The Stacks project

Lemma 89.2.4. If $(A, \mathfrak m, \kappa )$ is a complete Noetherian local domain of dimension $2$, then every modification of $\mathop{\mathrm{Spec}}(A)$ is projective over $A$.

Proof. By More on Morphisms of Spaces, Lemma 76.43.6 it suffices to show that the special fibre of any modification $X$ of $\mathop{\mathrm{Spec}}(A)$ has dimension $\leq 1$. This follows from Lemma 89.2.3. $\square$

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