Proposition 7

Let A be a presentable ring.

a) The ring $A$ is noetherian and catenary.

b) Let M be a finitely generated A-module. The mapping

of $\operatorname{Spec}(A)$ into $\mathbf{Z}$ is upper semicontinuous.

c) Let M be a finitely generated $A$-module. The set of $p \in \operatorname{Spec}(A)$ such that the $A_p$-module $M_p$ is Macaulay is a dense open set. Its intersection with $\operatorname{Supp}(M)$ is dense in $\operatorname{Supp}(M)$.

Choose a regular ring R and a surjective homomorphism $R \to A$.

a) The ring R is a Macaulay ring (No. 2, Cor. 2 of Prop. 4), hence A is catenary ($§ 2$, No. 2, Prop. 2 and VIII, § 1, No. 3, Rem. 2).

b) The R-module M is finitely generated and of projective dimension $< +\infty$ (No. 1, Prop. 1). Let us identify $\operatorname{Spec}(A)$ with a closed subset of $\operatorname{Spec}(R)$. Then the function $p \mapsto \dim_{A_p}(M_p) - \operatorname{prof}_{A_p}(M_p)$ on $\operatorname{Spec}(A)$ is the restriction of the function $q \mapsto \dim_{R_q}(M_q) - \operatorname{prof}_{R_q}(M_q)$ on $\operatorname{Spec}(R)$; it is then enough to apply Cor. 4 of Th. 1 of § 3, No. 5.

c) This is proved as in part c) of loc. cit.