Definition 29.17.1. Let $S$ be a scheme. Assume $S$ is locally Noetherian. We say $S$ is universally catenary if for every morphism $X \to S$ locally of finite type the scheme $X$ is catenary.
29.17 Universally catenary schemes
Recall that a topological space $X$ is called catenary if for every pair of irreducible closed subsets $T \subset T'$ there exist a maximal chain of irreducible closed subsets
\[ T = T_0 \subset T_1 \subset \ldots \subset T_ e = T' \]
and every such chain has the same length. See Topology, Definition 5.11.4. Recall that a scheme is catenary if its underlying topological space is catenary. See Properties, Definition 28.11.1.
This is a “better” notion than catenary as there exist Noetherian schemes which are catenary but not universally catenary. See Examples, Section 110.19. Many schemes are universally catenary, see Lemma 29.17.5 below.
Recall that a ring $A$ is called catenary if for any pair of prime ideals $\mathfrak p \subset \mathfrak q$ there exists a maximal chain of primes
\[ \mathfrak p = \mathfrak p_0 \subset \ldots \subset \mathfrak p_ e = \mathfrak q \]
and all of these have the same length. See Algebra, Definition 10.105.1. We have seen the relationship between catenary schemes and catenary rings in Properties, Section 28.11. Recall that a ring $A$ is called universally catenary if $A$ is Noetherian and for every finite type ring map $A \to B$ the ring $B$ is catenary. See Algebra, Definition 10.105.3. Many interesting rings which come up in algebraic geometry satisfy this property.
Lemma 29.17.2. Let $S$ be a locally Noetherian scheme. The following are equivalent
$S$ is universally catenary,
there exists an open covering of $S$ all of whose members are universally catenary schemes,
for every affine open $\mathop{\mathrm{Spec}}(R) = U \subset S$ the ring $R$ is universally catenary, and
there exists an affine open covering $S = \bigcup U_ i$ such that each $U_ i$ is the spectrum of a universally catenary ring.
Moreover, in this case any scheme locally of finite type over $S$ is universally catenary as well.
Proof. By Lemma 29.15.5 an open immersion is locally of finite type. A composition of morphisms locally of finite type is locally of finite type (Lemma 29.15.3). Thus it is clear that if $S$ is universally catenary then any open and any scheme locally of finite type over $S$ is universally catenary as well. This proves the final statement of the lemma and that (1) implies (2).
If $\mathop{\mathrm{Spec}}(R)$ is a universally catenary scheme, then every scheme $\mathop{\mathrm{Spec}}(A)$ with $A$ a finite type $R$-algebra is catenary. Hence all these rings $A$ are catenary by Algebra, Lemma 10.105.2. Thus $R$ is universally catenary. Combined with the remarks above we conclude that (1) implies (3), and (2) implies (4). Of course (3) implies (4) trivially.
To finish the proof we show that (4) implies (1). Assume (4) and let $X \to S$ be a morphism locally of finite type. We can find an affine open covering $X = \bigcup V_ j$ such that each $V_ j \to S$ maps into one of the $U_ i$. By Lemma 29.15.2 the induced ring map $\mathcal{O}(U_ i) \to \mathcal{O}(V_ j)$ is of finite type. Hence $\mathcal{O}(V_ j)$ is catenary. Hence $X$ is catenary by Properties, Lemma 28.11.2. $\square$
Lemma 29.17.3. Let $S$ be a locally Noetherian scheme. The following are equivalent:
$S$ is universally catenary, and
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. $\square$
Lemma 29.17.4. Let $S$ be a locally Noetherian scheme. Then $S$ is universally catenary if and only if the irreducible components of $S$ are universally catenary.
Proof. Omitted. For the affine case, please see Algebra, Lemma 10.105.8. $\square$
Lemma 29.17.5. The following types of schemes are universally catenary.
Any scheme locally of finite type over a field.
Any scheme locally of finite type over a Cohen-Macaulay scheme.
Any scheme locally of finite type over $\mathbf{Z}$.
Any scheme locally of finite type over a $1$-dimensional Noetherian domain.
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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