Remark 1

Under the hypotheses of Prop. 2, it may happen that two irreducible components $X_1$ and $X_2$ of $\mathrm{Supp}(M)$ have an intersection $Y$ reduced to a point and that one has $\dim X_1 \neq \dim X_2$ and $\dim X_2 \neq \mathrm{codim}(Y, X_2)$ (cf. exercise 4). However, this cannot happen when the ring $A$ is local, as is shown by the following corollary.