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. $\square$

Proof. In this paragraph we use absolute Noetherian approximation to reduce to the case of algebraic spaces of finite presentation over $\mathop{\mathrm{Spec}}(\mathbf{Z})$. We may view $X$ as an algebraic space over $\mathop{\mathrm{Spec}}(\mathbf{Z})$, see Spaces, Definition 65.16.2 and Properties of Spaces, Definition 66.3.1. Thus we may apply Limits of Spaces, Proposition 70.8.1. It follows that we can find an affine morphism $X \to X_0$ with $X_0$ of finite presentation over $\mathbf{Z}$. If we can prove the lemma for $X_0$, then we can pull back the stratification and the centers of the blowing ups to $X$ and get the result for $X$; this uses that the resolution property goes up along affine morphisms (Derived Categories of Spaces, Lemma 75.28.3) and that the strict transform of an affine morphism is affine – details omitted. This reduces us to the case discussed in the next paragraph.

Assume $X$ is of finite presentation over $\mathbf{Z}$. Then $X$ is Noetherian and $|X|$ is a Noetherian topological space (with finitely many irreducible components) of finite dimension. Hence we may use induction on $\dim (|X|)$. By Lemma 76.57.1 there exists a dense open $U \subset X$ and a $U$-admissible blowing up $X' \to X$ such that $X'$ has the resolution property. Set $Z_0 = X$ and let $Z_1 \subset X$ be the reduced closed subspace with $|Z_1| = |X| \setminus |U|$. By induction we find an integer $t \geq 0$ and a filtration

by closed subspaces, where $Z_{1, 0} \to Z_1$ is a thickening and there exist $(Z_{1, i} \setminus Z_{1, i + 1})$-admissible blowing ups $Z'_{1, i} \to Z_{1, i}$ such that $Z'_{1, i}$ has the resolution property. Since $Z_1$ is reduced, we have $Z_1 = Z_{1, 0}$. Hence we can set $Z_ i = Z_{1, i - 1}$ and $Z'_ i = Z'_{1, i - 1}$ for $i \geq 1$ and the lemma is proved. $\square$