Loading web-font TeX/Math/Italic

The Stacks project

Lemma 15.116.6. Let A \subset B be an extension of discrete valuation rings. Assume

  1. the extension L/K of fraction fields is separable,

  2. B is Nagata, and

  3. there exists a solution for A \subset B.

Then there exists a separable solution for A \subset B.

Proof. The lemma is trivial if the characteristic of K is zero; thus we may and do assume that the characteristic of K is p > 0.

Let K_2/K be a solution for A \to B. We will use induction on the inseparable degree [K_2 : K]_ i (Fields, Definition 9.14.7) of K_2/K. If [K_2 : K]_ i = 1, then K_2 is separable over K and we are done. If not, then there exists a subfield K_2/K_1/K such that K_2/K_1 is purely inseparable of degree p (Fields, Lemmas 9.14.6 and 9.14.5). By Lemma 15.116.5 there exists a separable extension K_3/K_1 which is a solution for A \subset B. Then [K_3 : K]_ i = [K_1 : K]_ i = [K_2 : K]_ i/p (Fields, Lemma 9.14.9) is smaller and we conclude by induction. \square


Comments (0)


Your email address will not be published.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.