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This is a followup to a previous question

http://mathoverflow.net/questions/13768/what-is-the-right-definition-of-the-picard-group-of-a-commutative-ring

where I was worried about the distinction between invertible modules and rank one projective modules over an arbitrary commutative ring. The answer is that I was worrying too much, because of the following theorem [Bourbaki, Commutative Algebra, Section II.5.2, Theorem 1]:

Let $R$ be a commutative ring and $M$ a finitely generated $R$-module. The following are equivalent:
(i) $M$ is projective.
(ii) $M$ is finitely presented and locally free in the weaker sense: $\forall \mathfrak{p} \in \operatorname{Spec}(R), \ M_{\mathfrak{p}} \cong R_{\mathfrak{p}}^{r(\mathfrak{p})}$.
(iii) $M$ is locally free in the weaker sense and its rank function $\mathfrak{p} \mapsto r(\mathfrak{p})$ is locally constant on $\operatorname{Spec}(R)$.
(iv) $M$ is locally free in the stronger sense: there exist $f_1,\ldots,f_n \in R$, generating the unit ideal, such that for each $i$, $M_{f_i}$ is a free $R_{f_i}$-module.
(v) For every maximal ideal $\mathfrak{m}$ of $R$, there exists $f \in R \setminus \mathfrak{m}$ such that $M_f$ is a free $R_f$-module.

Note that this

This answers my previous question, because the rank function of an invertible module is identically one.

In order to really feel like I understand what's going on here, I would like to see an example of a finitely generated locally free [in the weaker sense of (ii) above] module which is not projective. Thus $R$ must be non-Noetherian. The wikipedia article on projective modules contains some nice information, in particular sketching an example of such a module over a Boolean ring. For a Boolean ring though the localization at every prime ideal is simply $\mathbb{Z}/2\mathbb{Z}$, so it is not too surprising that there are more locally free modules than projectives.

I would like to see an example with $R$ an integral domain, if possible. It would be especially nice if you can give a reference to one of the standard texts on commutative algebra which contains such an example or at least a citation of such an example.

1

# A finitely generated, locally free module over a domain which is not projective?

This is a followup to a previous question

http://mathoverflow.net/questions/13768/what-is-the-right-definition-of-the-picard-group-of-a-commutative-ring

where I was worried about the distinction between invertible modules and rank one projective modules over an arbitrary commutative ring. The answer is that I was worrying too much, because of the following theorem [Bourbaki, Commutative Algebra, Section II.5.2, Theorem 1]:

Let $R$ be a commutative ring and $M$ a finitely generated $R$-module. The following are equivalent:
(i) $M$ is projective.
(ii) $M$ is finitely presented and locally free in the weaker sense: $\forall \mathfrak{p} \in \operatorname{Spec}(R), \ M_{\mathfrak{p}} \cong R_{\mathfrak{p}}^{r(\mathfrak{p})}$.
(iii) $M$ is locally free in the weaker sense and its rank function $\mathfrak{p} \mapsto r(\mathfrak{p})$ is locally constant on $\operatorname{Spec}(R)$.
(iv) $M$ is locally free in the stronger sense: there exist $f_1,\ldots,f_n \in R$, generating the unit ideal, such that for each $i$, $M_{f_i}$ is a free $R_{f_i}$-module.
(v) For every maximal ideal $\mathfrak{m}$ of $R$, there exists $f \in R \setminus \mathfrak{m}$ such that $M_f$ is a free $R_f$-module.

Note that this answers my previous question, because the rank function of an invertible module is identically one.

In order to really feel like I understand what's going on here, I would like to see an example of a finitely generated locally free [in the weaker sense of (ii) above] module which is not projective. Thus $R$ must be non-Noetherian. The wikipedia article on projective modules contains some nice information, in particular sketching an example of such a module over a Boolean ring. For a Boolean ring though the localization at every prime ideal is simply $\mathbb{Z}/2\mathbb{Z}$, so it is not surprising that there are more locally free modules than projectives.

I would like to see an example with $R$ an integral domain, if possible. It would be especially nice if you can give a reference to one of the standard texts on commutative algebra which contains such an example or at least a citation of such an example.