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Concept
Catenary ring
Summary
In mathematics, a commutative ring R is catenary if for any pair of prime ideals p, q, any two strictly increasing chains
p = p0 ⊂ p1 ⊂ ... ⊂ 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 algebras over it are catenary rings.
The word 'catenary' is derived from the Latin word catena, which means "chain".
There is the following chain of inclusions.
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
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 , then equality also holds.
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.
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>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.
Official source
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