## 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…)

## How to tell if a ring is NoetherianAugust 9, 2009

Posted by Akhil Mathew in algebra, commutative algebra.
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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.

## 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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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.