The Stacks project

Lemma 33.14.1. Let $X$ be a scheme which is locally of finite type over $k$. Then

  1. for any closed point $x \in X$ the extension $\kappa (x)/k$ is algebraic, and

  2. $X$ is a Jacobson scheme (Properties, Definition 28.6.1).

Proof. A scheme is Jacobson if and only if it has an affine open covering by Jacobson schemes, see Properties, Lemma 28.6.3. The property on residue fields at closed points is also local on $X$. Hence we may assume that $X$ is affine. In this case the result is a consequence of the Hilbert Nullstellensatz, see Algebra, Theorem 10.34.1. It also follows from a combination of Morphisms, Lemmas 29.16.8, 29.16.9, and 29.16.10. $\square$

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