Proposition 37.80.3. Let X be a quasi-compact and quasi-separated scheme. There exists a morphism f : Y \to X which is of finite presentation, proper, and completely decomposed (Definition 37.78.1) such that the scheme Y has an ample family of invertible modules.
Proof. By Limits, Proposition 32.5.4 there exists an affine morphism X \to X_0 where X_0 is a scheme of finite type over \mathbf{Z}. Below we produce a morphism Y_0 \to X_0 with all the desired properties. Then setting Y = X \times _{X_0} Y_0 and f equal to the projection f : Y \to X we conclude. To see this observe that f is of finite presentation (Morphisms, Lemma 29.21.4), f is proper (Morphisms, Lemma 29.41.5), f is completely decomposed (Lemma 37.78.3). Finally, since Y \to Y_0 is affine (as the base change of X \to X_0) we see that Y has an ample family of invertible modules by Lemma 37.79.2. This reduces us to the case discussed in the next paragraph.
Assume X is of finite type over \mathbf{Z}. In particular \dim (X) < \infty . We will argue by induction on \dim (X). If \dim (X) = 0, then X is affine and has the resolution property. In general, there exists a dense open U \subset X and a U-admissible blowing up X' \to X such that X' has an ample family of invertible modules, see Lemma 37.80.2. Since f : X' \to X is an isomorphism over U we see that every point of U lifts to a point of X' with the same residue field. Let Z = X \setminus U with the reduced induced scheme structure. Then \dim (Z) < \dim (X) as U is dense in X (see above). By induction we find a proper, completely decomposed morphism W \to Z such that W has an ample family of invertible modules. Then it follows that Y = X' \amalg W \to X is the desired morphism. \square
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