The Stacks project

Definition 15.109.1. A Noetherian local ring $A$ is formally catenary if for every minimal prime $\mathfrak p \subset A$ the spectrum of $A^\wedge /\mathfrak p A^\wedge $ is equidimensional.

Comments (2)

Comment #3608 by Kestutis Cesnavicius on

This is not the usual definition of formal catenarity. Usually, one would require this to hold for any , not only the minimal ones. It would be good to mention the equivalence with the usual definition.

Comment #3719 by on

OK, yes, now I see sometimes people do this thing you say. Actually, I took my definition of formal catenarity from a paper about formally cattenary rings, but somehow now I cannot find it. Anyway, it seems that the definition as given here is the more useful one, since this allows one to state Ratliff's result in optimal form. And of course as you say it is equivalent. I have added a short discussion of the implication just after the definition here.

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View Definition 15.109.1 as pdf