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

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

Tags: algebra, commutative algebra, Hilbert basis theorem, localization, Noetherian rings

7 comments

Tags: algebra, commutative algebra, Hilbert basis theorem, localization, Noetherian rings

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 be a Noetherian ring. Then the polynomial ring is also Noetherian.

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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

11 comments

Tags: algebra, commutative algebra, generic freeness, localization

11 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

4 comments

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

4 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.