The Stacks project

Proof. By Limits of Spaces, Lemma 70.16.3 there exists a surjective, finite, and finitely presented morphism $f : Y \to X$ where $Y$ is a scheme and a quasi-compact dense open $U \subset X$ such that $f^{-1}(U) \to U$ is finite étale. By More on Morphisms, Lemma 37.80.2 there is a quasi-compact dense open $V \subset Y$ and a $V$-admissible blowing up $Y' \to Y$ such that $Y'$ has an ample family of invertible modules. After shrinking $U$ we may assume that $f^{-1}(U) \subset V$ (details omitted). Hence $f' : Y' \to X$ is finite étale over $U$ and in particular, the morphism $(f')^{-1}(U) \to U$ is finite locally free. By Lemma 76.39.2 there is a $U$-admissible blowing up $X' \to X$ such that the strict transform $Y''$ of $Y'$ is finite locally free over $X'$. Picture

Since $g : Y'' \to Y'$ is a blowing up (Divisors on Spaces, Lemma 71.18.3) in the inverse image of the center of $X' \to X$, we see that $g : Y'' \to Y'$ is projective and that there exists some $g$-ample invertible module on $Y''$. Hence by More on Morphisms, Lemma 37.79.1 we see that $Y''$ has an ample family of invertible modules. Hence $Y''$ has the resolution property, see Derived Categories of Schemes, Lemma 36.36.7. We conclude that $X'$ has the resolution property by Derived Categories of Spaces, Lemma 75.28.4. $\square$