Catenary ring

In mathematics, a commutative ring "R" is catenary if for any pair of prime ideals

:"p", "q", any two strictly increasing chains

:"p"="p"₀ ⊂"p"₁ ... ⊂"p"_"n"= "q" of prime ideals

are contained in maximal strictly increasing chains from "p" to "q" of the same (finite) length. In other words, there is a well-defined function from pairs of prime ideals to natural numbers, attaching to "p" and "q" the length of any such maximal chain. In a geometric situation, in which the dimension of an algebraic variety attached to a prime ideal will decrease as the prime ideal becomes bigger, there is reason to believe that the length of such a chain will conform to "n" = difference in dimensions, with dimension decrementing by 1 at each step.

A ring is called universally catenary if all finitely generated rings over it are catenary.

The word 'catenary' is derived from the Latin word "catena", which means "chain".

Examples

Almost all Noetherian rings that appear in algebraic geometry are universally catenary.In particular the following rings are universally catenary:
*Complete Noetherian local rings
*Dedekind domains (and fields)
*Cohen-Macaulay rings
*Any localization of a universally catenary ring
*Any finitely generated algebra over a universally catenary ring.

It is very hard to construct examples of rings that are not universally catenary. The first example was found by Masayoshi Nagata in 1956 (see Nagata (1962) page 203 example 2), who produced a Noetherian local domain that was catenary but not universally catenary.

References

*H. Matsumura, "Commutative algebra" ISBN 0-8053-7026-9.
*Nagata, Masayoshi "Local rings." Interscience Tracts in Pure and Applied Mathematics, No. 13 Interscience Publishers a division of John Wiley & Sons,New York-London 1962, reprinted by R. E. Krieger Pub. Co (1975) ISBN 0882752286