The Stacks project

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.

Definition 28.11.1. Let $S$ be a scheme. We say $S$ is catenary if the underlying topological space of $S$ is catenary.

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

  1. $S$ is catenary,

  2. there exists an open covering of $S$ all of whose members are catenary schemes,

  3. for every affine open $\mathop{\mathrm{Spec}}(R) = U \subset S$ the ring $R$ is catenary, and

  4. 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:

  1. $S$ is catenary, and

  2. locally in the Zariski topology there exists a dimension function on $S$ (see Topology, Definition 5.20.1).

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

  1. $X$ is catenary, and

  2. 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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