The Stacks project

Lemma 33.14.2. Let $X$ be a scheme over a field $k$. For any field extension $k \subset K$ whose cardinality is large enough we have

  1. for any closed point $x \in X_ K$ the extension $K \subset \kappa (x)$ is algebraic, and

  2. $X_ K$ 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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