More properties of almost Cohen-Macaulay rings

Abstract

Some interesting properties of almost Cohen-Macaulay rings are investigated and a Serre type property connected with this class of rings is studied.

A note on almost Cohen-Macaulay monomial ideals

“…It is clear that all CM monomial ideals are aCM. Several authors studied almost Cohen-Macaulay modules (see for example [2], [5], [12], [13], [14], [16], [17] and [18]). For a square-free monomial ideal I of R, we may consider the simplicial complex ∆ for which I = I ∆ is the Stanley-Reisner ideal of ∆ and K[∆] = R/I ∆ is the Stanley-Reisner ring.…”

Almost Cohen-Macaulay bipartite graphs and connected in codimension two

“…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].…”

Flat local morphisms of rings with prescribed depth and dimension

“…While preparing [3], the present author was looking for an example of a flat local ring homomorphism of noetherian local rings u : (A, m) → (B, n) such that A and B/mB are almost Cohen-Macaulay, while B is not almost Cohen-Macaulay. This means that one should construct for example such a morphism with depth(B) = depth(A) = 0, dim(B) = 2 and dim(A) = 1.…”

“…Note that actually the flatness of the homomorphism u is the non-trivial point in the construction. After asking several people without obtaining a satisfactory answer, he decided to let it as an open question in [3]. The answer came soon, an example with the desired features being constructed by Tabaâ [6].…”

“…Remark 2.6 If (A, m) is a noetherian local ring, we call the Cohen-Macaulay defect of A, the natural number cmd(A) = dim(A)−depth(A). Thus A is Cohen-Macaulay if and only if cmd(A) = 0 and A is almost Cohen-Macaulay if and only if cmd(A) ≤ 1(see[3]).…”