The Stacks project

Lemma 33.36.6. Let $p > 0$ be a prime number. Let $S$ be a scheme in characteristic $p$. Let $X$ be a scheme over $S$. Then the relative frobenius $F_{X/S} : X \to X^{(p)}$ is a universal homeomorphism, is integral, and induces purely inseparable residue field extensions.

Proof. By Lemma 33.36.3 the morphisms $F_ X : X \to X$ and the base change $h : X^{(p)} \to X$ of $F_ S$ are universal homeomorphisms. Since $h \circ F_{X/S} = F_ X$ we conclude that $F_{X/S}$ is a universal homeomorphism. By Morphisms, Lemmas 29.45.5 and 29.10.2 we conclude that $F_{X/S}$ has the other properties as well. $\square$

Comments (0)

There are also:

  • 4 comment(s) on Section 33.36: Frobenii

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Your email address will not be published.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