The Stacks project

Lemma 27.11.2. 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 10.104.2. $\square$


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