The Stacks project

Lemma 35.33.5. Consider a commutative diagram of morphisms of schemes

\[ \xymatrix{ U' \ar[r] \ar[d] & V' \ar[d] \\ U \ar[r] & V } \]

with étale vertical arrows and a point $v' \in V'$ mapping to $v \in V$. Then the morphism of fibres $U'_{v'} \to U_ v$ is étale.

Proof. Note that $U'_ v \to U_ v$ is étale as a base change of the étale morphism $U' \to U$. The scheme $U'_ v$ is a scheme over $V'_ v$. By Morphisms, Lemma 29.36.7 the scheme $V'_ v$ is a disjoint union of spectra of finite separable field extensions of $\kappa (v)$. One of these is $v' = \mathop{\mathrm{Spec}}(\kappa (v'))$. Hence $U'_{v'}$ is an open and closed subscheme of $U'_ v$ and it follows that $U'_{v'} \to U'_ v \to U_ v$ is étale (as a composition of an open immersion and an étale morphism, see Morphisms, Section 29.36). $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 35.33: Properties of morphisms of germs local on source-and-target

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Your email address will not be published.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 04NI. Beware of the difference between the letter 'O' and the digit '0'.