Proposition 13

Let A be a noetherian ring, M a finitely generated A-module, p and q two prime ideals of Supp(M) with p ⊂ q. One has

More precisely, for every saturated chain of prime ideals p ⊂ p_1 ⊂ ... ⊂ p_{r−1} ⊂ q, one has \text{prof}_{A_q}(M_q) \leq \text{prof}_{A_p}(M_p) + r.

Put p = \text{prof}_{A_p}(M_p), and let us prove the second inequality. It is obvious if p = +∞; in the contrary case one has \text{Ext}^p_{A_p}(κ(p), M_p) ≠ 0, whence

$Ext_{A_q}^{p+r}(\kappa(q), M_q) \neq 0$ by lemma 3, which implies $prof_{A_q}(M_q) \leq p + r$. Since $\dim(A_q/pA_q)$ is the least upper bound of the lengths of saturated chains of prime ideals with endpoints $p$ and $q$, the first assertion is a consequence of the second.