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
$h$ is of finite presentation,
$h$ is normal, and
$T$ 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
all fibres of $V \to S'$ are geometrically integral,
$V_{s'} = T \times _ s s'$.
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. $\square$
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