A Generalization of the Taylor Complex Construction

Herzog, Jürgen

doi:10.1080/00927870601139500

Cited by 30 publications

(19 citation statements)

References 4 publications

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“…Kalai and Meshulam [32] obtain the following powerful result. This result was later extended to arbitrary (not necessarily squarefree) monomial ideals by Herzog [29]. Theorem 3.6.…”

Section: Regularity and Inductionmentioning

confidence: 91%

Regularity of Squarefree Monomial Ideals

Harbourne

2014

Springer Proceedings in Mathematics &Amp; Statistics

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Abstract. We survey a number of recent studies of the Castelnuovo-Mumford regularity of squarefree monomial ideals. Our focus is on bounds and exact values for the regularity in terms of combinatorial data from associated simplicial complexes and/or hypergraphs.Dedicated to Tony Geramita, a great teacher, colleague and friend. IntroductionCastelnuovo-Mumford regularity (or simply regularity) is an important invariant in commutative algebra and algebraic geometry that governs the computational complexity of ideals, modules and sheaves. Computing or finding bounds for the regularity is a difficult problem. Many simply stated questions and conjectures have been verified only in very special cases. For instance, the Eisenbud-Goto conjecture, which states that the regularity of a projective variety is bounded by the difference between its degree and codimension, has been proved only for arithmetic Cohen-Macaulay varieties, curves and surfaces.The class of squarefree monomial ideals is a classical object in commutative algebra, with strong connections to topology and combinatorics, which continues to inspire much of current research. During the last few decades, advances in computer technology and speed of computation have drawn many researchers' attention toward problems and questions involving this class of ideals. Investigating the regularity of squarefree monomial ideals has evolved to be a highly active research topic in combinatorial commutative algebra.In this paper, we survey a number of recent studies on the regularity of squarefree monomial ideals. Even for this class of ideals, the list of results on regularity is too large to exhaust. Our focus will be on studies that find bounds and/or compute the regularity of squarefree monomial ideals in terms of combinatorial data from associated simplicial complexes and hypergraphs. Our aim is to provide readers with an adequate overall picture of the problems, results and techniques, and to demonstrate similarities and differences between these studies. At the same time, we hope to motivate interested researchers, and especially graduate students, to start working in this area. We shall showcase results in a timeline to exhibit (if possible) trends in developing new and/or more general results from previous ones.The paper is structured as follows. In the next section we collect basic notation and terminology from commutative algebra and combinatorics that will be used in the paper. This section provides readers who are new to the research area the necessary background to start. More advanced readers can skip this section and go directly to Section 3. In Section 1 arXiv:1310.7912v2 [math.AC]

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“…Kalai and Meshulam [32] obtain the following powerful result. This result was later extended to arbitrary (not necessarily squarefree) monomial ideals by Herzog [29]. Theorem 3.6.…”

Section: Regularity and Inductionmentioning

confidence: 91%

Regularity of Squarefree Monomial Ideals

Harbourne

2014

Springer Proceedings in Mathematics &Amp; Statistics

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“…Theorem 4 was conjectured by Terai [31]. Herzog [20] later generalized the result to monomial ideals that are not square-free. In the context of edge ideals, Theorem 4 says that if G 1 , .…”

Section: Proof Of Theoremmentioning

confidence: 99%

Matchings, coverings, and Castelnuovo-Mumford regularity

Woodroofe¹

2014

J. Commut. Algebra

120

104

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We show that the co-chordal cover number of a graph G gives an upper bound for the Castelnuovo-Mumford regularity of the associated edge ideal. Several known combinatorial upper bounds of regularity for edge ideals are then easy consequences of covering results from graph theory, and we derive new upper bounds by looking at additional covering results.Comment: 12 pages; v4 has minor changes for publicatio

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“…Kalai and Meshulam gave a topological proof of Theorem 3.1 via the correspondence in Lemma 2.3. Theorem 3.1 was later extended to arbitrary (not necessarily square-free) monomial ideals by Herzog [11], who used algebraic techniques.…”

Section: Regularity and Collagesmentioning

confidence: 99%

Results on the regularity of square-free monomial ideals

Harbourne

Woodroofe

2014

Advances in Applied Mathematics

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Abstract. In a 2008 paper, the first author and Van Tuyl proved that the regularity of the edge ideal of a graph G is at most one greater than the matching number of G. In this note, we provide a generalization of this result to any square-free monomial ideal. We define a 2-collage in a simple hypergraph to be a collection of edges with the property that for any edge E of the hypergraph, there exists an edge F in the 2-collage such that |E \ F | ≤ 1. The Castelnuovo-Mumford regularity of the edge ideal of a simple hypergraph is bounded above by a multiple of the minimum size of a 2-collage. We also give a recursive formula to compute the regularity of a vertex-decomposable hypergraph. Finally, we show that regularity in the graph case is bounded by a certain statistic based on maximal packings of nondegenerate star subgraphs.

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A Generalization of the Taylor Complex Construction

Abstract: Abstract. Given multigraded free resolutions of two monomial ideals we construct a multigraded free resolution of the sum of the two ideals.

Cited by 30 publications

References 4 publications

Regularity of Squarefree Monomial Ideals

Regularity of Squarefree Monomial Ideals

Matchings, coverings, and Castelnuovo-Mumford regularity

Results on the regularity of square-free monomial ideals

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