The Stacks project

Lemma 10.44.4. Let $k$ be a field. Let $S$ be a $k$-algebra. The following are equivalent:

  1. $k' \otimes _ k S$ is reduced for every finite purely inseparable extension $k'$ of $k$,

  2. $k^{1/p} \otimes _ k S$ is reduced,

  3. $k^{perf} \otimes _ k S$ is reduced, where $k^{perf}$ is the perfect closure of $k$,

  4. $\overline{k} \otimes _ k S$ is reduced, where $\overline{k}$ is the algebraic closure of $k$, and

  5. $S$ is geometrically reduced over $k$.

Proof. Note that any finite purely inseparable extension $k'/k$ embeds in $k^{perf}$. Moreover, $k^{1/p}$ embeds into $k^{perf}$ which embeds into $\overline{k}$. Thus it is clear that (5) $\Rightarrow $ (4) $\Rightarrow $ (3) $\Rightarrow $ (2) and that (3) $\Rightarrow $ (1).

We prove that (1) $\Rightarrow $ (5). Assume $k' \otimes _ k S$ is reduced for every finite purely inseparable extension $k'$ of $k$. Let $K/k$ be an extension of fields. We have to show that $K \otimes _ k S$ is reduced. By Lemma 10.43.4 we reduce to the case where $K/k$ is a finitely generated field extension. Choose a diagram

\[ \xymatrix{ K \ar[r] & K' \\ k \ar[u] \ar[r] & k' \ar[u] } \]

as in Lemma 10.42.4. By assumption $k' \otimes _ k S$ is reduced. By Lemma 10.43.6 it follows that $K' \otimes _ k S$ is reduced. Hence we conclude that $K \otimes _ k S$ is reduced as desired.

Finally we prove that (2) $\Rightarrow $ (5). Assume $k^{1/p} \otimes _ k S$ is reduced. Then $S$ is reduced. Moreover, for each localization $S_{\mathfrak p}$ at a minimal prime $\mathfrak p$, the ring $k^{1/p}\otimes _ k S_{\mathfrak p}$ is a localization of $k^{1/p} \otimes _ k S$ hence is reduced. But $S_{\mathfrak p}$ is a field by Lemma 10.25.1, hence $S_{\mathfrak p}$ is geometrically reduced by Lemma 10.44.2. It follows from Lemma 10.43.7 that $S$ is geometrically reduced. $\square$


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Your email address will not be published.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 030V. Beware of the difference between the letter 'O' and the digit '0'.