The Stacks project

Proof. Part (1) follows from the definition of a prime divisor (Divisors, Definition 31.26.2) and the definition of a dimension function (Topology, Definition 5.20.1). Let $\xi \in Z$ be the generic point of an irreducible component $Z$ of an effective Cartier divisor $D \subset X$. Then $\dim (\mathcal{O}_{D, \xi }) = 0$ and $\mathcal{O}_{D, \xi } = \mathcal{O}_{X, \xi }/(f)$ for some nonzerodivisor $f \in \mathcal{O}_{X, \xi }$ (Divisors, Lemma 31.15.2). Then $\dim (\mathcal{O}_{X, \xi }) = 1$ by Algebra, Lemma 10.60.13. Hence $Z$ is as in (1) by Properties, Lemma 28.10.3 and the proof is complete. $\square$

Proof. This lemma is a special case of Varieties, Lemma 33.17.2. Here is a direct argument in this case. By Cohomology of Schemes, Lemma 30.21.2 it suffices to prove that $f^{-1}(\{ \xi \} )$ is finite. We replace $Y$ by an affine open, say $Y = \mathop{\mathrm{Spec}}(R)$. Note that $R$ is Noetherian, as $Y$ is assumed locally Noetherian. Since $f$ is proper it is quasi-compact. Hence we can find a finite affine open covering $X = U_1 \cup \ldots \cup U_ n$ with each $U_ i = \mathop{\mathrm{Spec}}(A_ i)$. Note that $R \to A_ i$ is a finite type injective homomorphism of domains such that the induced extension of fraction fields is finite. Thus the lemma follows from Algebra, Lemma 10.113.2. $\square$