Lemma 32.18.4. Notation and assumptions as in Situation 32.8.1. If

1. $f$ has relative dimension $d$, and

2. $f_0$ is locally of finite presentation,

then there exists an $i \geq 0$ such that $f_ i$ has relative dimension $d$.

Proof. By Lemma 32.18.1 we may assume all fibres of $f_0$ have dimension $\leq d$. By Morphisms, Lemma 29.28.6 the set $U_0 \subset X_0$ of points $x \in X_0$ such that the dimension of the fibre of $X_0 \to Y_0$ at $x$ is $\leq d - 1$ is open and retrocompact in $X_0$. Hence the complement $E = X_0 \setminus U_0$ is constructible. Moreover the image of $X \to X_0$ is contained in $E$ by Morphisms, Lemma 29.28.3. Thus for $i \gg 0$ we have that the image of $X_ i \to X_0$ is contained in $E$ (Lemma 32.4.10). Then all fibres of $X_ i \to Y_ i$ have dimension $d$ by the aforementioned Morphisms, Lemma 29.28.3. $\square$

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