The Stacks project

Lemma 33.6.9. Let $k$ be a field. Let $X$ be a scheme over $k$. Let $x \in X$. Assume $X$ locally Noetherian and geometrically reduced at $x$. Then there exists an open neighbourhood $U \subset X$ of $x$ which is geometrically reduced over $k$.

Proof. Assume $X$ locally Noetherian and geometrically reduced at $x$. By Properties, Lemma 28.29.8 we can find an affine open neighbourhood $U \subset X$ of $x$ such that $R = \mathcal{O}_ X(U) \to \mathcal{O}_{X, x}$ is injective. By Lemma 33.6.2 the assumption means that $\mathcal{O}_{X, x}$ is geometrically reduced over $k$. By Algebra, Lemma 10.43.2 this implies that $R$ is geometrically reduced over $k$, which in turn implies that $U$ is geometrically reduced. $\square$

Comments (1)

Comment #766 by Keenan Kidwell on

The equality should be .

There are also:

  • 2 comment(s) on Section 33.6: Geometrically reduced schemes

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Your email address will not be published.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 0360. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 33.6.9 as pdf