Corollary 1

Suppose the ideal J to be of finite type and JM $\neq$ M. Then $\operatorname{prof}_A(J; M)$ is finite and $\leq$ Card(I) ; in order that it be equal to Card(I), it is necessary and sufficient that the family x be completely secant for M (A, X, p. 157, Def. 2).

Suppose first that I is finite, and let r denote its cardinal. The A-module $H^r(x, M)$ is canonically isomorphic to $H_0(x, M)$, itself isomorphic to $M/JM$ (A, X, p. 155); the inequality $\operatorname{prof}_A(J; M) \leq r$ therefore follows from Theorem 1. For equality to hold, it is necessary and sufficient that the A-module $H^i(x, M)$ be zero for $i < r$, which signifies that the family x is completely secant for M (A, X, p. 157).

From the foregoing, $\operatorname{prof}_A(J; M)$ is finite; it remains to prove that if the family x is completely secant for M, the set I is finite. Now the condition $H_1(x, M) = 0$ (A, X, p. 157, Def. 2) implies that one has an exact sequence

where the image of $\partial_2$ is contained in $JM^{(I)}$. By tensor product with $A/J$, one deduces from this an A-linear isomorphism of $(M/JM)^{(I)}$ onto $JM/J^2M$. Now the latter module is of finite type, since J and M are so; as $M/JM$ is not zero, it follows that the set I is finite.