The Stacks project

Lemma 33.10.3. Let $k$ be a field. Let $X$ be a scheme over $k$. The following are equivalent

  1. $X$ is geometrically normal,

  2. $X_{k'}$ is a normal scheme for every field extension $k'/k$,

  3. $X_{k'}$ is a normal scheme for every finitely generated field extension $k'/k$,

  4. $X_{k'}$ is a normal scheme for every finite purely inseparable field extension $k'/k$,

  5. for every affine open $U \subset X$ the ring $\mathcal{O}_ X(U)$ is geometrically normal (see Algebra, Definition 10.165.2), and

  6. $X_{k^{perf}}$ is a normal scheme.

Proof. Assume (1). Then for every field extension $k'/k$ and every point $x' \in X_{k'}$ the local ring of $X_{k'}$ at $x'$ is normal. By definition this means that $X_{k'}$ is normal. Hence (2).

It is clear that (2) implies (3) implies (4).

Assume (4) and let $U \subset X$ be an affine open subscheme. Then $U_{k'}$ is a normal scheme for any finite purely inseparable extension $k'/k$ (including $k = k'$). This means that $k' \otimes _ k \mathcal{O}(U)$ is a normal ring for all finite purely inseparable extensions $k'/k$. Hence $\mathcal{O}(U)$ is a geometrically normal $k$-algebra by definition. Hence (4) implies (5).

Assume (5). For any field extension $k'/k$ the base change $X_{k'}$ is gotten by gluing the spectra of the rings $\mathcal{O}_ X(U) \otimes _ k k'$ where $U$ is affine open in $X$ (see Schemes, Section 26.17). Hence $X_{k'}$ is normal. So (1) holds.

The equivalence of (5) and (6) follows from the definition of geometrically normal algebras and the equivalence (just proved) of (3) and (4). $\square$

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Your email address will not be published.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 038O. Beware of the difference between the letter 'O' and the digit '0'.