Corollary 4

Let $A$ be a noetherian Macaulay ring and $M$ a finitely generated $A$-module of finite projective dimension.

a) Let $p \in \operatorname{Spec}(A)$; let us denote by $\mathscr{C}(p)$ the set of irreducible components of $\operatorname{Supp}(M)$ containing $p$. We have

b) The mapping $p \mapsto \dim_{A_p}(M_p) - \operatorname{prof}_{A_p}(M_p)$ of $\operatorname{Spec}(A)$ into $\overline{\mathbf{Z}}$ is upper semi-continuous.

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

a) We may suppose that $p \in \operatorname{Supp}(M)$. Put $\varphi(p) = \dim(A_p) - \dim_{A_p}(M_p)$. By cor. 2 above, we have

Since $A$ is a Macaulay ring, we have

and consequently

b) Let $p \in \operatorname{Spec}(A)$, and let $F$ be the union of the irreducible components of $\operatorname{Supp}(M)$ which do not contain $p$. For every element $q$ of $\operatorname{Spec}(A) - F$, we have $\mathscr{C}(q) \subset \mathscr{C}(p)$, whence $\varphi(q) \geq \varphi(p)$ by the formula above. Consequently the function $\varphi$ is lower semi-continuous; assertion b) then follows from prop. 3 of No. 2.

c) Let $U$ be the set of elements $p$ of $\operatorname{Spec}(A)$ such that $M_p$ is Cohen-Macaulay. The condition $p \in U$ is equivalent to $\dim(M_p) - \operatorname{prof}(M_p) \leq 0$, so that $U$ is open by b). Since $U$ contains $\operatorname{Spec}(A) - \operatorname{Supp}(M)$, it is enough to prove that $U \cap \operatorname{Supp}(M)$ is dense in $\operatorname{Supp}(M)$. For every minimal prime ideal $p$ of $\operatorname{Supp}(M)$, the $A_p$-module $M_p$ is of finite length (IV, § 2, No. 5, Corollary 2 of Proposition 7 and § 1, No. 3, Corollary 1 of Proposition 7), hence Cohen-Macaulay; consequently $U$ meets all the irreducible components of $\operatorname{Supp}(M)$. One concludes by means of Proposition 1 of II, § 4, No. 1.