The Stacks project

Theorem 41.12.3. Let $\varphi : X \to Y$ be a morphism of schemes. Let $x \in X$. Let $V \subset Y$ be an affine open neighbourhood of $\varphi (x)$. If $\varphi $ is étale at $x$, then there exist exists an affine open $U \subset X$ with $x \in U$ and $\varphi (U) \subset V$ such that we have the following diagram

\[ \xymatrix{ X \ar[d] & U \ar[l] \ar[d] \ar[r]_-j & \mathop{\mathrm{Spec}}(R[t]_{f'}/(f)) \ar[d] \\ Y & V \ar[l] \ar@{=}[r] & \mathop{\mathrm{Spec}}(R) } \]

where $j$ is an open immersion, and $f \in R[t]$ is monic.

Proof. This is equivalent to Morphisms, Lemma 29.36.14 although the statements differ slightly. See also, Varieties, Lemma 33.18.3 for a variant for unramified morphisms. $\square$

Comments (0)

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Your email address will not be published.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 025C. Beware of the difference between the letter 'O' and the digit '0'.