Proposition 8

Let A be a noetherian ring, F a closed subset of $\mathrm{Spec}(A)$, and M a finitely generated A-module. One has

This is clear if $\mathrm{prof}_F(M) = +\infty$ (Remark 2). If $\mathrm{prof}_F(M) = 0$, there exists a prime ideal $p \in \mathrm{Ass}(M) \cap F$ (Remark 2); one has $pA_p \in \mathrm{Ass}(M_p)$ (IV, § 1, No. 2, Prop. 5), hence $\mathrm{prof}_{A_p}(M_p) = 0$ (Remark 2 of No. 1), whence the proposition in this case.

Suppose $0 < \mathrm{prof}_F(M) < +\infty$; let J be an ideal of A such that $V(J) = F$, and let x be an element of J such that the homothety $x_M$ is injective (loc. cit.). For each prime ideal p, the homothety $x_{M_p}$ is injective. By Prop. 7 of No. 4, we therefore have

One then concludes by induction on the integer $\mathrm{prof}_F(M)$.