The Stacks project

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. $\square$

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