arXiv 1603.013312016-03-04math.AC

Hypergraphs with high projective dimension and 1-dimensional Hypergraphs

Kuei-Nuan Lin, Paolo Mantero

备注：: 22 pages, 9 figures, 2 algorithm

英文摘要：

We prove a sufficient and a necessary condition for a square-free monomial ideal $J$ associated to a (dual) hypergraph to have projective dimension equal to the minimal number of generators of $J$ minus 2. We also provide an effective explicit procedure to compute the projective dimension of 1-dimensional hypergraphs $\mathcal{H}$ when each connected component contains at most one cycle. An algorithm to compute the projective dimension is also included. Applications of these results are given; they include, for instance, computing the projective dimension of monomial ideals whose associated hypergraph has a spanning Ferrers graph.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Codismantlability and projective dimension of the Stanley-Reisner ring of special hypergraphs

In this paper firstly, we generalize the concept of codismantlable graphs to hypergraphs and show that some special vertex decomposable hypergraphs are codismantlable. Then we generalize the concept of bouquet in graphs to hypergraphs to extend some combinatorial invariants of graphs about disjointness of a set of bouquets. We use these invariants to characterize the projective dimension of Stanley-Reisner ring of special hypergraphs in some sense.

2016-01-04math.ACmath.CO

Projective Dimension of String and Cycle Hypergraphs

We present a closed formula and a simple algorithmic procedure to compute the projective dimension of square-free monomial ideals associated to string or cycle hypergraphs. As an application, among these ideals we characterize all the Cohen-Macaulay ones.

2014-05-08math.ACmath.CO

Projective dimension of Hypergraphs

Given a square-free monomial ideal $I$, satisfying certain hypotheses, in a polynomial ring $R$ over a field $\mathbb{K}$, we compute the projective dimension of $I$. Specifically, we focus on the cases where the 1-skeleton of an associated hypergraph is either a string or a cycle. We investigate the impact on the projective dimension when higher dimensional edges are removed. We prove that the higher dimensional edge either has no effect on the projective dimension or the projective dimension only goes up by one with the extra higher dimensional edge.

2021-02-28math.ACmath.CO

Lattices and Hypergraphs associated to square-free monomial ideals

Given a square-free monomial ideal $I$ in a polynomial ring $R$ over a field $\mathbb{K}$, one can associate it with its LCM-lattice and its hypergraph. In this short note, we establish the connection between the LCM-lattice and the hypergraph, and in doing so we provide a sufficient condition for removing higher dimension edges of the hypergraph without impacting the projective dimension of the square-free monomial ideal. We also offer algorithms to compute the projective dimension of a class of square-free monomial ideals built using the new result and previous results of Lin-Mantero.

2019-09-21math.AC

通过组合不变量对超图边理想的代数不变量的若干解释

Some interpretations for algebraic invariants of edge ideals of hypergraphs via combinatorial invariants

本文引入并比较几类类似匹配数的超图组合不变量，用以刻画和界定边理想的分次 Betti 数、投射维数与 CM 正则度等代数不变量。

2020-01-24math.AC

超图品种的matroid分层，其实现空间以及离散条件独立模型

Matroid stratifications of hypergraph varieties, their realization spaces, and discrete conditional independence models

本文研究了超图品种的matroid分层及其实现空间，揭示了超图品种的不可约分解结构，并展示了其在条件独立模型中的应用。

2022-09-08math.ACmath.AGmath.CO

更多相关论文收起更多论文

Quasi-projective dimension

In this paper, we introduce a new homological invariant called quasi-projective dimension, which is a generalization of projective dimension. We discuss various properties of quasi-projective dimension. Among other things, we prove the following. (1) Over a quotient of a regular local ring by a regular sequence, every finitely generated module has finite quasi-projective dimension. (2) The Auslander--Buchsbaum formula and the depth formula for modules of finite projective dimension remain valid for modules of finite quasi-projective dimension. (3) Several results on vanishing of Tor and Ext ho…

2021-02-12math.AC

Hypergraphs and the Regularity of Square-free Monomial Ideals

We define a new combinatorial object, which we call a labeled hypergraph, uniquely associated to any square-free monomial ideal. We prove several upper bounds on the regularity of a square-free monomial ideal in terms of simple combinatorial properties of its labeled hypergraph. We also give specific formulas for the regularity of square-free monomial ideals with certain labeled hypergraphs. Furthermore, we prove results in the case of one-dimensional labeled hypergraphs.

2013-03-29math.ACmath.CO

Conditional probabilities via line arrangements and point configurations

We study the connection between probability distributions satisfying certain conditional independence (CI) constraints, and point and line arrangements in incidence geometry. To a family of CI statements, we associate a polynomial ideal whose algebraic invariants are encoded in a hypergraph. The primary decompositions of these ideals give a characterisation of the distributions satisfying the original CI statements. Classically, these ideals are generated by 2-minors of a matrix of variables, however, in the presence of hidden variables, they contain higher degree minors. This leads to the stu…

2021-03-31math.ACmath.AGmath.CO

Associated Primes of the Square of the Alexander Dual of Hypergraphs

The purpose of this paper is to provide methods for determining the associated primes of the square of the Alexander dual of the edge ideal for an m-hypergraph H. We prove a general method for detecting associated primes of the square of the Alexander dual of the edge ideal based on combinatorial conditions on the m-hypergraph. Also, we demonstrate a more efficient combinatorial criterion for detecting the non-existence of non-minimal associated primes. In investigating 3-hypergraphs, we prove a surprising extension of the previously discovered results for 2-hypergraphs (simple graphs). For 2-…

2009-01-12math.ACmath.CO

拟投射维数与 Gorenstein 投射维数

Quasi-projective dimension and Gorenstein projective dimension

Yunnan Normal University（云南师范大学）

本文从 Gorenstein 同调代数角度研究拟投射维数，证明其在特定条件下与 Gorenstein 投射维数一致，并引入拟 Gorenstein 投射维数作为其类比。

2026-09-09math.ACmath.RAmath.RT