Lemma 31.13.5. Let $S$ be a scheme. Let $D \subset S$ be an effective Cartier divisor. Let $s \in D$. If $\dim _ s(S) < \infty$, then $\dim _ s(D) < \dim _ s(S)$.

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. $\square$

Comment #736 by Keenan Kidwell on

The $X$ in the second line of the proof should be $s$.

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