Proposition 9

Let $A$ be a ring and $M$ a finitely generated $A$-module.

a) For every $p \in \mathrm{Supp}(M)$, one has $\dim_{A_p}(M_p) = \mathrm{codim}(V(p), \mathrm{Supp}(M))$.

b) $\dim_A(M)$ is the least upper bound of the $\dim_{A_p}(M_p)$, where $p$ ranges over $\mathrm{Spec}(A)$ (resp. where $p$ ranges over the set of maximal ideals of $A$ belonging to $\mathrm{Supp}(M)$).

c) Let $M'$ be a finitely generated submodule of $M$; then

a) Let $a$ be the annihilator of $M$; then the annihilator of the $A_p$-module $M_p$ is $aA_p$ (II, § 2, n° 4, formule (9)), whence $\dim_{A_p}(M_p) = \dim(A_p/aA_p)$. One concludes by prop. 7, b) of n° 3.

b) This follows at once from a) and from the fact that $\dim_{A_p}(M_p) = -\infty$ if $p$ does not belong to $\mathrm{Supp}(M)$.

c) One has $\mathrm{Supp}(M) = \mathrm{Supp}(M') \cup \mathrm{Supp}(M/M')$ (II, § 4, n° 4, prop. 16), and one applies prop. 1 of n° 1.