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Let $A$ be a commutative noetherian ring, and let $M$ be a finitely generated projective $A$-module. It is well known and easy to prove that $A$ is locally free in the sense that for every $p \in\operatorname{Spec} A$, the module $M_p$ is a free $A_p$-module.

Is it true that projectives are also locally free in the following (more geometric?) sense:

There are elements $f_1,\dots,f_n \in A$ such that $(f_1,\dots,f_n) = 1$, and such that $M_{f_i}$ is a free $A_{f_i}$-module for all $1\le i \le n$.

Is this true? if so, can you provide a reference or explain how to prove it?


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2 Answers 2

up vote 7 down vote accepted

Yes, this is true. See this Math Overflow question for a precise statement and a reference to its proof in Bourbaki's Commutative Algebra.

This result is also stated in my commutative algebra notes, but the proof is not unfortunately not yet written up there. I certainly hope that this will be remedied soon though, as I will be teaching a course out of these notes starting on Monday. When the proof gets written, I will update this answer with a page number.

Added: Here is something in the MO answer that I decided was worth a comment here. For finitely generated modules, this stronger version of local freeness is actually equivalent to projectivity, whereas the weaker "pointwise local freeness" is subtly weaker in general.

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Thank you for the reference. For the intrested reader, I should mention that Bourbaki proves the following much stronger result: If $M$ is a finitely presented module and if for some prime $p$, the module $M_p$ is free, then there exists a neighborhood $D(f)$ of $p$ such that $M_f$ is free over $A_f$, so that "being free at a point" implies "being free at a neigborhood of a point". – the L Jan 8 '11 at 16:51
@anonymous: Dear anonymous, more generally, if $M,N$ are finitely presented modules and $M_p, N_p$ are isomorphic, then $M_f \simeq N_f$ for some $f \in A-p$. This is explained in more detail (and generality) in EGA IV-8, where it is proved that (in a certain sense) if $\{R_\alpha\}$ is an inductive system of rings, then the category of f.p. modules over $\varinjlim R_\alpha$ is the "colimit of the categories of modules over $R_\alpha$," which has many useful applications in reducing questions about arbitrary f.p. modules to questions about finitely generated modules over noetherian rings. – Akhil Mathew Jan 15 '11 at 18:40
(contd.) For instance, Grothendieck's generic flatness lemma (if $A$ noetherian integral domain, $B$ finite $A$-algebra, $M$ finite $B$-module, then there is $f \in A$ such that $M_f$ is flat over $A$) works for finitely presented modules because any such module must "descend" to a finitely presented module over a noetherian (e.g. finitely generated) subring. – Akhil Mathew Jan 15 '11 at 18:41

For future reference, I have written up a constructive and reasonably self-contained, if somewhat dense, proof (one page). The basic idea is to first verify that idempotent matrices over local rings are equivalent to diagonal matrices with entries $1$ and $0$, thus showing that finitely generated projective modules over local rings are free.

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Well, I think there are simpler proofs for finitely generated projective modules over local rings are free. Never mind, +1. – user26857 May 8 at 16:40
@user26857: I would be interested to hear them! Are they constructive in the technical sense that they avoid the use of the law of excluded middle? I only know of my write-up and the one in Mulvey's gem Intuitionistic algebra and representations of rings. – Ingo Blechschmidt May 8 at 16:54
@user26857: Thanks! The problem is that a proof beginning with "Consider a minimal system of generators" is not constructive, since constructively one cannot show that any inhabited subset of the natural numbers contains a minimal element. In some cases though, the problem can be avoided by a clever rewording; I didn't try that in this case hard enough yet. – Ingo Blechschmidt May 8 at 18:03

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