In this section we prove a number of variants of Chow's lemma dealing with morphisms between non-Noetherian algebraic spaces. The Noetherian versions are Lemma 76.40.3 and Lemma 76.40.5.
Lemma 76.41.1. Let S be a scheme. Let Y be a quasi-compact and quasi-separated algebraic space over S. Let f : X \to Y be a separated morphism of finite type. Then there exists a commutative diagram
where X' \to X is proper surjective, X' \to \overline{X}' is an open immersion, and \overline{X}' \to Y is proper and representable morphism of algebraic spaces.
Proof. By Limits of Spaces, Proposition 70.11.7 we can find a closed immersion X \to X_1 where X_1 is separated and of finite presentation over Y. Clearly, if we prove the assertion for X_1 \to Y, then the result follows for X. Hence we may assume that X is of finite presentation over Y.
We may and do replace the base scheme S by \mathop{\mathrm{Spec}}(\mathbf{Z}). Write Y = \mathop{\mathrm{lim}}\nolimits _ i Y_ i as a directed limit of quasi-separated algebraic spaces of finite type over \mathop{\mathrm{Spec}}(\mathbf{Z}), see Limits of Spaces, Proposition 70.8.1. By Limits of Spaces, Lemma 70.7.1 we can find an index i \in I and a scheme X_ i \to Y_ i of finite presentation so that X = Y \times _{Y_ i} X_ i. By Limits of Spaces, Lemma 70.6.9 we may assume that X_ i \to Y_ i is separated. Clearly, if we prove the assertion for X_ i over Y_ i, then the assertion holds for X. The case X_ i \to Y_ i is treated by Lemma 76.40.3. \square
Lemma 76.41.2. Let S be a scheme. Let f : X \to Y be a morphism of algebraic spaces over S. Assume f separated of finite type, and Y separated and quasi-compact. Then there exists a commutative diagram
where X' \to X is proper surjective morphism and the morphism X' \to \mathbf{P}^ n_ Y is an immersion.
Proof. By Limits of Spaces, Proposition 70.11.7 we can find a closed immersion X \to X_1 where X_1 is separated and of finite presentation over Y. Clearly, if we prove the assertion for X_1 \to Y, then the result follows for X. Hence we may assume that X is of finite presentation over Y.
We may and do replace the base scheme S by \mathop{\mathrm{Spec}}(\mathbf{Z}). Write Y = \mathop{\mathrm{lim}}\nolimits _ i Y_ i as a directed limit of quasi-separated algebraic spaces of finite type over \mathop{\mathrm{Spec}}(\mathbf{Z}), see Limits of Spaces, Proposition 70.8.1. By Limits of Spaces, Lemma 70.5.9 we may assume that Y_ i is separated for all i. By Limits of Spaces, Lemma 70.7.1 we can find an index i \in I and a scheme X_ i \to Y_ i of finite presentation so that X = Y \times _{Y_ i} X_ i. By Limits of Spaces, Lemma 70.6.9 we may assume that X_ i \to Y_ i is separated. Clearly, if we prove the assertion for X_ i over Y_ i, then the assertion holds for X. The case X_ i \to Y_ i is treated by Lemma 76.40.5. \square
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