Lemma 37.18.4. The property $\mathcal{P}(f)=$“the fibres of $f$ are locally Noetherian” is local in the fppf topology on the source and the target.

**Proof.**
Let $f : X \to Y$ be a morphism of schemes. Let $\{ \varphi _ i : Y_ i \to Y\} _{i \in I}$ be an fppf 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

Moreover, as $\varphi _ i$ is of finite presentation the field extension $\kappa (y) \subset \kappa (y_ i)$ is finitely generated. Hence in this situation we have that $X_ y$ is locally Noetherian if and only if $X_{i, y_ i}$ is locally Noetherian, see Varieties, Lemma 33.11.1. This fact implies locality on the target.

Let $\{ X_ i \to X\} $ be an fppf covering of $X$. Let $y \in Y$. In this case $\{ X_{i, y} \to X_ y\} $ is an fppf covering of the fibre. Hence the locality on the source follows from Descent, Lemma 35.13.1. $\square$

## 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Your email address will not be published.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)