Lemma 33.8.13 (04KY)—The Stacks project

Processing math: 100%

Table of contents
Part 2: Schemes
Chapter 33: Varieties
Section 33.8: Geometrically irreducible schemes
Lemma 33.8.13 (cite)

numberstags

Lemma 33.8.13. Let $k$ be a field, with separable algebraic closure $\overline{k}$ . Let $X$ be a scheme over $k$ . The fibres of the map

$\text{IrredComp}(X_{\overline{k}}) \longrightarrow \text{IrredComp}(X)$

of Lemma 33.8.10 are exactly the orbits of $\text{Gal}(\overline{k}/k)$ under the action of Lemma 33.8.12.

Proof. Let $T \subset X$ be an irreducible component of $X$ . Let $\eta \in T$ be its generic point. By Lemmas 33.8.9 and 33.8.10 the generic points of irreducible components of $\overline{T}$ which map into $T$ map to $\eta$ . By Algebra, Lemma 10.47.14 the Galois group acts transitively on all of the points of $X_{\overline{k}}$ mapping to $\eta$ . Hence the lemma follows. $\square$

Comments (0)

There are also:

2 comment(s) on Section 33.8: Geometrically irreducible schemes

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