Corollary 2

Suppose the ideal J to be of finite type, and let J' be an ideal of A such that V(J) \supset V(J').

a) One has prof_A(J ; M) \leq prof_A(J' ; M).

b) If the ideal J' is of finite type and if V(J) = V(J'), one has prof_A(J ; M) = prof_A(J' ; M).

By II, § 4, No. 3, cor. 2 of prop. 11 and § 2, No. 6, prop. 15, there exists an integer m > 0 such that J^m \subset J'. The assertion a) therefore follows from cor. 1 and assertion b) is deduced from it.

Cor. 2 may fail when the ideal J is not of finite type (exercise 2).