Proposition 53.13.7 (0CD2)—The Stacks project

Loading web-font TeX/Math/Italic

Table of contents
Part 3: Topics in Scheme Theory
Chapter 53: Algebraic Curves
Section 53.13: Inseparable maps
Proposition 53.13.7 (cite)

numberstags

Proposition 53.13.7. Let $k$ be a field of characteristic $p > 0$ . Let $f : X \to Y$ be a nonconstant morphism of proper smooth curves over $k$ . Then we can factor $f$ as

$X \longrightarrow X^{(p^ n)} \longrightarrow Y$

where $X^{(p^ n)} \to Y$ is a nonconstant morphism of proper smooth curves inducing a separable field extension $k(X^{(p^ n)})/k(Y)$ , we have

$X^{(p^ n)} = X \times _{\mathop{\mathrm{Spec}}(k), F_{\mathop{\mathrm{Spec}}(k)}^ n} \mathop{\mathrm{Spec}}(k),$

and $X \to X^{(p^ n)}$ is the $n$ -fold relative frobenius of $X$ .

Proof. By Fields, Lemma 9.14.6 there is a subextension $k(X)/E/k(Y)$ such that $k(X)/E$ is purely inseparable and $E/k(Y)$ is separable. By Theorem 53.2.6 this corresponds to a factorization $X \to Z \to Y$ of $f$ with $Z$ a nonsingular proper curve. Apply Lemma 53.13.4 to the morphism $X \to Z$ to conclude. $\square$

Comments (0)

There are also:

7 comment(s) on Section 53.13: Inseparable maps

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).

Comments (0)

Post a comment