Lemma 37.35.6. Let $X \to S$ be a morphism of schemes. Let $x \in X$ with image $s \in S$. Let $(V, v) \to (X_ s, x)$ be an étale neighbourhood. Then there exists an étale neighbourhood $(U, u) \to (X, x)$ such that there exists a morphism $(U_ s, u) \to (V, v)$ of étale neighbourhoods of $(X_ s, x)$ which is an open immersion.

** Lift étale neighbourhood of point on fibre to total space. **

**Proof.**
We may assume $X$, $V$, and $S$ affine. Say the morphism $X \to S$ is given by $A \to B$ the point $x$ by a prime $\mathfrak q \subset B$, the point $s$ by $\mathfrak p = A \cap \mathfrak q$, and the morphism $V \to X_ s$ by $B \otimes _ A \kappa (\mathfrak p) \to C$. Since $\kappa (\mathfrak p)$ is a localization of $A/\mathfrak p$ there exists an $f \in A$, $f \not\in \mathfrak p$ and an étale ring map $B \otimes _ A (A/\mathfrak p)_ f \to D$ such that

See Algebra, Lemma 10.143.3 part (9). After replacing $A$ by $A_ f$ and $B$ by $B_ f$ we may assume $D$ is étale over $B \otimes _ A A/\mathfrak p = B/\mathfrak p B$. Then we can apply Algebra, Lemma 10.143.10. This proves the lemma. $\square$

## 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Your email address will not be published.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.

## Comments (0)

There are also: