Algebraic Varieties, Catenary Rings

Catenary Curve

If you're looking for the catenary curve, it is unrelated to the catenary ring. The catenary curve is the hyperbolic cosine: (exp(x)+exp(-x))/2, as demonstrated by a hanging chain, or the St. Louis Arch. This curve satisfies certain constraints, as shown by the calculus of variations. That's really all I know about catenary curves, and when I learn more, I'll write a page about it. Meantime, let's return to catenary rings.

Building a Chain of Primes Above and Below P

Let K be an arbitrary field, and let B be a transcendental extension of K followed by an integral extension. Also, B is an integral domain. For notational convenience, let Bu be the transcendental subring inside B, namely K adjoin x1 through xn. Thus B has transcendent degree n.

Let f be a ring homomorphism from B onto D, such that the kernel is prine. Hence D is also an integral domain. Restrict f to Bu, giving a ring homomorphism from a purely transcendental extension into an integral domain. Call the image Du, and note that this is a finitely generated K algebra. Apply noether normalization, and find indeterminants y1 through yl mapping faithfully into D, while yl+1 through yn map to 0. This is the map from Bt onto Dt, with Bu integral over Bt and mapping to Du integral over Dt, and B integral over Bu mapping to D integral over Du. In other words, D is a transcendental extension of degree l followed by an integral extension, followed by an integral extension. We know that such a ring has dimension l.

Build a chain of l+1 primes in D, starting with 0 and ending with a maximal ideal. This pulls back to a chain of prime ideals in B, starting with the kernel of f and ending with a maximal ideal.

How about the prime ideals inside the kernel? Start with 0 and adjoin yl+1 through yn-1, one at a time. This builds an ascending chain of prime ideals inside the subring Bu. Turn this around so that it is a descending chaing of prime ideals. The largest prime ideal is generated by yl+1 through yn-1, and the last prime ideal is 0. Remember that B is integral over Bu, and is an integral domain. Lift this descending chain to a descending chain in B. All these prime ideals are properly contained in the kernel of f. This because the kernel is a prime ideal lying over yl+1 through yn.

Combine the chain containing the kernel and the chain inside the kernel to build a chain of length n+1 inside B. Since the dimension of B equals its transcendent degree, this is the longest possible chain. It has to run from 0 up to some maximal ideal. Every prime ideal in B participates in a chain of length n+1.

You might be wondering why B has to be an integral domain. The chain of primes inside the kernel was lifted up to B in descending order. Descending chains are harder to lift than ascending chains. The larger ring, in this case B, must be an integral domain, and the smaller ring must be integrally closed. In this case the smaller ring is K adjoin several indeterminants, which is a ufd, which is integrally closed.

Filling in the Gaps

Assume B contains a chain of prime ideals. Let P1 be the least of these and let D = B/P1. Use the above theorem to find prime ideals of B inside P1. Map all the larger prime ideals into D. By induction, we can fill in the (shorter) chain in D. Pull these prime ideals back to B and our chain is complete. Thus every chain of primes embeds in a chain of length n+1. New primes slip between old ones, as dictated by their dimensions.

Catenary Ring

The ring R is catenary if it has a finite dimension n, and every chain of primes in R embeds in a chain of length n+1. By the above, a finitely generated K algebra that is an integral domain is catenary.

Chains of Varieties

Start with a chain of varieties in Cn. This corresponds to a chain of prime ideals in the ring of polynomials K[x1,x2,…xn], which is a catenary ring. Therefore the chain of varieties is part of a chain of length n+1, culminating in the entire space Cn.