The Stacks project

Remark 53.19.9 (Trivial quadratic extension). Let $k$ be a field. Let $(A, \mathfrak m, \kappa )$ be as in Remark 53.19.8 and let $\kappa '/\kappa $ be the associated separable algebra of degree $2$. Then the following are equivalent

  1. $\kappa ' \cong \kappa \times \kappa $ as $\kappa $-algebra,

  2. the form $q$ of Lemma 53.19.3 can be chosen to be $xy$,

  3. $A$ has two branches,

  4. the extension $A'/A$ of Lemma 53.19.4 has two maximal ideals, and

  5. $A^\wedge \cong \kappa [[x, y]]/(xy)$ as a $k$-algebra.

The equivalence between these conditions has been shown in the proof of Lemma 53.19.4. If $X$ is a locally Noetherian $k$-scheme and $x \in X$ is a point such that $\mathcal{O}_{X, x} = A$, then this means exactly that there are two points $x_1, x_2$ of the normalization $X^\nu $ lying over $x$ and that $\kappa (x) = \kappa (x_1) = \kappa (x_2)$.

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Your email address will not be published.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