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.


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.