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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