The Stacks project

Proof. Assume $\dim _ s(S) < \infty $. Let $U = \mathop{\mathrm{Spec}}(A) \subset S$ be an affine open neighbourhood of $s$ such that $\dim (U) = \dim _ s(S)$ and such that $D = V(f)$ for some nonzerodivisor $f \in A$ (see Lemma 31.13.2). Recall that $\dim (U)$ is the Krull dimension of the ring $A$ and that $\dim (U \cap D)$ is the Krull dimension of the ring $A/(f)$. Then $f$ is not contained in any minimal prime of $A$. Hence any maximal chain of primes in $A/(f)$, viewed as a chain of primes in $A$, can be extended by adding a minimal prime. $\square$