Normal Rings and Serre’s Criterion
It’s been a grip since I’ve posted, and I probably won’t start posting more regularly for awhile, but I typed an email today which I may as well put on here. It’s about normal rings which are not necessarily domains.
—-
In his book, Matsumura admits that most people just use the word normal to mean an integrally closed domain. However, he still follows the more general treatment of Serre and Grothendieck. This is useful in that in simplifies some of the statements in Eisenbud’s book, and is not that much more difficult. Here’s all you need to know:
Let’s assume all rings are Noetherian.
Defn: A ring R is normal is R_p is a normal domain for every prime ideal p. (Note that by definition, a local normal ring is a domain)
This implies, using the chinese remainder theorem, that if the minimal primes of R are {p1,…pn} then R is a product of normal domains as follows:
R = R/p_1 x … R/p_n
In fact, we have the following “structure theorem”
If R is noetherian, TFAE
1) R is normal
2) R is a finite product of normal domains
3) R is reduced and integrally closed in Quot(R)
4) R satisfies Serre’s conditions R1 and S2
Computationally, there is no nice algorithm for computing the integral closure of a ring. There are methods implemented in Macaulay, but they are very ad hoc and even now, Eisenbud and Stillman are trying to improve these things. However, a general strategy that one can use to compute integral closures (by hand) is:
1) Find all “reasonable” elements integral over the ring
2) Show that R adjoin these guys is integrally closed. This can usually be done by showing that the ring is isomorphic to a polynomial ring (or a product of polynomial rings) Anything harder than this would be unreasonable to do by hand.
In case that R is a domain, implicit in the above is that if the integral closure is a polynomial ring then R itself must be a subring of a polynomial ring, and sometimes you might even be able to eyeball that.
For example, you can check (using a Grobner basis for instance) that k[t^2,t^3] is isomorphic to k[x,y]/(y^2-x^3), and thus this integral closure is equal to k[t].
More generally the integral closure of k[f_1,.... f_n] in k[x_1,.... x_m] (if the f_i are monomial) will be given by some combinatorial subset in R^n (See Eisenbud Ch.4)
I’ve found computing normalizations to be one of the most frustrating experiences, but once you figure out the basics, they’re not so bad.
