Lemma 33.36.9. Let $k$ be a field of characteristic $p > 0$. Let $X$ be a scheme over $k$. Then $X$ is geometrically reduced if and only if $X^{(p)}$ is reduced.
Proof. Consider the absolute frobenius $F_ k : k \to k$. Then $F_ k(k) = k^ p$ in other words, $F_ k : k \to k$ is isomorphic to the embedding of $k$ into $k^{1/p}$. Thus the lemma follows from Lemma 33.6.4. $\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)
There are also: