Definition 10.105.1. A ring $R$ is said to be catenary if for any pair of prime ideals $\mathfrak p \subset \mathfrak q$, there exists an integer bounding the lengths of all finite chains of prime ideals $\mathfrak p = \mathfrak p_0 \subset \mathfrak p_1 \subset \ldots \subset \mathfrak p_ e = \mathfrak q$ and all maximal such chains have the same length.

Comment #5869 by Bjorn Poonen on

As worded, the definition sounds as if it is requiring the finite maximal chains to have equal length, while imposing no condition on infinite maximal chains. I suspect that the definition should instead say

A ring $R$ is said to be catenary if for every pair of prime ideals $\mathfrak{p} \subset \mathfrak{q}$, there exists an integer bounding the lengths of all finite chains of primes $\mathfrak{p} = \mathfrak{p}_0 \subset \mathfrak{p}_1 \subset \cdots \subset \mathfrak{p}_e = \mathfrak{q}$, and the maximal such chains all have the same length.

Less importantly, I wonder also whether it would be clearer to define "saturated" chain and to use that word instead of "maximal" since I could imagine that some people could misunderstand maximal to mean maximal-length (even though they would quickly realize that that could be not be what was meant).

Even less important: Note that it is better to use \cdots instead of \ldots in the chain.

Comment #6081 by on

OK, I changed the definition as you suggested. To make sure I added a definition of "chain of primes" in a ring because this is used all over the place. (The definition was implicit in the definition of the dimension of a ring.) See changes here.

Side comment: I dislike cdots and I like ldots, so I consistently use ldots for everything.

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