## 42.13 Preparation for flat pullback

Recall that a morphism $f : X \to Y$ which is locally of finite type is said to have relative dimension $r$ if every nonempty fibre is equidimensional of dimension $r$. See Morphisms, Definition 29.29.1.

Lemma 42.13.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \to Y$ be a morphism. Assume $f$ is flat of relative dimension $r$. For any closed subset $Z \subset Y$ we have

$\dim _\delta (f^{-1}(Z)) = \dim _\delta (Z) + r.$

provided $f^{-1}(Z)$ is nonempty. If $Z$ is irreducible and $Z' \subset f^{-1}(Z)$ is an irreducible component, then $Z'$ dominates $Z$ and $\dim _\delta (Z') = \dim _\delta (Z) + r$.

Proof. It suffices to prove the final statement. We may replace $Y$ by the integral closed subscheme $Z$ and $X$ by the scheme theoretic inverse image $f^{-1}(Z) = Z \times _ Y X$. Hence we may assume $Z = Y$ is integral and $f$ is a flat morphism of relative dimension $r$. Since $Y$ is locally Noetherian the morphism $f$ which is locally of finite type, is actually locally of finite presentation. Hence Morphisms, Lemma 29.25.10 applies and we see that $f$ is open. Let $\xi \in X$ be a generic point of an irreducible component of $X$. By the openness of $f$ we see that $f(\xi )$ is the generic point $\eta$ of $Z = Y$. Note that $\dim _\xi (X_\eta ) = r$ by assumption that $f$ has relative dimension $r$. On the other hand, since $\xi$ is a generic point of $X$ we see that $\mathcal{O}_{X, \xi } = \mathcal{O}_{X_\eta , \xi }$ has only one prime ideal and hence has dimension $0$. Thus by Morphisms, Lemma 29.28.1 we conclude that the transcendence degree of $\kappa (\xi )$ over $\kappa (\eta )$ is $r$. In other words, $\delta (\xi ) = \delta (\eta ) + r$ as desired. $\square$

Here is the lemma that we will use to prove that the flat pullback of a locally finite collection of closed subschemes is locally finite.

Lemma 42.13.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \to Y$ be a morphism. Assume $\{ Z_ i\} _{i \in I}$ is a locally finite collection of closed subsets of $Y$. Then $\{ f^{-1}(Z_ i)\} _{i \in I}$ is a locally finite collection of closed subsets of $X$.

Proof. Let $U \subset X$ be a quasi-compact open subset. Since the image $f(U) \subset Y$ is a quasi-compact subset there exists a quasi-compact open $V \subset Y$ such that $f(U) \subset V$. Note that

$\{ i \in I \mid f^{-1}(Z_ i) \cap U \not= \emptyset \} \subset \{ i \in I \mid Z_ i \cap V \not= \emptyset \} .$

Since the right hand side is finite by assumption we win. $\square$

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