Corollary 1

Retain the hypotheses of theorem 1.

a) One has $dp_A(M) \leq \operatorname{prof}(A)$. For there to be equality, it is necessary and sufficient that the maximal ideal $m_A$ be associated to $M$.

b) One has $\operatorname{prof}_A(M) \leq \operatorname{prof}(A)$. For there to be equality, it is necessary and sufficient that $M$ be free.

a) In fact, “ $\operatorname{prof}_A(M) = 0$ ” is equivalent to “ $m_A \in \operatorname{Ass}(A)$ ” ($§ 1$, No. 1, remark 2).

b) In fact, “ $dp_A(M) = 0$ ” is equivalent to “ $M$ is free ”.