Corollary

Let $A$ be a noetherian ring and $M$ a finitely generated $A$-module. The following conditions are equivalent: (i) the $A$-module $M$ is Macaulay; (ii) one has $\mathrm{prof}_{A_p}(M_p) = \dim_{A_p}(M_p)$ for every $p \in \mathrm{Supp}(M)$; (iii) one has $\mathrm{prof}_F(M) = \mathrm{codim}(\mathrm{Supp}(M) \cap F', \mathrm{Supp}(M))$ for every closed subset $F$ of $\mathrm{Spec}(A)$; (iv) one has $\mathrm{prof}_A(p; M) = \dim_{A_p}(M_p)$ for every $p \in \mathrm{Supp}(M)$.

(i) $\Rightarrow$ (ii): this follows from Proposition 1. (ii) $\Rightarrow$ (iii): by prop. 8 of § 1, No. 5, $\mathrm{prof}_F(M)$ is the greatest lower bound of the integers $\mathrm{prof}(M_p)$ for $p$ ranging over $\mathrm{Supp}(M) \cap F$. If $M$ is Macaulay, then for such an ideal $p$ one has the equalities $\mathrm{prof}(M_p) = \dim(M_p) = \mathrm{codim}(V(p), \mathrm{Supp}(M))$ (VIII, § 1, No. 4, prop. 9), whence (iii). (iii) $\Rightarrow$ (iv): it is enough to take $F = V(p)$. (iv) $\Rightarrow$ (i): this follows from the inequalities $\mathrm{prof}_A(p; M) \leq \mathrm{prof}(M_p) \leq \dim(M_p)$, valid for every $p \in \mathrm{Supp}(M)$ ($§ 1$, No. 5, remark 3 and No. 4, cor. 2 of th. 2).