On 2-Buchsbaum complexes

Miyazaki, Mitsuhiro

doi:10.1215/kjm/1250520019

Cited by 6 publications

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“…This is just due to the fact that a simplicial (d−1)-sphere is 2-Cohen-Macaulay and (d − 1)-Cohen-Macaulayness implies (d − 1)-connectedness. This corollary also generalizes a result of Hibi [7] (see [10,Introduction]) which says that if ∆ is a Cohen-Macaulay complex of dimension d − 1, then the (d − 2)-skeleton of ∆ is 2-Cohen-Macaulay. Example 3.12.…”

Section: Definition 31 Let K Be a Field For Positive Integer K Andsupporting

confidence: 73%

“…Therefore, we have dim (τ ) = d − 1. Now we are ready to give one of the main results of this paper which generalizes results due to Hibi [7] and Miyazaki [10].…”

Section: Definition 31 Let K Be a Field For Positive Integer K Andsupporting

confidence: 56%

“…(see[10, Lemma 4.2]) Let ∆ be a pure complex of dimension (d−1) with vertex set V . Then for all positive integers k the following are equivalent:(i) ∆ is k-Buchsbaum.…”

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confidence: 99%

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A generalization of k-Cohen–Macaulay simplicial complexes

2012

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For a positive integer k and a non-negative integer t a class of simplicial complexes, to be denoted by k-CMt, is introduced. This class generalizes two notions for simplicial complexes: being k-Cohen-Macaulay and k-Buchsbaum. In analogy with the Cohen-Macaulay and Buchsbaum complexes, we give some characterizations of CMt(=1-CMt) complexes, in terms of vanishing of some homologies of its links and, in terms of vanishing of some relative singular homologies of the geometric realization of the complex and its punctured space. We show that a complex is k-CMt if and only if the links of its nonempty faces are k-CM t−1 . We prove that for an integer s ≤ d, the (d−s−1)-skeleton of a (d−1)-dimensional k-CMt complex is (k+s)-CMt. This result generalizes Hibi's result for Cohen-Macaulay complexes and Miyazaki's result for Buchsbaum complexes.2000 Mathematics Subject Classification. 05C75, 13H10.

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Section: Definition 31 Let K Be a Field For Positive Integer K Andsupporting

confidence: 73%

“…Therefore, we have dim (τ ) = d − 1. Now we are ready to give one of the main results of this paper which generalizes results due to Hibi [7] and Miyazaki [10].…”

Section: Definition 31 Let K Be a Field For Positive Integer K Andsupporting

confidence: 56%

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A generalization of k-Cohen–Macaulay simplicial complexes

2012

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“…To this end let k ∈ [n] be any vertex of . Since is 2-Buchsbaum, lk k is 2-CM by Miyazaki [11,Lemma 4.2]. Thus, we may apply Lemma 3.1.…”

Section: Lemma 31 Let Be a (D − 1)-dimensional Buchsbaum Complex Anmentioning

confidence: 90%

Level algebras through Buchsbaum* manifolds

Nagel

2010

Collect. Math.

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Stanley-Reisner rings of Buchsbaum* complexes are studied by means of their quotients modulo a linear system of parameters. The socle of these quotients is computed. Extending a recent result by Novik and Swartz for orientable homology manifolds without boundary, it is shown that modulo a part of their socle these quotients are level algebras. This provides new restrictions on the face vectors of Buchsbaum* complexes.

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“…When i 0 the condition K ∈ A i is topologically invariant by Proposition 3.10. The condition K ∈ A 1 (i.e., K is a 2-CM-complex) is also topologically invariant (see the introduction in [19]). However, when i > 1, the topological invariance may no longer hold, as can be seen from the following example.…”

Section: Example 52 Notice That the Simplexmentioning

confidence: 99%

Topological applications of Stanley-Reisner rings of simplicial complexes

Айзенберг

2013

Trans. Moscow Math. Soc.

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Methods of commutative and homological algebra yield information on the Stanley-Reisner ring k[K] of a simplicial complex K. Consider the following problem: describe topological properties of simplicial complexes with given properties of the ring k[K]. It is known that for a simplicial complex K = ∂P * , where P * is a polytope dual to the simple polytope P of dimension n, the depth of depth k[K] equals n. A recent construction allows us to associate a simplicial complex K P to any convex polytope P. As a consequence, one wants to study the properties of the rings k[K P ]. In this paper, we report on the obtained results for both of these problems. In particular, we characterize the depth of k[K] in terms of the topology of links in the complex K and prove that depth k[K P ] = n for all convex polytopes P of dimension n. We obtain a number of relations between bigraded betti numbers of the complexes K P. We also establish connections between the above questions and the notion of a k-Cohen-Macaulay complex, which leads to a new filtration on the set of simplicial complexes.

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On 2-Buchsbaum complexes

Cited by 6 publications

References 0 publications

A generalization of k-Cohen–Macaulay simplicial complexes

A generalization of k-Cohen–Macaulay simplicial complexes

Level algebras through Buchsbaum* manifolds

Topological applications of Stanley-Reisner rings of simplicial complexes

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