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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

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

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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 is finitely generated. Similarly, a module is **Noetherian** if every submodule is finitely generated. I will summarize the basic facts below briefly.

Proposition 1A module is Noetherian iff every ascending chainstabilizes.

*Proof:* Indeed, we can write ; then is finitely generated, and each generator lies in some ; thus for some large , contains all the generators of and hence . The other implication is similar.

Proposition 2Submodules and quotient modules of Noetherian modules are Noetherian.

*Proof:* In the quotient case, take inverse images in the initial module. In the submodule case, take images in the bigger initial module.

Proposition 3Given an exact sequence , if are Noetherian, then is Noetherian.

*Proof:* Given an ascending series , the ascending series and stabilize eventually. Now use:

Lemma 4Given an exact sequence as above, if , and , then .

Indeed, given , we can write for some ; then , so , i.e. .

The following is the form in which Noetherian hypotheses are frequently used:

Proposition 5Let be Noetherian; then any finitely generated -module is Noetherian.

*Proof:* By induction on the rank and Proposition 3, any finitely generated *free* -module is Noetherian; now any finitely generated -module is a quotient of such.

There are more interesting facts about Noetherian rings in general, which I will get to in another post. What we need here is the following *filtration or dévissage lemma*:

Proposition 6 (Dévissage)Let be a finitely generated module over a Noetherian ring . Then there is a filtrationsuch that each quotient for prime ideals .

*Proof:* The proof illustrates a useful technique that one often uses with Noetherian rings, an induction of sorts.

Consider the set defined as

Then has a *maximal* element . (Otherwise, we’d have an infinite strictly ascending collection of submodules.) There is thus such a filtration of . If we can find a submodule (coming from some ) with for some prime , then we would have by piecing together the filtration of with . But is bigger than , contradiction!

So we will be done if we show:

Lemma 7Let be a module over a Noetherian ring . Then contains a submodule isomorphic to , for prime.

To prove this, we need to find an such that is prime. Choose such that is maximal; we may do this since is Noetherian. Then, for any , we have

I claim that is prime. So suppose and yet . Then . Thus , contradiction.

**Generic Freeness **

To state this result, I’ll need the notion of *localization*, which I don’t have time to define here yet; fortunately, Rigorous Trivialties has already covered it well.

The result is as follows:

Theorem 8Let be a Noetherian integral domain, a finitely generated -module. Then there there exists with a free -module.

The result implies the corresponding fact over schemes:

Corollary 9Let be a Noetherian integral scheme. Then if is a coherent sheaf on , there is an open dense such that the restriction is free.

*Proof:* Take an open affine ; since is irreducible, is dense. Say , where is a Noetherian domain. Then is the sheaf associated to a finitely generated -module . We can find with free over ; then one takes .

There is a more general formulation of this generic freeness lemma, which I’ll talk about next (as well as the proof, which goes by dévissage.)

Proposition 6 is so useful (especially in the local case). I used it many times last quarter.

Yes, though it took me a while to actually understand the significance, even though it was used quite a few times in commutative algebra books, and probably will be referred to again on this blog.

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