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Dedekind domains August 31, 2009

Posted by Akhil Mathew in algebra, algebraic number theory, commutative algebra, number theory.
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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 {\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…)

DVRs II August 30, 2009

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

The finite presentation trick and completions August 27, 2009

Posted by Akhil Mathew in algebra, category theory, commutative algebra.
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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 {A} be a Noetherian ring, and {I} an ideal. If we take all completions with respect to the {I}-adic topology,      

 

\displaystyle \hat{M} = \hat{A} \otimes_A M

for any f.g. {A}-module {M}.   (more…)

Completions of rings and modules August 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…)

Gradings, filtrations, and gr August 18, 2009

Posted by Akhil Mathew in algebra, commutative algebra.
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5 comments

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.

 Gradings

 The idea of a graded ring is necessary to define projective space. 

Definition 1    A graded ring is ring {A} together with a decomposition

 \displaystyle A = \bigoplus_{n=-\infty}^\infty A_n \ \mathrm{as \ abelian \ groups},

such that {A_i \cdot A_j \subset A_{i+j}}. The set {A_i} is said to consist of homogeneous elements of degree {i}. (more…)

A prime ideal criterion for being Noetherian August 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

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

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

There are special cases where we can conclude a subring of a Noetherian ring is Noetherian.

(more…)

How to tell if a ring is Noetherian August 9, 2009

Posted by Akhil Mathew in algebra, commutative algebra.
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7 comments

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 {A} be a Noetherian ring. Then the polynomial ring {A[X]} is also Noetherian.

  (more…)

Generic freeness I July 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.

(more…)

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