Corollary

Let $A$ be a noetherian local ring and $M$ a nonzero finitely generated Macaulay $A$-module.

a) All maximal chains of irreducible closed subsets of $\mathrm{Supp}(M)$ have length equal to $\dim(M)$.

b) For every closed subset $X$ of $\mathrm{Supp}(M)$, one has

c) All irreducible components of $\mathrm{Supp}(M)$ have the same dimension.

d) For every ideal $J$ of $A$, one has

a) A maximal chain of irreducible closed subsets of $\mathrm{Supp}(M)$ has as smallest element $\{m_A\}$ and as greatest element an irreducible component $X$ of

Supp(M). Its length is equal to the codimension of $\{ m_A \}$ in X (Prop. 2, c)); by Prop. 2, b) applied to the closed subsets $\{ m_A \} \subset X$, this is equal to $\operatorname{codim}(\{ m_A \}, \operatorname{Supp}(M))$, that is, to $\dim(M)$.

b) This is a consequence of a) when the subset X is irreducible (VIII, § 1, No. 2, prop. 5); the general case follows from VIII, § 1, No. 1, prop. 1 and § 1, No. 2, remark 4.

c) This is a consequence of b).

d) One has $\operatorname{prof}_A(J; M) = \operatorname{codim}(\operatorname{Supp}(M) \cap V(J), \operatorname{Supp}(M))$ by the cor. to Prop. 1 of No. 1. It then suffices to apply b) with $X = \operatorname{Supp}(M) \cap V(J) = \operatorname{Supp}(M/JM)$ (II, § 4, No. 4, cor. to Prop. 18).