Lemma 33.14.2. Let X be a scheme over a field k. For any field extension K/k whose cardinality is large enough we have
for any closed point x \in X_ K the extension \kappa (x)/K is algebraic, and
X_ K is a Jacobson scheme (Properties, Definition 28.6.1).
Proof. Choose an affine open covering X = \bigcup U_ i. By Algebra, Lemma 10.35.12 and Properties, Lemma 28.6.2 there exist cardinals \kappa _ i such that U_{i, K} has the desired properties over K if \# (K) \geq \kappa _ i. Set \kappa = \max \{ \kappa _ i\} . Then if the cardinality of K is larger than \kappa we see that each U_{i, K} satisfies the conclusions of the lemma. Hence X_ K is Jacobson by Properties, Lemma 28.6.3. The statement on residue fields at closed points of X_ K follows from the corresponding statements for residue fields of closed points of the U_{i, K}. \square
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