Characterization of rings and modules with Serre condition \((S_n)\) (Q1808749)

Let \((R, \mathfrak m)\) denote a local ring. Let \(M\) be a finitely generated \(R\)-module. A well-known characterization says that \(M\) is a Cohen-Macaulay module if and only if \(M/(a_1, \ldots, a_i)M\), \(i = 0, \ldots, \dim_R M -1,\) is unmixed (with respect to height) for every (equivalently some) system of parameters \(a_1, \ldots, a_d\), \(d = \dim_R M,\) of \(M\). This result is generalized to a characterization of Serre's condition \((S_n)\) saying that \( \dim M_{\mathfrak p} \geq \min \{n, \text{ height}_M \mathfrak p \}\) for every prime ideal \(\mathfrak p \in \text{ Supp}_R M.\) This was already observed (at least in the case of \(M = R\)) by \textit{H. Matsumura} [``Commutative algebra'' (Benjamin, New York 1970; Zbl 0211.06501); see p. 125]. Then the authors prove a corresponding result for the unmixedness of subsystems of parameters with respect to the dimension and call the corresponding condition \((S'_n).\) Reviewers remark: This \((S'_n)\) condition means nothing else but \[ \dim M_{\mathfrak p} \geq \min \{n, \dim_R M - \dim R/\mathfrak p \} \] for every prime ideal \(\mathfrak p \in \text{Supp}_R M.\)