Proposition 8

Let $A$ be a noetherian ring, $M$ a finitely generated $A$-module. Then $\dim_A(M) \leq \operatorname{di}_A(M)$.

Let $r$ be a positive integer less than $\dim_A(M)$. There exists a saturated chain of prime ideals $p \subset p_1 \subset \ldots \subset p_{r-1} \subset q$ such that $p$ is a minimal element of the support of $M$; then the $A_p$-module $M_p$ is of finite length, so that $\operatorname{Hom}_{A_p}(\kappa(p), M_p) \neq 0$, hence $\operatorname{Ext}^r_{A_q}(\kappa(q), M_q) \neq 0$ (§ 1, No. 7, Lemma 3), which implies $\operatorname{di}_A(M) \geq r$ (No. 3, Proposition 6).