Regularity, depth and arithmetic rank of bipartite edge ideals

Kummini, Manoj

doi:10.1007/s10801-009-0171-6

Cited by 89 publications

(81 citation statements)

References 13 publications

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“…(1) G is a sequentially Cohen-Macaulay bipartite graph (see [48]); (2) G is an unmixed bipartite graph (see [36]); (3) G is a very well-covered graph (see [38]); (4) G is a C 5 -free vertex decomposable graph (see [34], the case where G is also C 4 -free was proved in [5]). …”

Section: Small Regularity and Computing Regularitymentioning

confidence: 99%

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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Section: Small Regularity and Computing Regularitymentioning

confidence: 99%

Regularity of Squarefree Monomial Ideals

Harbourne

2014

Springer Proceedings in Mathematics &Amp; Statistics

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“…Bounds on the regularity of edge ideals have been studied by a number of researchers (see [1], [2], [3], [4], [5], [6], [7], [8] [9], [11], [12], [13]). For example, Fröberg (see [3]) has shown that, when I(G) is the edge ideal of a graph whose complement does not have any induced cycle of size greater than or equal to four, then I(G) has linear minimal free resolution.…”

Section: Introductionmentioning

confidence: 99%

The regularity of powers of edge ideals

Banerjee

2014

J Algebr Comb

162

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“…One goal of this research program is to relate algebraic properties and invariants of a squarefree monomial ideal to combinatorial properties and statistics of the corresponding simple hypergraph. In this note, we will examine the Castelnuovo-Mumford regularity of such ideals, which has been previously studied in work including [5,7,9,14,16,20,22].…”

Section: Introductionmentioning

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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Regularity, depth and arithmetic rank of bipartite edge ideals

Cited by 89 publications

References 13 publications

Regularity of Squarefree Monomial Ideals

Regularity of Squarefree Monomial Ideals

The regularity of powers of edge ideals

Results on the regularity of square-free monomial ideals

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