Proposition 15.52.3. The following types of rings are excellent:

fields,

Noetherian complete local rings,

$\mathbf{Z}$,

Dedekind domains with fraction field of characteristic zero,

finite type ring extensions of any of the above.

Proposition 15.52.3. The following types of rings are excellent:

fields,

Noetherian complete local rings,

$\mathbf{Z}$,

Dedekind domains with fraction field of characteristic zero,

finite type ring extensions of any of the above.

**Proof.**
See Propositions 15.50.12 and 15.48.7 to see that these rings are G-rings and have J-2. Any Cohen-Macaulay ring is universally catenary, see Algebra, Lemma 10.105.9. In particular fields, Dedekind rings, and more generally regular rings are universally catenary. Via the Cohen structure theorem we see that complete local rings are universally catenary, see Algebra, Remark 10.160.9.
$\square$

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