Lemma 33.8.9. Let $K/k$ be an extension of fields. Let $X$ be a scheme over $k$. For every irreducible component $T$ of $X$ the inverse image $T_ K \subset X_ K$ is a union of irreducible components of $X_ K$.

Proof. Let $T \subset X$ be an irreducible component of $X$. The morphism $T_ K \to T$ is flat, so generalizations lift along $T_ K \to T$. Hence every $\xi \in T_ K$ which is a generic point of an irreducible component of $T_ K$ maps to the generic point $\eta$ of $T$. If $\xi ' \leadsto \xi$ is a specialization in $X_ K$ then $\xi '$ maps to $\eta$ since there are no points specializing to $\eta$ in $X$. Hence $\xi ' \in T_ K$ and we conclude that $\xi = \xi '$. In other words $\xi$ is the generic point of an irreducible component of $X_ K$. This means that the irreducible components of $T_ K$ are all irreducible components of $X_ K$. $\square$

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