Catenary ring

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

p, q,

any two strictly increasing chains

p=p0p1 ... ⊂pn= q of prime ideals

are contained in maximal strictly increasing chains from p to q of the same (finite) length. 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, the length of such a chain n is usually the difference in dimensions.

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".

Dimension formula[edit]

Suppose that A is a Noetherian domain and B is a domain containing A that is finitely generated over A. If P is a prime ideal of B and p its intersection with A, then

\text{height}(P)\le \text{height}(p)+ \text{tr.deg.}_A(B) - \text{tr.deg.}_{\kappa(p)}(\kappa(P)).

The dimension formula for universally catenary rings says that equality holds if A is universally catenary. Here κ(P) is the residue field of P and tr.deg. means the transcendence degree (of quotient fields). In fact, when A is not universally catenary, but B=A[x_1,\dots,x_n], then equality also holds. [1]

Examples[edit]

Almost all Noetherian rings that appear in algebraic geometry are universally catenary. In particular the following rings are universally catenary:

A ring that is catenary but not universally catenary[edit]

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

Nagata's example is as follows. Choose a field k and a formal power series zi>0aixi in the ring S of formal power series in x over k such that z and x are algebraically independent.

Define z1 = z and zi+1=zi/x–ai.

Let R be the (non-Noetherian) ring generated by x and all the elements zi.

Let m be the ideal (x), and let n be the ideal generated by x–1 and all the elements zi. These are both maximal ideals of R, with residue fields isomorphic to k. The local ring Rm is a regular local ring of dimension 1 (the proof of this uses the fact that z and x are algebraically independent) and the local ring Rn is a regular Noetherian local ring of dimension 2.

Let B be the localization of R with respect to all elements not in either m or n. Then B is a 2-dimensional Noetherian semi-local ring with 2 maximal ideals, mB (of height 1) and nB (of height 2).

Let I be the Jacobson radical of B, and let A = k+I. The ring A is a local domain of dimension 2 with maximal ideal I, so is catenary because all 2-dimensional local domains are catenary. The ring A is Noetherian because B is Noetherian and is a finite A-module. However A is not universally catenary, because if it were then the ideal mB of B would have the same height as mBA by the dimension formula for universally catenary rings, but the latter ideal has height equal to dim(A)=2.

Nagata's example is also a quasi-excellent ring, so gives an example of a quasi-excellent ring that is not an excellent ring.

References[edit]