Lemma 76.57.2. Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. There exists a t \geq 0 and closed subspaces
X \supset Z_0 \supset Z_1 \supset \ldots \supset Z_ t = \emptyset
such that Z_ i \to X is of finite presentation, Z_0 \subset X is a thickening, and for each i = 0, \ldots t - 1 there exists a (Z_ i \setminus Z_{i - 1})-admissible blowing up Z'_ i \to Z_ i such that Z'_ i has the resolution property.
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
Z_1 \supset Z_{1, 0} \supset Z_{1, 1} \supset \ldots \supset Z_{1, t} = \emptyset
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
Comments (0)