The Stacks project

Lemma 29.17.3. Let $S$ be a locally Noetherian scheme. The following are equivalent:

  1. $S$ is universally catenary, and

  2. all local rings $\mathcal{O}_{S, s}$ of $S$ are universally catenary.

Proof. Assume that all local rings of $S$ are universally catenary. Let $f : X \to S$ be locally of finite type. We know that $X$ is catenary if and only if $\mathcal{O}_{X, x}$ is catenary for all $x \in X$. If $f(x) = s$, then $\mathcal{O}_{X, x}$ is essentially of finite type over $\mathcal{O}_{S, s}$. Hence $\mathcal{O}_{X, x}$ is catenary by the assumption that $\mathcal{O}_{S, s}$ is universally catenary.

Conversely, assume that $S$ is universally catenary. Let $s \in S$. We may replace $S$ by an affine open neighbourhood of $s$ by Lemma 29.17.2. Say $S = \mathop{\mathrm{Spec}}(R)$ and $s$ corresponds to the prime ideal $\mathfrak p$. Any finite type $R_{\mathfrak p}$-algebra $A'$ is of the form $A_{\mathfrak p}$ for some finite type $R$-algebra $A$. By assumption (and Lemma 29.17.2 if you like) the ring $A$ is catenary, and hence $A'$ (a localization of $A$) is catenary. Thus $R_{\mathfrak p}$ is universally catenary. $\square$

Comments (2)

Comment #231 by on

The first word of the second paragraph should be Conversely, not Conversly.

Comment #232 by on

Fixed, see here. Also for the other typos you found. Thanks!

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