Catenary ring

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. ==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 (and regular local rings) • Any localization of a universally catenary ring • Any finitely generated algebra over a universally catenary ring. A ring that is catenary but not universally catenary It is delicate to construct examples of Noetherian rings that are not universally catenary. The first example was found by , 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 z=Σi>0''a'i'x''i 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 mB∩A 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. == See also ==