Example 10.35.16. Here is a simple example that shows Lemma 10.35.14 is false if $R$ is Jacobson but we localize at infinitely many elements. Namely, let $R = \mathbf{Z}$ and consider the localization $(R \setminus \{ 0\} )^{-1}R \cong \mathbf{Q}$ of $R$ at the set of all nonzero elements. Clearly the map $\mathbf{Z} \to \mathbf{Q}$ maps the closed point of $\mathop{\mathrm{Spec}}(\mathbf{Q})$ to the generic point of $\mathop{\mathrm{Spec}}(\mathbf{Z})$.

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