Lemma 32.13.1.slogan Assumptions and notation as in Situation 32.8.1. If
f is proper, and
f_0 is locally of finite type,
then there exists an i such that f_ i is proper.
Here is a first application of Chow's lemma.
Lemma 32.13.1.slogan Assumptions and notation as in Situation 32.8.1. If
f is proper, and
f_0 is locally of finite type,
then there exists an i such that f_ i is proper.
Proof. By Lemma 32.8.6 we see that f_ i is separated for some i \geq 0. Replacing 0 by i we may assume that f_0 is separated. Observe that f_0 is quasi-compact, see Schemes, Lemma 26.21.14. By Lemma 32.12.1 we can choose a diagram
where X_0' \to \mathbf{P}^ n_{Y_0} is an immersion, and \pi : X_0' \to X_0 is proper and surjective. Introduce X' = X_0' \times _{Y_0} Y and X_ i' = X_0' \times _{Y_0} Y_ i. By Morphisms, Lemmas 29.41.4 and 29.41.5 we see that X' \to Y is proper. Hence X' \to \mathbf{P}^ n_ Y is a closed immersion (Morphisms, Lemma 29.41.7). By Morphisms, Lemma 29.41.9 it suffices to prove that X'_ i \to Y_ i is proper for some i. By Lemma 32.8.5 we find that X'_ i \to \mathbf{P}^ n_{Y_ i} is a closed immersion for i large enough. Then X'_ i \to Y_ i is proper and we win. \square
Lemma 32.13.2. Let f : X \to S be a proper morphism with S quasi-compact and quasi-separated. Then X = \mathop{\mathrm{lim}}\nolimits X_ i is a directed limit of schemes X_ i proper and of finite presentation over S such that all transition morphisms and the morphisms X \to X_ i are closed immersions.
Proof. By Proposition 32.9.6 we can find a closed immersion X \to Y with Y separated and of finite presentation over S. By Lemma 32.12.1 we can find a diagram
where Y' \to \mathbf{P}^ n_ S is an immersion, and \pi : Y' \to Y is proper and surjective. By Lemma 32.9.4 we can write X = \mathop{\mathrm{lim}}\nolimits X_ i with X_ i \to Y a closed immersion of finite presentation. Denote X'_ i \subset Y', resp. X' \subset Y' the scheme theoretic inverse image of X_ i \subset Y, resp. X \subset Y. Then \mathop{\mathrm{lim}}\nolimits X'_ i = X'. Since X' \to S is proper (Morphisms, Lemmas 29.41.4), we see that X' \to \mathbf{P}^ n_ S is a closed immersion (Morphisms, Lemma 29.41.7). Hence for i large enough we find that X'_ i \to \mathbf{P}^ n_ S is a closed immersion by Lemma 32.4.20. Thus X'_ i is proper over S. For such i the morphism X_ i \to S is proper by Morphisms, Lemma 29.41.9. \square
Lemma 32.13.3. Let f : X \to S be a proper morphism with S quasi-compact and quasi-separated. Then there exists a directed set I, an inverse system (f_ i : X_ i \to S_ i) of morphisms of schemes over I, such that the transition morphisms X_ i \to X_{i'} and S_ i \to S_{i'} are affine, such that f_ i is proper, such that S_ i is of finite type over \mathbf{Z}, and such that (X \to S) = \mathop{\mathrm{lim}}\nolimits (X_ i \to S_ i).
Proof. By Lemma 32.13.2 we can write X = \mathop{\mathrm{lim}}\nolimits _{k \in K} X_ k with X_ k \to S proper and of finite presentation. Next, by absolute Noetherian approximation (Proposition 32.5.4) we can write S = \mathop{\mathrm{lim}}\nolimits _{j \in J} S_ j with S_ j of finite type over \mathbf{Z}. For each k there exists a j and a morphism X_{k, j} \to S_ j of finite presentation with X_ k \cong S \times _{S_ j} X_{k, j} as schemes over S, see Lemma 32.10.1. After increasing j we may assume X_{k, j} \to S_ j is proper, see Lemma 32.13.1. The set I will be consist of these pairs (k, j) and the corresponding morphism is X_{k, j} \to S_ j. For every k' \geq k we can find a j' \geq j and a morphism X_{j', k'} \to X_{j, k} over S_{j'} \to S_ j whose base change to S gives the morphism X_{k'} \to X_ k (follows again from Lemma 32.10.1). These morphisms form the transition morphisms of the system. Some details omitted. \square
Lemma 32.13.4. Let S be a scheme. Let X = \mathop{\mathrm{lim}}\nolimits X_ i be a directed limit of schemes over S with affine transition morphisms. Let Y \to X be a morphism of schemes over S. If Y \to X is proper, X_ i quasi-compact and quasi-separated, and Y locally of finite type over S, then Y \to X_ i is proper for i large enough.
Proof. Choose a closed immersion Y \to Y' with Y' proper and of finite presentation over X, see Lemma 32.13.2. Then choose an i and a proper morphism Y'_ i \to X_ i such that Y' = X \times _{X_ i} Y'_ i. This is possible by Lemmas 32.10.1 and 32.13.1. Then after replacing i by a larger index we have that Y \to Y'_ i is a closed immersion, see Lemma 32.4.16. \square
Recall the scheme theoretic support of a finite type quasi-coherent module, see Morphisms, Definition 29.5.5.
Lemma 32.13.5. Assumptions and notation as in Situation 32.8.1. Let \mathcal{F}_0 be a quasi-coherent \mathcal{O}_{X_0}-module. Denote \mathcal{F} and \mathcal{F}_ i the pullbacks of \mathcal{F}_0 to X and X_ i. Assume
f_0 is locally of finite type,
\mathcal{F}_0 is of finite type,
the scheme theoretic support of \mathcal{F} is proper over Y.
Then the scheme theoretic support of \mathcal{F}_ i is proper over Y_ i for some i.
Proof. We may replace X_0 by the scheme theoretic support of \mathcal{F}_0. By Morphisms, Lemma 29.5.3 this guarantees that X_ i is the support of \mathcal{F}_ i and X is the support of \mathcal{F}. Then, if Z \subset X denotes the scheme theoretic support of \mathcal{F}, we see that Z \to X is a universal homeomorphism. We conclude that X \to Y is proper as this is true for Z \to Y by assumption, see Morphisms, Lemma 29.41.9. By Lemma 32.13.1 we see that X_ i \to Y is proper for some i. Then it follows that the scheme theoretic support Z_ i of \mathcal{F}_ i is proper over Y by Morphisms, Lemmas 29.41.6 and 29.41.4. \square
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