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The Stacks project

Lemma 32.18.5. Let S be a quasi-compact and quasi-separated scheme. Let f : X \to S be a morphism of finite presentation. Let d \geq 0 be an integer. If Z \subset X be a closed subscheme such that \dim (Z_ s) \leq d for all s \in S, then there exists a closed subscheme Z' \subset X such that

  1. Z \subset Z',

  2. Z' \to X is of finite presentation, and

  3. \dim (Z'_ s) \leq d for all s \in S.

Proof. By Proposition 32.5.4 we can write S = \mathop{\mathrm{lim}}\nolimits S_ i as the limit of a directed inverse system of Noetherian schemes with affine transition maps. By Lemma 32.10.1 we may assume that there exist a system of morphisms f_ i : X_ i \to S_ i of finite presentation such that X_{i'} = X_ i \times _{S_ i} S_{i'} for all i' \geq i and such that X = X_ i \times _{S_ i} S. Let Z_ i \subset X_ i be the scheme theoretic image of Z \to X \to X_ i. Then for i' \geq i the morphism X_{i'} \to X_ i maps Z_{i'} into Z_ i and the induced morphism Z_{i'} \to Z_ i \times _{S_ i} S_{i'} is a closed immersion. By Lemma 32.18.1 we see that the dimension of the fibres of Z_ i \to S_ i all have dimension \leq d for a suitable i \in I. Fix such an i and set Z' = Z_ i \times _{S_ i} S \subset X. Since S_ i is Noetherian, we see that X_ i is Noetherian, and hence the morphism Z_ i \to X_ i is of finite presentation. Therefore also the base change Z' \to X is of finite presentation. Moreover, the fibres of Z' \to S are base changes of the fibres of Z_ i \to S_ i and hence have dimension \leq d. \square


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