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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…)

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 II July 30, 2009

Posted by Akhil Mathew in algebra, commutative algebra.
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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 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.

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Generic freeness I July 29, 2009

Posted by Akhil Mathew in algebra, commutative algebra.
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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 {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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