Cohen–Macaulay modifications of the facet ideal of a simplicial complex

Abstract

We define the chordal simplicial complex by using the definition of chordal clutter introduced by Woodroofe. We show that the facet ideal of the chordal simplicial complex is Cohen–Macaulay if and only if it is unmixed. Moreover, we prove that the facet ideal of a chordal simplicial complex has infinitely many nontrivial Cohen–Macaulay modifications.

Cohen-Macaulay Growing Graphs

“…Order the faces of ∆ in terms of increasing dimensions. If two faces have the same dimension, then order them by the ordering of variables defined in equation (2). Then, associated to each face F of ∆, consider the block of facets of ∆ ′ associated to F , ordered as in (2).…”

“…Here arises a natural question: Under what conditions, squarefree Cohen-Macaulay monomial ideals does there exist at least one nontrivial Cohen-Macaulay modification, or do exist infinitely many nontrivial Cohen-Macaulay modifications? The answer to this question has been given for several classes of ideal in the following papers [1], [2], [4], [5], [6]. In this section, we discuss the conditions for which the facet ideal of the chordal simplicial complex [Definition 4.3] admits non trivial modifications.…”

“…At the end of this section, we give a complete characterization of all Cohen-Macaulay chordal graphs by adding one more equivalent condition to a well known characterization of Cohen-Macaulay chordal graphs due to Herzog, Hibi and Zheng [18]. In the final section, we discuss Cohen-Macaulay modifications, which are discussed in [1], [2], [4], [5] and [6]. Here we link the concept of growing graphs[Definition 3.1] to the grafted simpicial complexes[Definition 4.7], defined by Sara Faridi in [12].…”