A prime ideal criterion for being NoetherianAugust 13, 2009

Posted by Akhil Mathew in algebra, commutative algebra.
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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 1

If ${I,J}$ are ideals of a commutative ring ${A}$, then we define

e, f, and the remainder theoremSeptember 12, 2009

Posted by Akhil Mathew in algebra, algebraic number theory, commutative algebra, number theory.
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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 ${A}$ with quotient field ${K}$; let ${L}$ be a finite separable extension of ${K}$, and ${B}$ the integral closure of ${A}$ in ${L}$. We know that ${B}$ 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 ${\mathfrak{p} \subset A}$, there is a prime ${\mathfrak{P} \subset B}$ lying above ${\mathfrak{p}}$. 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 ${\mathfrak{p}B \subset B,}$ say ${\mathfrak{p}B = \mathfrak{P}_1^{e_1} \dots \mathfrak{P}_g^{e_g}}$. The primes ${\mathfrak{P}_i}$ contain ${\mathfrak{p}B}$ and consequently lie above ${\mathfrak{p}}$. Conversely, any prime of ${B}$ containing ${\mathfrak{p}B}$ must lie above ${\mathfrak{p}}$, since if ${I}$ is an ideal in a Dedekind domain contained in a prime ideal ${P}$, then ${P}$ occurs in the prime factorization of ${I}$ (to see this, localize and work in a DVR). (more…)

Dedekind domainsAugust 31, 2009

Posted by Akhil Mathew in algebra, algebraic number theory, commutative algebra, number theory.
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Today’s (quick) topic focuses on Dedekind domains. These come up when you take the ring of integers in any finite extension of ${\mathbb{Q}}$ (i.e. number fields). In these, you don’t necessarily have unique factorization. But you do have something close, which makes these crucial.

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

Generic freeness IIJuly 30, 2009

Posted by Akhil Mathew in algebra, commutative algebra.
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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 1 Let ${A}$ be a Noetherian integral domain, ${M}$ a finitely generated ${A}$-module. Then there there exists ${f \in A - \{0\}}$ with ${M_f}$ a free ${A_f}$-module.

Generic freeness IJuly 29, 2009

Posted by Akhil Mathew in algebra, commutative algebra.
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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 ${A}$ is finitely generated. Similarly, a module is Noetherian if every submodule is finitely generated. I will summarize the basic facts below briefly.

DVRs IIAugust 30, 2009

Posted by Akhil Mathew in algebra, algebraic number theory, commutative algebra, number theory.
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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 ${R}$ is Noetherian, integrally closed, and has a unique nonzero prime ideal ${\mathfrak{m}}$, then ${R}$ is a DVR. Conversely, any DVR has those properties. (more…)

Completions of rings and modulesAugust 25, 2009

Posted by Akhil Mathew in algebra, commutative algebra.
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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 ${A}$ with a descending filtration (satisfying ${A_0 =A}$), which implies the ${A_i}$ are ideals; as we saw, the completion will also be a ring. Most often, there will be an ideal ${I}$ such that ${A_i = I^i}$, i.e. the filtration is ${I}$-adic. We have a completion functor from filtered rings to rings, sending ${A \rightarrow \hat{A}}$. Given a filtered ${A}$-module ${M}$, there is a completion ${\hat{M}}$, which is also a ${\hat{A}}$-module; this gives a functor from filtered ${A}$-modules to ${\hat{A}}$-modules. (more…)

Integrality, invariant theory for finite groups, and more tools for Noetherian testingAugust 11, 2009

Posted by Akhil Mathew in algebra, commutative algebra.
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In general, it is not true that a subring of a Noetherian ring is Noetherian. For instance, let ${A := k[X_1, X_2, \dots]}$ be the polynomial ring in infinitely many variables over a field ${k}$. Then ${A}$ is not Noetherian because of the ascending chain
$\displaystyle (X_0) \subset (X_0, X_1) \subset (X_0, X_1, X_2) \subset \dots.$
However, the quotient field of ${A}$ is Noetherian. This applies to any non-Noetherian integral domain.