Dedekind domains August 31, 2009Posted by Akhil Mathew in algebra, algebraic number theory, commutative algebra, number theory.
Tags: Dedekind domains, discrete valuation rings, Krull dimension, localization, Noetherian rings, unique factorization
Today’s (quick) topic focuses on Dedekind domains. These come up when you take the ring of integers in any finite extension of (i.e. number fields). In these, you don’t necessarily have unique factorization. But you do have something close, which makes these crucial.
DVRs II August 30, 2009Posted by Akhil Mathew in algebra, algebraic number theory, commutative algebra, number theory.
Tags: discrete valuation rings, Noetherian rings, PIDs, prime ideals, UFDs
Earlier I went over the definition and first properties of a discrete valuation ring. Today, it’s time to say how we can tell a ring is a DVR–it turns out to be not too bad, which is nice because the properties we need in this criterion are often easier to work with than the existence of some discrete valuation.
Today’s result is:
Theorem 1 If the domain is Noetherian, integrally closed, and has a unique nonzero prime ideal , then is a DVR. Conversely, any DVR has those properties. (more…)
The finite presentation trick and completions August 27, 2009Posted by Akhil Mathew in algebra, category theory, commutative algebra.
Tags: abelian categories, completions, finite presentations, flatness, Noetherian rings
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The previous post got somewhat detailed and long, so today’s will be somewhat lighter. I’ll use completions to illustrate a well-known categorical trick using finite presentations.
The finite presentation trick
Our goal here is:
Theorem 1 Let be a Noetherian ring, and an ideal. If we take all completions with respect to the -adic topology,
for any f.g. -module . (more…)
Completions of rings and modules August 25, 2009Posted by Akhil Mathew in algebra, commutative algebra.
Tags: Artin-Rees lemma, completions, exact functors, Hilbert basis theorem, Noetherian rings
So, we saw in the previous post that completion can be defined generally for abelian groups. Now, to specialize to rings and modules.
The case in which we are primarily interested comes from a ring with a descending filtration (satisfying ), which implies the are ideals; as we saw, the completion will also be a ring. Most often, there will be an ideal such that , i.e. the filtration is -adic. We have a completion functor from filtered rings to rings, sending . Given a filtered -module , there is a completion , which is also a -module; this gives a functor from filtered -modules to -modules. (more…)
Gradings, filtrations, and gr August 18, 2009Posted by Akhil Mathew in algebra, commutative algebra.
Tags: filtered modules, filtered rings, gr, graded modules, graded rings, Noetherian rings
Bourbaki has a whole chapter in Commutative Algebra devoted to “graduations, filtrations, and topologies,” which indicates the importance of these concepts. That’s the theme for the next few posts I’ll do here, although I will (of course) be more concise.
In general, all rings will be commutative.
The idea of a graded ring is necessary to define projective space.
Definition 1 A graded ring is ring together with a decomposition
such that . The set is said to consist of homogeneous elements of degree . (more…)
A prime ideal criterion for being Noetherian August 13, 2009Posted by Akhil Mathew in algebra, commutative algebra.
Tags: commutative algebra, Noetherian rings, prime ideals
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This post, the third in the mini-series so far, gives one more criterion for when a ring is Noetherian. I also discuss how prime ideals tend to crop up in commutative algebra.
Why prime ideals are important
As discussed in the end of my previous post and in the comments, ideals satisfying some property and maximal with respect to it are often prime. To prove these results, we often use the following convenient notation:
Definition 1If are ideals of a commutative ring , then we define
Integrality, invariant theory for finite groups, and more tools for Noetherian testing August 11, 2009Posted by Akhil Mathew in algebra, commutative algebra.
Tags: algebra, integrality, invariant theory, Noetherian rings
There are quite a few more tools to tell whether a ring is Noetherian. In this post, I’ll discuss another basic tool: integrality. I’ll discuss the application to invariant theory for finite groups.
In general, it is not true that a subring of a Noetherian ring is Noetherian. For instance, let be the polynomial ring in infinitely many variables over a field . Then is not Noetherian because of the ascending chain
However, the quotient field of is Noetherian. This applies to any non-Noetherian integral domain.
There are special cases where we can conclude a subring of a Noetherian ring is Noetherian.
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.
Generic freeness I July 29, 2009Posted by Akhil Mathew in algebra, commutative algebra.
Tags: commutative algebra, devissage, generic freeness, localization, Noetherian rings, schemes
There is a useful fact in algebraic geometry that if you have a coherent sheaf over a Noetherian integral scheme, then it is locally free on some dense open subset. That is the content of today’s post, although I will use the language of commutative algebra than that of schemes (except at the end), to keep the presentation as elementary as possible. The goal is to get the generic freeness in a restricted case. Later, I’ll discuss the full “generic freeness” lemma of Grothendieck.
Noetherian Rings and Modules
All rings are assumed commutative in this post.
As I have already mentioned, a ring is Noetherian if each ideal of is finitely generated. Similarly, a module is Noetherian if every submodule is finitely generated. I will summarize the basic facts below briefly.