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The Stacks project

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


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