The Stacks project

Lemma 29.20.7. The following types of schemes are excellent.

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

  2. Any scheme locally of finite type over a Noetherian complete local ring.

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

  4. Any scheme locally of finite type over a Dedekind ring of characteristic zero.

Proof. By Lemmas 29.20.5 and 29.20.6 we only need to show that the rings mentioned above are excellent. For this see More on Algebra, Proposition 15.53.3. $\square$

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