Lemma 2

Let $\overline{\mathscr{G}}$ be the image of $\mathscr{G}$ in $A/\mathrm{Ann}(M)$. One has

When moreover $SM \neq M$, one has

Let us denote by $\alpha$ the annihilator of $M$. By the cor. to Prop. 18 of II, § 4, No. 4, the support of the A-module $M/SM$ is $V(\mathscr{G} + \alpha)$. Its codimension in $\mathrm{Supp}(M)$ is therefore equal to the codimension of $V(\mathscr{G} + \alpha)$ in $V(\alpha)$, or again to the codimension of $V((\mathscr{G} + \alpha)/\alpha)$ in $\mathrm{Spec}(A/\alpha)$, which is none other than the height of $\overline{\mathscr{G}}$.

Suppose $SM \neq M$; the inequality $\mathrm{ht}(\overline{\mathscr{G}}) \leq \mathrm{Card}(S)$ is obvious when $S$ is infinite, and follows from Prop. 4 b) of VIII, § 3, No. 3 when $S$ is finite.