Exercise 10, § 4

Let $A$ be a noetherian ring, $M$ a finitely generated $A$-module. For every integer $n \geqslant 0$, denote by $X_n$ the set of elements $p$ of $\mathrm{Spec}(A)$ such that $\dim_{A_p}(M_p) - \mathrm{prof}_{A_p}(M_p) \geqslant n$. Let $k$ be an integer $\geqslant 0$. (a) In order that $M$ satisfy the property $S_k$ ($§ 1$, exer. 8), it is necessary and sufficient that one have $\mathrm{codim}(X_n, \mathrm{Supp}(M)) \geqslant n + k$ for every $n \geqslant 0$. (b) Suppose that $X_n$ is closed in $\mathrm{Spec}(A)$ for every $n \geqslant 0$. Let $p$ be an element of $\mathrm{Supp}(M)$. In order that the $A_p$-module $M_p$ satisfy the property $S_k$ ($§ 1$, exer. 8), it is necessary and sufficient that one have $\mathrm{codim}(X'_n, \mathrm{Supp}(M)) \geqslant n + k$ for every $n \geqslant 0$ and every component $X'_n$ of $X_n$ containing $p$. (c) Suppose that the ring $A$ is presentable. Deduce from (b) that the set of prime ideals $p$ of $A$ such that $M_p$ satisfies the property $S_k$ is open in $\mathrm{Spec}(A)$.