Exercise 9, § 6

Let $A$ be a ring and $M$ an a-module of finite presentation. For every $\mathfrak{p} \in \mathrm{Spec}(A)$, put $\kappa(\mathfrak{p}) = A_\mathfrak{p}/\mathfrak{p} \cdot A_\mathfrak{p}$. Prove that for every $k \in \mathbf{N}$, the set of $\mathfrak{p} \in \mathrm{Spec}(A)$ such that $\dim_{\kappa(\mathfrak{p})}(M \otimes_A \kappa(\mathfrak{p})) \geq k$ $^1$ is a closed subset of $\mathrm{Spec}(A)$. (One may take a presentation $A^m \xrightarrow{u} A^n \to M \to 0$ of M and consider the minors of the matrix of $u$.)