa and Terai [39] have generalized the Eagon-Reiner's theorem by showing that Δ ∨ satisfies the Serre's condition (S r ) if and only if the minimal free resolution of K[Δ] is linear in the first r steps. Since the Serre's condition (S r ) implies CM d−r , the above proposition is also a generalization of Yanagawa's result (see [9,Corollary 3.7]). According to Yanagawa's result, if…”

Section: Proposition 25 ([11 Theorem 28])mentioning

confidence: 59%

“…Note that the converse is false. Indeed, it is enough to think about a disconnected Buchsbaum simplicial complex Δ which is CM 1 and does not even satisfy the Serre's condition (S 2 ) (see [9,Remark 3.3] and [37, Remark 2.1]).…”

Section: Proposition 25 ([11 Theorem 28])mentioning

confidence: 99%

“…The next one is also a characterization for bi-CM t chordal graphs. Proposition 3.8 ([9,Theorem 4.8]) Let G be a chordal graph and let t be a nonnegative integer. Then, the following conditions hold true.…”

Section: And Only If G Is CM T and The Edge Ideal I (G) Of G Has A Linear Resolutionmentioning

confidence: 99%

“…It is easy to see that if G is either a bipartite graph or a chordal graph, then G can only have chordless four-cycles (see, for example, [9,Lemmas 4.2 and 4.7]). Combining this fact with the previous proposition, one can obtain the following result.…”

Section: And Only If G Is CM T and The Edge Ideal I (G) Of G Has A Linear Resolutionmentioning

confidence: 99%

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A Brief Survey on Pure Cohen–Macaulayness in a Fixed Codimension

et al. 2021

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A concept of Cohen-Macaulay in codimension t is defined and characterized for arbitrary finitely generated modules and coherent sheaves by Miller, Novik, and Swartz in 2011. Soon after, Haghighi, Yassemi, and Zaare-Nahandi defined and studied CM t simplicial complexes, which is the pure version of the abovementioned concept and naturally generalizes both Cohen-Macaulay and Buchsbaum properties. The purpose of this paper is to survey briefly recent results of CM t simplicial complexes.

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Section: Proposition 25 ([11 Theorem 28])mentioning

confidence: 59%

Section: Proposition 25 ([11 Theorem 28])mentioning

confidence: 99%

Section: And Only If G Is CM T and The Edge Ideal I (G) Of G Has A Linear Resolutionmentioning

confidence: 99%

Section: And Only If G Is CM T and The Edge Ideal I (G) Of G Has A Linear Resolutionmentioning

confidence: 99%

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A Brief Survey on Pure Cohen–Macaulayness in a Fixed Codimension

et al. 2021

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“…[8, Theorem 3.1]. Let ∆ be a simplicial complex on on n vertices, ∆ ∨ its Alexander dual and I ∆ ⊂ R the Stanley-Reisner ideal of ∆.…”

mentioning

confidence: 99%

Simplicial complexes of small codimension

Varbaro¹,

Zaare-Nahandi

2019

Proc. Amer. Math. Soc.

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We show that a Buchsbaum simplicial complex of small codimension must have large depth. More generally, we achieve a similar result for CMt simplicial complexes, a notion generalizing Buchsbaum-ness, and we prove more precise results in the codimension 2 case. Along the paper, we show that the CMt property is a topological invariant of a simplicial complex.2010 Mathematics Subject Classification. 13H10, 13F55.

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On the dimension of dual modules of local cohomology and the Serre's condition for the unmixed Stanley–Reisner ideals of small height

et al. 2023

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A generalization of Eagon–Reiner’s theorem and a characterization of bi-CM_t bipartite and chordal graphs

Cited by 4 publications

References 15 publications

A Brief Survey on Pure Cohen–Macaulayness in a Fixed Codimension

A Brief Survey on Pure Cohen–Macaulayness in a Fixed Codimension

Simplicial complexes of small codimension

On the dimension of dual modules of local cohomology and the Serre's condition for the unmixed Stanley–Reisner ideals of small height

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A generalization of Eagon–Reiner’s theorem and a characterization of bi-CMt bipartite and chordal graphs

Cited by 4 publications

References 15 publications

A Brief Survey on Pure Cohen–Macaulayness in a Fixed Codimension

A Brief Survey on Pure Cohen–Macaulayness in a Fixed Codimension

Simplicial complexes of small codimension

On the dimension of dual modules of local cohomology and the Serre's condition for the unmixed Stanley–Reisner ideals of small height

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A generalization of Eagon–Reiner’s theorem and a characterization of bi-CM_t bipartite and chordal graphs