Resolutions of Facet Ideals

By Xinxian Zheng. 2003

151 citations recorded by the source corpus.

Abstract In this paper we study the resolution of a facet ideal associated with a special class of simplicial complexes introduced by Faridi. These simplicial complexes are called trees, and are a generalization (to higher dimensions) of the concept of a tree in graph theory. We show that the Koszul homology of the facet ideal I of a tree is generated by the homology classes of monomial cycles, determine the projective dimension and the regularity of I if the tree is 1-dimensional, show that the graded Betti numbers of I satisfy an alternating sum property if the tree is connected in codimension 1, and classify all trees whose facet ideal has a linear resolution.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/80a001a2e514171c8e40d0a9ec84e2896b140d6c

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In Algebra and Number Theory

In Computational Theory and Mathematics

Citations in the corpus

Non-vanishingness of Betti numbers of edge ideals (2011, methodology): “In Section 2, we recall some definitions on graphs and results by Zheng [16], Hà and Van Tuyl [7], and Katzman [11] mentioned above.”
Algebraic study on Cameron-Walker graphs (2013, background): “, chordal graphs, unmixed bipartite graphs and sequentially Cohen–Macaulay bipartite graphs ([9, 12, 15]; see also [7, 13, 19, 20]).”
Resolutions of square-free monomial ideals via facet ideals: a survey (2006, methodology): “For instance, Jacques [31] provided formulas for the graded Betti numbers of edge ideals of special classes of graphs including cycles and complete graphs; Roth and the second author [37] calculated the graded Betti numbers in the linear strand of edge ideals of graphs having no minimal cycle of len”
On vertex decomposable simplicial complexes and their Alexander duals (2013, background): “and [19] this description of regularity has been extended to chordal graphs and shellable bipartite graphs, respectively. Also, Kimura in [11] extended the characterization of projective dimension in [23] to chordal graphs and showed that, when G is a chordal graph, pd(R/I(G)) = bight(I(G)). In [12]”
Simplicial trees are sequentially Cohen–Macaulay ☆ (2003, methodology): “In particular, in [ F2 ] we show that a tree is unmixed if and only if it is \grafted".”
Regularity of powers of bipartite graphs (2016, unknown): “In particular, there has been active research on bounding the homological invariants of edge ideals in terms of the combinatorial invariants of the associated graphs, see for example [3,10,12,15,17,19,22,23,26,28,32,33].”
Edge ideals: algebraic and combinatorial properties (2010, background): “15 shows that the regularity of R/I(G) equals im(G) for any forest G, which was first proved by Zheng [118].”
Standard graded vertex cover algebras, cycles and leaves (2006, background): “The notion good leaf was introduced in the thesis of Zheng [26].”
A Note on Rees Algebras and the MFMC Property (2005, methodology): “This notion has been studied by Faridi [10] and Zheng [18].”
Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers (2006, background): “Our definition of properly-connected is similar to (but not equivalent to) what Zheng [33, Definition 3.14] called the intersection property for a simplicial complex.”
Splittings of monomial ideals (2008, background): “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.”
Regularity, depth and arithmetic rank of bipartite edge ideals (2009, background): “Zheng [19] who showed that if I is the edge ideal of a tree (an acyclic graph) then regR/I is the maximum size of a pairwise disconnected set of edges [19, Theorem 2.”
Sequentially Cohen–Macaulay bipartite graphs: vertex decomposability and regularity (2009, background): “We recover Zheng’s [25] formula for the regularity of the edge ideals of trees as corollary.”
A CLASS OF HYPERGRAPHS THAT GENERALIZES CHORDAL GRAPHS (2008, unknown): “See for instance [5], [7], [10], [11], [12], [13], [14], [16], [19].”