Lemma 2

Let $A$ be a noetherian local ring and $q$ an ideal of $A$. The following conditions are equivalent:

(i) $q$ is an ideal of definition of $A$; (ii) there exists an integer $n \geqslant 0$ such that $m_A^n \subset q \subset m_A$;

(iii) one has q ≠ A and A/q is an A-module of finite length ; (iv) V(q) is equal to { m_A } (in other words, m_A is the only prime ideal of A containing q).

In fact, the equivalence of (i), (ii) and (iv) was proved in III, § 2, No. 5, and the equivalence of (i) and (iii) follows from IV, § 2, No. 5, Corollary 2 to Proposition 9.

The corollary to Theorem 1 therefore makes it possible to state the following scholium: