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A prime ideal criterion for being Noetherian
*August 13, 2009*

*Posted by Akhil Mathew in algebra, commutative algebra.*

Tags: commutative algebra, Noetherian rings, prime ideals

1 comment so far

Tags: commutative algebra, Noetherian rings, prime ideals

1 comment so far

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

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e, f, and the remainder theorem
*September 12, 2009*

*Posted by Akhil Mathew in algebra, algebraic number theory, commutative algebra, number theory.*

Tags: Chinese remainder theorem, Dedekind domains, ramification

1 comment so far

Tags: Chinese remainder theorem, Dedekind domains, ramification

1 comment so far

So, now to the next topic in introductory algebraic number theory: ramification. This is a measure of how primes “split.” (*No, definitely wrong word there…)*

**e and f**

Fix a Dedekind domain with quotient field ; let be a finite separable extension of , and the integral closure of in . We know that is a Dedekind domain.

(By the way, I’m now assuming that readers have been following the past few posts or so on these topics.)

Given a prime , there is a prime lying above . I hinted at the proof in the previous post, but to save time and avoid too much redundancy I’ll refer interested readers to this post.

Now, we can do a prime factorization of say . The primes contain and consequently lie above . Conversely, any prime of containing must lie above , since if is an ideal in a Dedekind domain contained in a prime ideal , then occurs in the prime factorization of (to see this, localize and work in a DVR). (more…)

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Dedekind domains
*August 31, 2009*

*Posted 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

2 comments

Tags: Dedekind domains, discrete valuation rings, Krull dimension, localization, Noetherian rings, unique factorization

2 comments

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.

Definition 1A(more…)Dedekind domainis a Noetherian integral domain that is integrally closed, and of Krull dimension one—that is, each nonzero prime ideal is maximal.

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Generic freeness II
*July 30, 2009*

*Posted by Akhil Mathew in algebra, commutative algebra.*

Tags: algebra, commutative algebra, generic freeness, localization

12 comments

Tags: algebra, commutative algebra, generic freeness, localization

12 comments

Today’s goal is to partially finish the proof of the generic freeness lemma; the more general case, with finitely generated algebras, will have to wait for a later time though.

Recall that our goal was the following:

Theorem 1Let be a Noetherian integral domain, a finitely generated -module. Then there there exists with a free -module.

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Generic freeness I
*July 29, 2009*

*Posted by Akhil Mathew in algebra, commutative algebra.*

Tags: commutative algebra, devissage, generic freeness, localization, Noetherian rings, schemes

5 comments

Tags: commutative algebra, devissage, generic freeness, localization, Noetherian rings, schemes

5 comments

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.

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DVRs II
*August 30, 2009*

*Posted by Akhil Mathew in algebra, algebraic number theory, commutative algebra, number theory.*

Tags: discrete valuation rings, Noetherian rings, PIDs, prime ideals, UFDs

3 comments

Tags: discrete valuation rings, Noetherian rings, PIDs, prime ideals, UFDs

3 comments

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 1If the domain is Noetherian, integrally closed, and has a unique nonzero prime ideal , then is a DVR. Conversely, any DVR has those properties. (more…)

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Completions of rings and modules
*August 25, 2009*

*Posted by Akhil Mathew in algebra, commutative algebra.*

Tags: Artin-Rees lemma, completions, exact functors, Hilbert basis theorem, Noetherian rings

2 comments

Tags: Artin-Rees lemma, completions, exact functors, Hilbert basis theorem, Noetherian rings

2 comments

So, we saw in the previous post that completion can be defined generally for abelian groups. Now, to specialize to rings and modules.

**Rings **

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…)

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Integrality, invariant theory for finite groups, and more tools for Noetherian testing
*August 11, 2009*

*Posted by Akhil Mathew in algebra, commutative algebra.*

Tags: algebra, integrality, invariant theory, Noetherian rings

6 comments

Tags: algebra, integrality, invariant theory, Noetherian rings

6 comments

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.

**Subrings **

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.