Lemma 89.2.1. Let $(A, \mathfrak m, \kappa )$ be a $2$-dimensional Noetherian local domain such that $U = \mathop{\mathrm{Spec}}(A) \setminus \{ \mathfrak m\} $ is a normal scheme. Then any modification $f : X \to \mathop{\mathrm{Spec}}(A)$ is a morphism as in (89.2.0.1).
Proof. Let $f : X \to S$ be a modification. We have to show that $f^{-1}(U) \to U$ is an isomorphism. Since every closed point $u$ of $U$ has codimension $1$, this follows from Spaces over Fields, Lemma 72.3.3. $\square$
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