Lemma 42.16.2. Let $f : X \to Y$ be a morphism of schemes. Let $\xi \in Y$ be a point. Assume that

1. $X$, $Y$ are integral,

2. $Y$ is locally Noetherian

3. $f$ is proper, dominant and $R(Y) \subset R(X)$ is finite, and

4. $\dim (\mathcal{O}_{Y, \xi }) = 1$.

Then there exists an open neighbourhood $V \subset Y$ of $\xi$ such that $f|_{f^{-1}(V)} : f^{-1}(V) \to V$ is finite.

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. $\square$

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