Definition 28.11.1. Let S be a scheme. We say S is catenary if the underlying topological space of S is catenary.
28.11 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 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.
Lemma 28.11.2. Let S be a scheme. The following are equivalent
S is catenary,
there exists an open covering of S all of whose members are catenary schemes,
for every affine open \mathop{\mathrm{Spec}}(R) = U \subset S the ring R is catenary, and
there exists an affine open covering S = \bigcup U_ i such that each U_ i is the spectrum of a catenary ring.
Moreover, in this case any locally closed subscheme of S is catenary as well.
Proof. Combine Topology, Lemma 5.11.5, and Algebra, Lemma 10.105.2. \square
Lemma 28.11.3. Let S be a locally Noetherian scheme. The following are equivalent:
S is catenary, and
locally in the Zariski topology there exists a dimension function on S (see Topology, Definition 5.20.1).
Proof. This follows from Topology, Lemmas 5.11.5, 5.20.2, and 5.20.4, Schemes, Lemma 26.11.1 and finally Lemma 28.5.5. \square
It turns out that a scheme is catenary if and only if its local rings are catenary.
Lemma 28.11.4. Let X be a scheme. The following are equivalent
X is catenary, and
for any x \in X the local ring \mathcal{O}_{X, x} is catenary.
Proof. Assume X is catenary. Let x \in X. By Lemma 28.11.2 we may replace X by an affine open neighbourhood of x, and then \Gamma (X, \mathcal{O}_ X) is a catenary ring. By Algebra, Lemma 10.105.4 any localization of a catenary ring is catenary. Whence \mathcal{O}_{X, x} is catenary.
Conversely assume all local rings of X are catenary. Let Y \subset Y' be an inclusion of irreducible closed subsets of X. Let \xi \in Y be the generic point. Let \mathfrak p \subset \mathcal{O}_{X, \xi } be the prime corresponding to the generic point of Y', see Schemes, Lemma 26.13.2. By that same lemma the irreducible closed subsets of X in between Y and Y' correspond to primes \mathfrak q \subset \mathcal{O}_{X, \xi } with \mathfrak p \subset \mathfrak q \subset \mathfrak m_{\xi }. Hence we see all maximal chains of these are finite and have the same length as \mathcal{O}_{X, \xi } is a catenary ring. \square
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