The Stacks project

Lemma 5.11.6. Let $X$ be a topological space. The following are equivalent:

  1. $X$ is catenary, and

  2. for every pair of irreducible closed subsets $Y \subset Y'$ we have $\text{codim}(Y, Y') < \infty $ and for every triple $Y \subset Y' \subset Y''$ of irreducible closed subsets we have

    \[ \text{codim}(Y, Y'') = \text{codim}(Y, Y') + \text{codim}(Y', Y''). \]

Proof. Omitted. $\square$

Comments (7)

Comment #344 by JuanPablo on

This seem false, if I am not misinterpreting something.

Take to be together with and additional point . Closed sets are of the form where , empty and the .

Then the closed irreducible sets are of the form and and .

So that is catenary but .

uhmm left and right brackets do not appear in the preview hopefully it is not unintelligible.

Comment #345 by JuanPablo on

oops I meant with brackets around .

Comment #346 by JuanPablo on

oops I meant with brackets around and .

I need to be more careful.

Comment #347 by on

Sorry, but your comment is too hard to read. It seems that brackets do not show up correctly in the comments. I will see if we can fix that.

Question: Is your example pointing out that the definition of catenary is incorrect? Because now that I look at it, it should include the condition that every chain of irreducible closed subsets between T and T' (as in the definition of catenary) can be filled out to a maximal chain and not just that every maximal chain has the same length. In other words, given irreducible closed subsets there exists an integer such that any finite chain between and should have length at most and any maximal chain should have length exactly . OK?

Comment #348 by JuanPablo on

Ahh..., I'm very sorry, I just reread the definition of catenary, please ignore my comments.(Which includes that all maximal chains have the same lenght).

Comment #351 by on

OK, but at least now, thanks to Pieter, the curly brackets work!

Comment #357 by on

Clarified the definition of catenary spaces, see here.

There are also:

  • 2 comment(s) on Section 5.11: Codimension and catenary spaces

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