Simplicial trees are sequentially Cohen–Macaulay ☆

By Sara Faridi. 2003

104 citations recorded by the source corpus.

Abstract This paper uses dualities between facet ideal theory and Stanley–Reisner theory to show that the facet ideal of a simplicial tree is sequentially Cohen–Macaulay. The proof involves showing that the Alexander dual (or the cover dual, as we call it here) of a simplicial tree is a componentwise linear ideal. We conclude with additional combinatorial properties of simplicial trees.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Pharmacology

View Paper: https://www.semanticscholar.org/paper/9e51136a3810f71fe904f020636f8b58c5ce2229

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Computational Theory and Mathematics

In Pharmacology

Citations in the corpus

Sequentially Cohen–Macaulay bipartite graphs: vertex decomposability and regularity (2009, background): “thor with Francisco and H`a [8]. The corollary is true because trees and CM bipartite graphs belong to the set of SCM bipartite graphs. Trees, which are always bipartite, were ﬁrst shown to be SCM in [6]; alternative proofs were given in [4, 9, 20, 24]. Corollary 3.4. IfG iseitheratreeoraCMbipartite”
On vertex decomposable simplicial complexes and their Alexander duals (2013, background): “monomial ideal I and combinatorial invariants of the Stanley-Reisner simplicial complex ∆I associated to I is of great interest and recently has been studied by many researchers (see for example [3], [4], [6], [7], [8], [11], [12], [13], [14], [15], [19], [23]). The edge ideal was ﬁrst considered by”
Algebraic Properties of the Path Ideal of a Tree (2009, background): “0 ⊂ M1 ⊂ ··· ⊂ Mr = M such that each Mi/Mi−1 for i = 1,...,r is Cohen-Macaulay, and the Krull dimensions of the quotients are increasing: dim(M1/M0) < dim(M2/M1) < ··· < dim(Mr/Mr−1). Faridi [7] showed that simplicial trees give rise to sequentially Cohen-Macaulay modules: Theorem 2.11 ([6,”
Resolutions of square-free monomial ideals via facet ideals: a survey (2006, background): “uced later by Faridi in [5]. To learn more about the properties of edge ideals, one should see [10, 14, 15, 16, 24, 26, 28, 27, 30, 32]; further properties of facet ideals can be found in the sequels [6, 7, 8] to Faridi’s paper cited above. Eliahou and Villarreal [4] provided one of the ﬁrst example”
Monomial ideals via square-free monomial ideals (2005, methodology): “We refer the reader to [2], [5], [6], [7], and [14] for more details and proofs in each of these topics.”
Edge ideals: algebraic and combinatorial properties (2010, background): “] over a ﬁeld K. The edge ideal of C, denoted by I(C), is the ideal of R generated by all monomials xe = Q x i∈e xi such that e ∈ E(C). Edge ideals of graphs and clutters were introduced in [109] and [39, 47, 53], respectively. The assignment C → I(C) establishes a natural one-to-one correspondence ”
Sequentially Cohen-Macaulay edge ideals (2005, background): “Remark 3.5. In [ 4 ], Faridi proved that if I(∆) is the facet ideal of simplicial forest ∆, then R/I(∆) is sequentially Cohen-Macaulay.”