Resolutions of Facet Ideals

Zheng, Xinxian

doi:10.1081/agb-120037222

Cited by 120 publications

(124 citation statements)

References 8 publications

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“…When restricted to simple graphs, Theorem 1.4(ii) also gives an interesting corollary, which was first proved by Zheng [33] in the special case that G was a forest. Corollary 1.7 (Corollary 6.9) Let G be a chordal graph.…”

Section: Theorem 11 (Theorem 32) Let H Be a Hypergraph With Two Or mentioning

confidence: 89%

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Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers

2007

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We use the correspondence between hypergraphs and their associated edge ideals to study the minimal graded free resolution of squarefree monomial ideals. The theme of this paper is to understand how the combinatorial structure of a hypergraph H appears within the resolution of its edge ideal I(H). We discuss when recursive formulas to compute the graded Betti numbers of I(H) in terms of its subhypergraphs can be obtained; these results generalize our previous work (Hà, H.T., Van Tuyl, A. in J. Algebra 309:405-425, 2007) on the edge ideals of simple graphs. We introduce a class of hypergraphs, which we call properly-connected, that naturally generalizes simple graphs from the point of view that distances between intersecting edges are "well behaved." For such a hypergraph H (and thus, for any simple graph), we give a lower bound for the regularity of I(H) via combinatorial information describing H and an upper bound for the regularity when H = G is a simple graph. We also introduce triangulated hypergraphs that are properly-connected hypergraphs generalizing chordal graphs. When H is a triangulated hypergraph, we explicitly compute the regularity of I(H) and show that the graded Betti numbers of I(H) are independent of the ground field. As a consequence, many known results about the graded Betti numbers of forests can now be extended to chordal graphs.

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Section: Theorem 11 (Theorem 32) Let H Be a Hypergraph With Two Or mentioning

confidence: 89%

“…In the case that H is also triangulated, we explicitly compute reg(I(H)). Our exact formula for reg(I(H)) generalizes Zheng's formula [33] for the regularity of I(H) when H = G is a forest.…”

Section: Properly-connected Hypergraphs and Regularitymentioning

confidence: 97%

Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers

2007

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“…The connection between regularity and disconnected edges first appeared in Zheng's paper [17] and was extended to the case of chordal graphs in [10, Corollary 6.9]. Kummini showed that the conclusion to Corollary 3.10 is still true if is a bipartite graph whose edge ideal is unmixed.…”

Section: } Both Of Which Are Cohen-macaulay Bipartite Graphsmentioning

confidence: 91%

Splittings of monomial ideals

Francisco

Harbourne

Tuyl

2009

Proc. Amer. Math. Soc.

126

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Abstract. We provide some new conditions under which the graded Betti numbers of a monomial ideal can be computed in terms of the graded Betti numbers of smaller ideals, thus complementing Eliahou and Kervaire's splitting approach. As applications, we show that edge ideals of graphs are splittable, and we provide an iterative method for computing the Betti numbers of the cover ideals of Cohen-Macaulay bipartite graphs. Finally, we consider the frequency with which one can find particular splittings of monomial ideals and raise questions about ideals whose resolutions are characteristic-dependent.

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“…After that, hypergraph algebras have been widely studied. See for instance [5], [7], [10], [11], [12], [13], [14], [16], [19]. In [10], the authors use certain connectedness properties to determine a class of hypergraphs such that the hypergraph algebras have linear resolutions.…”

Section: (K ) ⊆ X (H ) and E (K ) ⊆ E (H ) If Y ⊆ X The Inducedmentioning

confidence: 99%

A class of hypergraphs that generalizes chordal graphs

Emtander¹

2010

MATH. SCAND.

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In this paper we introduce a class of hypergraphs that we call chordal. We also extend the definition of triangulated hypergraphs, given by H. T. Hà and A. Van Tuyl, so that a triangulated hypergraph, according to our definition, is a natural generalization of a chordal (rigid circuit) graph. R. Fröberg has showed that the chordal graphs corresponds to graph algebras, R/I (G), with linear resolutions. We extend Fröberg's method and show that the hypergraph algebras of generalized chordal hypergraphs, a class of hypergraphs that includes the chordal hypergraphs, have linear resolutions. The definitions we give, yield a natural higher dimensional version of the well known flag property of simplicial complexes. We obtain what we call d-flag complexes.

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Resolutions of Facet Ideals

Cited by 120 publications

References 8 publications

Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers

Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers

Splittings of monomial ideals

A class of hypergraphs that generalizes chordal graphs

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