Cohen–Macaulay properties of square-free monomial ideals

Abstract

In this paper, we study simplicial complexes as higher-dimensional graphs in order to produce algebraic statements about their facet ideals. We introduce a large class of square-free monomial ideals with Cohen-Macaulay quotients, and a criterion for the Cohen-Macaulayness of facet ideals of simplicial trees. Along the way, we generalize several concepts from graph theory to simplicial complexes.

Resolutions of Facet Ideals

“…Since Γ is a pure tree and connected in codimension 1, by Proposition 1.12, lk Γ G is connected. Let 4 } be an irredundant chain between {x 1 , x 2 } and {x 3 , x 4 } in lk Γ G. Then the subcomplex F, F 1 , . .…”

Cohen-Macaulay edge ideal whose height is half of the number of vertices

“…They are a generalization of grafted graphs introduced by Faridi [4]. If G is an unmixed B-grafted graph, then we have 2 height I(G) = V (G).…”

Cohen-Macaulay clutters with combinatorial optimization properties and parallelizations of normal edge ideals

“…Set S = K[X ∪Y ] and I = I(C ). The clutter C is a grafting of C as defined by Faridi in [10]. Then I is Cohen-Macaulay by [10,Theorem 8.2].…”

“…Edge ideals of clutters also correspond to simplicial complexes via the Stanley-Reisner correspondence [25] and to facet ideals [9,32]. The Cohen-Macaulay property of edge ideals has been recently studied in [3,10,16,22,27] using a combinatorial approach based on the notions of shellability, linear quotients, unmixedness, acyclicity and transitivity of digraphs, and the König property.…”

“…The study of edge ideals from the combinatorial optimization point of view was initiated in [1,26] and continued in [6,8,13,14,15,31], see also [17]. The Cohen-Macaulay and normality properties are two of the most interesting properties an edge ideal can have, see [2,10,25,30] and [19,28] respectively.…”