Lemma 29.16.8. Let S be a scheme. The following are equivalent:
the scheme S is Jacobson,
S_{\text{ft-pts}} is the set of closed points of S,
for all T \to S locally of finite type closed points map to closed points, and
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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