Lemma 33.8.10. Let K/k be an extension of fields. Let X be a scheme over k. For every irreducible component \overline{T} \subset X_ K the image of \overline{T} in X is an irreducible component in X. This defines a canonical map
which is surjective.
Lemma 33.8.10. Let K/k be an extension of fields. Let X be a scheme over k. For every irreducible component \overline{T} \subset X_ K the image of \overline{T} in X is an irreducible component in X. This defines a canonical map
which is surjective.
Proof. Consider the diagram
where \overline{K} is the separable algebraic closure of K, and where \overline{k} is the separable algebraic closure of k. By Lemma 33.8.7 the morphism X_{\overline{K}} \to X_{\overline{k}} induces a bijection between irreducible components. Hence it suffices to show the lemma for the morphisms X_{\overline{k}} \to X and X_{\overline{K}} \to X_ K. In other words we may assume that K = \overline{k}.
The morphism p : X_{\overline{k}} \to X is integral, flat and surjective. Flatness implies that generalizations lift along p, see Morphisms, Lemma 29.25.9. Hence generic points of irreducible components of X_{\overline{k}} map to generic points of irreducible components of X. Integrality implies that p is universally closed, see Morphisms, Lemma 29.44.7. Hence we conclude that the image p(\overline{T}) of an irreducible component is a closed irreducible subset which contains a generic point of an irreducible component of X, hence p(\overline{T}) is an irreducible component of X. This proves the first assertion. If T \subset X is an irreducible component, then p^{-1}(T) =T_ K is a nonempty union of irreducible components, see Lemma 33.8.9. Each of these necessarily maps onto T by the first part. Hence the map is surjective. \square
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