Lemma 37.47.3. Assumption and notation as in Lemma 37.47.2. In addition to properties (1) – (6) we may also arrange it so that
$S'$, $Y'$, $X'$ are affine.
Lemma 37.47.3. Assumption and notation as in Lemma 37.47.2. In addition to properties (1) – (6) we may also arrange it so that
$S'$, $Y'$, $X'$ are affine.
Proof. Note that if $Y'$ is affine, then $X'$ is affine as $\pi $ is finite. Choose an affine open neighbourhood $U' \subset S'$ of $s'$. Choose an affine open neighbourhood $V' \subset h^{-1}(U')$ of $y'$. Let $W' = h(V')$. This is an open neighbourhood of $s'$ in $S'$, see Morphisms, Lemma 29.34.10, contained in $U'$. Choose an affine open neighbourhood $U'' \subset W'$ of $s'$. Then $h^{-1}(U'') \cap V'$ is affine because it is equal to $U'' \times _{U'} V'$. By construction $h^{-1}(U'') \cap V' \to U''$ is a surjective smooth morphism whose fibres are (nonempty) open subschemes of geometrically integral fibres of $Y' \to S'$, and hence geometrically integral. Thus we may replace $S'$ by $U''$ and $Y'$ by $h^{-1}(U'') \cap V'$. $\square$
Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)