Lemma 37.46.4. Let $h : Y \to S$ be a morphism of schemes. Let $s \in S$ be a point. Let $T \subset Y_ s$ be an open subscheme. Assume

1. $h$ is of finite presentation,

2. $h$ is normal, and

3. $T$ is geometrically irreducible over $\kappa (s)$.

Then we can find an affine elementary étale neighbourhood $(S', s') \to (S, s)$ and a quasi-compact open $V \subset Y_{S'}$ such that

1. all fibres of $V \to S'$ are geometrically integral,

2. $V_{s'} = T \times _ s s'$.

Proof. Apply Lemma 37.46.2 to find an affine elementary étale neighbourhood $(S', s') \to (S, s)$ and a quasi-compact open $V \subset Y_{S'}$ such that all fibres of $V \to S'$ are geometrically connected and $V_{s'} = T \times _ s s'$. As $V$ is an open of the base change of $h$ all fibres of $V \to S'$ are geometrically normal, see Lemma 37.20.2. In particular, they are geometrically reduced. To finish the proof we have to show they are geometrically irreducible. But, if $t \in S'$ then $V_ t$ is of finite type over $\kappa (t)$ and hence $V_ t \times _{\kappa (t)} \overline{\kappa (t)}$ is of finite type over $\overline{\kappa (t)}$ hence Noetherian. By choice of $S' \to S$ the scheme $V_ t \times _{\kappa (t)} \overline{\kappa (t)}$ is connected. Hence $V_ t \times _{\kappa (t)} \overline{\kappa (t)}$ is irreducible by Properties, Lemma 28.7.6 and we win. $\square$

## 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Your email address will not be published.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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 057J. Beware of the difference between the letter 'O' and the digit '0'.