The Stacks project

Lemma 29.16.8. Let $S$ be a scheme. The following are equivalent:

  1. the scheme $S$ is Jacobson,

  2. $S_{\text{ft-pts}}$ is the set of closed points of $S$,

  3. for all $T \to S$ locally of finite type closed points map to closed points, and

  4. for all $T \to S$ locally of finite type closed points $t \in T$ map to closed points $s \in S$ with $\kappa (s) \subset \kappa (t)$ finite.

Proof. We have trivially (4) $\Rightarrow $ (3) $\Rightarrow $ (2). Lemma 29.16.7 shows that (2) implies (1). Hence it suffices to show that (1) implies (4). Suppose that $T \to S$ is locally of finite type. Choose $t \in T$ closed and let $s \in S$ be the image. Choose affine open neighbourhoods $\mathop{\mathrm{Spec}}(R) = U \subset S$ of $s$ and $\mathop{\mathrm{Spec}}(A) = V \subset T$ of $t$ with $V$ mapping into $U$. The induced ring map $R \to A$ is of finite type (see Lemma 29.15.2) and $R$ is Jacobson by Properties, Lemma 28.6.3. Thus the result follows from Algebra, Proposition 10.35.19. $\square$

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