The Stacks project

Lemma 29.16.10. The following types of schemes are Jacobson.

  1. Any scheme locally of finite type over a field.

  2. Any scheme locally of finite type over $\mathbf{Z}$.

  3. Any scheme locally of finite type over a $1$-dimensional Noetherian domain with infinitely many primes.

  4. A scheme of the form $\mathop{\mathrm{Spec}}(R) \setminus \{ \mathfrak m\} $ where $(R, \mathfrak m)$ is a Noetherian local ring. Also any scheme locally of finite type over it.

Proof. We will use Lemma 29.16.9 without mention. The spectrum of a field is clearly Jacobson. The spectrum of $\mathbf{Z}$ is Jacobson, see Algebra, Lemma 10.35.6. For (3) see Algebra, Lemma 10.61.4. For (4) see Properties, Lemma 28.6.4. $\square$

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