The Stacks project

Lemma 32.7.16. Let $k \subset K$ be a finite extension of fields and let $X$ be a scheme over $k$. Denote by $p : X_ K \to X$ the projection morphism. For every connected component $T$ of $X_ K$ the image $p(T)$ is a connected component of $X$.

Proof. The image $p(T)$ is contained in some connected component $X'$ of $X$. Consider $X'$ as a closed subscheme of $X$ in any way. Then $T$ is also a connected component of $X'_ K = p^{-1}(X')$ and we may therefore assume that $X$ is connected. The morphism $p$ is open (Morphisms, Lemma 28.22.4), closed (Morphisms, Lemma 28.42.7) and the fibers of $p$ are finite sets (Morphisms, Lemma 28.42.10). Thus we may apply Topology, Lemma 5.7.7 to conclude. $\square$


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