Corollary 1

For every $M$-regular sequence $(x_1, \ldots, x_r)$ of elements of $J$, the following properties are equivalent:

(i) one has $r = \operatorname{prof}_A(J; M)$ ; (ii) the sequence $(x_1, \ldots, x_r)$ is maximal among the $M$-regular sequences formed of elements of $J$ ; (iii) the $A$-module $M/(x_1M + \ldots + x_rM)$ has a nonzero element annihilated by $J$ ; (iv) one has $\operatorname{Ass}(M/(x_1M + \ldots + x_rM)) \cap V(J) \neq \varnothing$.

The equivalence of (i) and (ii) follows from Theorem 2; the equivalence of (ii), (iii) and (iv) follows from Remark 2 of No. 1 applied to the $A$-module $M/(x_1M + \ldots + x_rM)$.