How to tell if a ring is Noetherian August 9, 2009Posted by Akhil Mathew in algebra, commutative algebra.
Tags: algebra, commutative algebra, Hilbert basis theorem, localization, Noetherian rings
I briefly outlined the definition and first properties of Noetherian rings and modules a while back. There are several useful and well-known criteria to tell whether a ring is Noetherian, as I will discuss in this post. Actually, I’ll only get to the first few basic ones here, though these alone give us a lot of tools for, say, algebraic geometry, when we want to show our schemes are relatively well-behaved. But there are plenty more to go.
Hilbert’s basis theorem
It is the following:
Theorem 1 (Hilbert) Let be a Noetherian ring. Then the polynomial ring is also Noetherian.
Pick an ideal ; we must show is finitely generated.
For each , consider the set of all polynomials in of degree , and let denote the set of their leading coefficients. Now is actually an ideal in because ; note also that if , because one can multiply an element of by and stay in . Since is Noetherian, the eventually stabilize at some , which must be finitely generated. Let be generators; by definition, these are the leading coefficients of some polynomials . Set .
The idea now is that given any polynomial , we can subtract multiples of to bring the degree down to less than . Suppose . Then the leading coefficient of lies in , say
has degree strictly less than that of , and we proceed inductively.
So the ‘s aren’t yet generators of , but almost there: we just have to find -generators of the -module , and we can pool these with the ‘s to get -generators of . But is a submodule of the finitely generated -module , hence finitely generated since is Noetherian. This completes the proof.
The basis theorem is actually often used in the following form:
Corollary 2 Let be a Noetherian ring and a finitely generated -algebra. Then is Noetherian.
Since the quotient ring of a Noetherian ring is Noetherian, we reduce to a polynomial ring; then it follows from Hilbert’s theorem and induction.
In algebraic geometry, this is important because it implies for instance that a scheme of finite type over a field is Noetherian, and is thus fairly well-behaved. As another example, it shows that a subvariety of —which by definition is defined by the zero set of some ideal in the polynomial ring —can be cut out by a finite number of polynomial equations: just take generators of that ideal.
Localization preserves the property of a ring being Noetherian:
Proposition 3 If is a Noetherian ring and a multiplicative subset, then is a Noetherian ring.
This follows from general facts about localization, namely that any ideal of is of the form for an ideal; hence since is finitely generated, so is .
There is a more interesting converse: If sufficiently many localizations are Noetherian, so is the original ring.
Theorem 4 Let be a ring with elements generating the unit ideal. If each localization is Noetherian, so is .
For each , there are localization maps . Given an ideal , we associate the ideals . By definition, these are defined as .
Indeed, the inclusion is clear from the definitions, but the other inclusion requires more checking. Suppose is such that for all ; this means that there are powers with
So, consider the ideal ; then . If we show that , then , and my claim will be proved.
Lemma 5 If generate the unit ideal in , so do for any .
Indeed, given a relation
raise it to a high power to get
and every term on the left, when expanded, will lie in the ideal generated by if is very large.
Return to the proof of the theorem. If we have a sequence of ideals, then for each , the each stabilize since is Noetherian; thus by (1), so do the .
This result implies the following (see e.g. Hartshorne, II.3):
Corollary 6 If is a locally Noetherian scheme, then is a Noetherian ring.
This is interesting because being locally Noetherian says something about an open affine cover via Noetherian rings. But this result says that any open affine subscheme comes from a Noetherian ring.
So, we got to some of the basic criteria. But there are other questions that arise. For instance, is a subring of a Noetherian ring Noetherian? In general, no, but there are important cases when we can say yes. I’ll discuss this in the next post.