Remark 2

Let M be a finitely generated Macaulay A-module, and $p$ an element of $\operatorname{Supp}(M)$. In view of Theorem 2 of § 1, No. 4, one has $\operatorname{prof}_A(p; M) < +\infty$, and there exists an M-regular sequence of length $\operatorname{prof}_A(p; M)$ consisting of elements of $p$. Let J denote the ideal of A generated by such a sequence; then the A-module $M/JM$ is Macaulay (No. 1, example 3) and $p$ is a minimal element of its support. In fact, $p$ contains J and therefore belong to the support of $M/JM$ (II, § 4, No. 4, cor. to Prop. 18); by Corollary 1 to Theorem 2 of § 1, No. 4, the ideal $p$ is contained in an element of $\operatorname{Ass}(M/JM)$, but every prime ideal associated with a finitely generated Macaulay module is a minimal element of its support (Prop. 2).