The Stacks project

Lemma 33.7.15. Let $k \subset K$ be an extension of fields. Let $X$ be a scheme over $k$. For every connected component $T$ of $X$ the inverse image $T_ K \subset X_ K$ is a union of connected components of $X_ K$.

Proof. This is a purely topological statement. Denote $p : X_ K \to X$ the projection morphism. Let $T \subset X$ be a connected component of $X$. Let $t \in T_ K = p^{-1}(T)$. Let $C \subset X_ K$ be a connected component containing $t$. Then $p(C)$ is a connected subset of $X$ which meets $T$, hence $p(C) \subset T$. Hence $C \subset T_ K$. $\square$


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