Property of Almost Cohen–Macaulay over Extension Modules

Abstract

Let (R, m) be a Cohen–Macaulay local ring of dimension d, C a canonical R-module and M an almost Cohen–Macaulay R-module of dimension n and of depth t. We prove that dim [Formula: see text] and if [Formula: see text] then [Formula: see text] is an almost Cohen–Macaulay R-module. In particular, if [Formula: see text] then HomR(M, C) is an almost Cohen–Macaulay R-module. In addition, with some conditions, we show that [Formula: see text] is also almost Cohen–Macaulay. Finally, we study the vanishing [Formula: see text] and [Formula: see text].

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].…”