A note on almost Cohen-Macaulay monomial ideals

Abstract

Let R = k[x1, . . . , xn] be the polynomial ring in n variables over a field k and let I be a monomial ideal of R. In this paper, we study almost Cohen-Macaulay simplicial complex. Moreover, we characterize the almost Cohen-Macaulay polymatroidal Veronese type and transversal polymatroidal ideals and furthermore we give some examples.

Almost Cohen-Macaulay bipartite graphs and connected in codimension two

“…, F m be the simplicial complex of independent sets of G. By Lemma 2.1 we have lk ∆ {x n } and lk ∆ {y n } are the simplicial complexs of independent sets of H and K respectively. Therefore by [18,Corollary 3.6], H and K are aCM.…”

“…Proof. Let H = G \ {u, v, w} be subgraph of G, by Lemma 2.1 we have lk ∆ {u} is the simplicial complexs of independent sets of H. By [18,Corollary 3.6], lk ∆ {u} is aCM. If Γ = F 1 , .…”

“…(=⇒). By [18,Theorem 3.4], it is enough to show that Hi (lk ∆ F ; k) = 0 for all F ∈ ∆(n − 1) and for all i < dim lk ∆ F − 1. We proceed by induction on dim lk ∆ F .…”

“…Since ∆ is aCM and d ≥ 3, we have ∆ is connected by [18, Lemma 3.2 and Theorem 3.4]. Therefore there exists a sequence of facets [18,Corollary 3.6]. By [13, Lemma 9.1.12] and working with induction on the dimension of ∆, we may assume that lk…”

“…We say that R/I(G) is aCM when depth R/I(G) ≥ dim R/I(G) − 1. The aCM modules has been studied in [9,15,16,14,3,19,20,21,18].…”