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The (combinatorial) dimension of a topological space is defined as the supremum of the lengths over all strictly ascending chains of closed irreducible subsets (e.g. Hartshorne). Can it also be defined similarly albeit using strictly descending chains? Are the two notions equivalent? If not, which one is more important and why?

Edited: Also, Atiyah-MacDonald define the Krull dimension of a ring using ascending chains of prime ideals while Matsumura uses descending chains. Any insights?

PS: Merry Christmas :)

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up vote 3 down vote accepted

In both the definition of combinatorial dimension and Krull dimension, you're looking at the supremum of lengths of finite chains of objects, so it doesn't really matter which definition you use. Reversing an ascending chain gives you a descending chain and vica versa, without changing the length.

But in other contexts, when possibly infinite chains are of interest, then the resulting notions might not be equivalent. For example, most Noetherian rings are not Artinian. Even though all ascending chains of ideals in a Noetherian ring are finite, they can still be arbitrarily long. The same can happen for ascending chains of prime ideals in a Noetherian ring (so the Krull dimension of a Noetherian ring can be infinite), although the counterexamples are more artificial.

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