Lemma 32.18.3. Notation and assumptions as in Situation 32.8.1. If

$f$ has relative dimension $d$, and

$f_0$ is locally of finite presentation,

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

Lemma 32.18.3. Notation and assumptions as in Situation 32.8.1. If

$f$ has relative dimension $d$, and

$f_0$ 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.
$\square$

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)