The Stacks project

Theorem 41.12.1. Let $f : A \to B$ be an étale homomorphism of local rings. Then there exist $f, g \in A[t]$ such that

  1. $B' = A[t]_ g/(f)$ is standard étale – see (a) and (b) above, and

  2. $B$ is isomorphic to a localization of $B'$ at a prime.

Proof. Write $B = B'_{\mathfrak q}$ for some finite type $A$-algebra $B'$ (we can do this because $B$ is essentially of finite type over $A$). By Lemma 41.11.2 we see that $A \to B'$ is étale at $\mathfrak q$. Hence we may apply Algebra, Proposition 10.144.4 to see that a principal localization of $B'$ is standard étale. $\square$

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Your email address will not be published.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