Remark 1

Suppose that A is a local noetherian integral domain of dimension d; let x be a nonzero element of m_A and let H = V(x). By Corollary 2 of Prop. 4, every irreducible component of H is of codimension 1 in X, hence of dimension $\leq d - 1$ ($§ 1$, No. 2, Prop. 3). By Corollary 2 of Prop. 2 of No. 1, H is of dimension $d - 1$, and one of these components is therefore of dimension $d - 1$; all are so if A is catenary. However, in general it may happen that there exists an irreducible component of H of dimension < $d - 1$ (cf. p. 87, Exerc. 7).