The Stacks project

Lemma 29.17.5. The following types of schemes are universally catenary.

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

  2. Any scheme locally of finite type over a Cohen-Macaulay scheme.

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

  4. Any scheme locally of finite type over a $1$-dimensional Noetherian domain.

  5. And so on.

Proof. All of these follow from the fact that a Cohen-Macaulay ring is universally catenary, see Algebra, Lemma 10.105.9. Also, use the last assertion of Lemma 29.17.2. Some details omitted. $\square$

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