The Stacks project

Lemma 37.45.4. 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 ├ętale 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 37.45.2 to find an affine elementary ├ętale 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 37.19.2. 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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