A prime ideal criterion for being Noetherian August 13, 2009Posted by Akhil Mathew in algebra, commutative algebra.
Tags: commutative algebra, Noetherian rings, prime ideals
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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 1If are ideals of a commutative ring , then we define
How to tell if a ring is Noetherian August 9, 2009Posted by Akhil Mathew in algebra, commutative algebra.
Tags: algebra, commutative algebra, Hilbert basis theorem, localization, Noetherian rings
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.
Generic freeness II July 30, 2009Posted by Akhil Mathew in algebra, commutative algebra.
Tags: algebra, commutative algebra, generic freeness, localization
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 be a Noetherian integral domain, a finitely generated -module. Then there there exists with a free -module.
Generic freeness I July 29, 2009Posted by Akhil Mathew in algebra, commutative algebra.
Tags: commutative algebra, devissage, generic freeness, localization, Noetherian rings, schemes
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.