Does Matsumura prove that finitely a generated algebra over a field is catenary?

Question

In Commutative ring theory, Matsumura states on page 30

Practically all the important Noetherian rings arising in applications are known to be catenary;

I think one of the important rings arising in algebraic geometry are finitely generated algebras (and localizations of those) over a field. However I didn't find a proof in Matsumura, that those are actually catenary.

I know that Eisenbud proves this in Commutative algebra with a view toward algebraic geometry, so I'm not looking for a reference, just curious if I missed this in Matsumura's book, or if this is really left out.

Answer 1

This is proven, but without drawing too much attention to it.

Theorem 17.9: Any quotient of a Cohen-Macaulay ring is universally catenary.

This appears in my copy on page 137. As a field is Cohen-Macaulay, the result follows.

Alternatively:

Theorem 31.7: For a noetherian local ring $A$, the following are equivalent:

• $A$ is formally catenary
• $A$ is universally catenary
• $A[x]$ is catenary.

(Page 252 in my copy.) So it remains to show that $k[x]$ for a field $k$ is catenary. This is trivial as $k[x]$ is a PID.