Lemma 37.20.5. The property $\mathcal{P}(f)=$“the fibres of $f$ are locally Noetherian and $f$ is normal” is local in the fppf topology on the target and local in the smooth topology on the source.

**Proof.**
We have $\mathcal{P}(f) = \mathcal{P}_1(f) \wedge \mathcal{P}_2(f) \wedge \mathcal{P}_3(f)$ where $\mathcal{P}_1(f)=$“the fibres of $f$ are locally Noetherian”, $\mathcal{P}_2(f)=$“$f$ is flat”, and $\mathcal{P}_3(f)=$“the fibres of $f$ are geometrically normal”. We have already seen that $\mathcal{P}_1$ and $\mathcal{P}_2$ are local in the fppf topology on the source and the target, see Lemma 37.20.4, and Descent, Lemmas 35.23.15 and 35.27.1. Thus we have to deal with $\mathcal{P}_3$.

Let $f : X \to Y$ be a morphism of schemes. Let $\{ \varphi _ i : Y_ i \to Y\} _{i \in I}$ be an fpqc covering of $Y$. Denote $f_ i : X_ i \to Y_ i$ the base change of $f$ by $\varphi _ i$. Let $i \in I$ and let $y_ i \in Y_ i$ be a point. Set $y = \varphi _ i(y_ i)$. Note that

Hence in this situation we have that $X_ y$ is geometrically normal if and only if $X_{i, y_ i}$ is geometrically normal, see Varieties, Lemma 33.10.4. This fact implies $\mathcal{P}_3$ is fpqc local on the target.

Let $\{ X_ i \to X\} $ be a smooth covering of $X$. Let $y \in Y$. In this case $\{ X_{i, y} \to X_ y\} $ is a smooth covering of the fibre. Hence the locality of $\mathcal{P}_3$ for the smooth topology on the source follows from Descent, Lemma 35.18.2. Combining the above the lemma follows. $\square$

## Post a comment

Your email address will not be published.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)