On almost Cohen-Macaulayness of quotient modules (Q1783581)

Let \((A,\mathfrak m)\) be a noetherian local ring and \(M\neq(0)\) be a finitely generated \(A\)-module of dimension \(d.\) It is said that \(M\) is almost Cohen-Maculay if \(\dim(M)\leq\mathrm{depth}(M)+1.\) It is shown that \(M\) is an almost Cohen-Macaulay \(A\)-module if and only if there exists a proper ideal \(\mathfrak a\) of \(A\) such that for \(n\) large enough \(M/\mathfrak a^nM\) is an almost Cohen-Macaulay \(A\)-module of dimension \(d\).