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What is the simplest example of a domain $R$ which is regular (in particular Noetherian) and factorial which admits a finitely generated projective module that is not free?

In fact I'll be at least somewhat happy with any example, since I can't think of one at the moment.

Some brief comments: $R$ needs to have Krull dimension greater than one or else it is a PID. The module in question needs to have rank greater than one, because the hypotheses force the Picard group to be equal to the divisor class group and the divisor class group to be trivial. And famously, by work of Quillen and Suslin, one cannot take $R$ to be a polynomial ring over a field. Oh yes, and of course $R$ can't be local (or even semilocal, I suppose). I'm already out of ideas...

P.S.: If you can get an easier example by removing the hypothesis of finite generation, I'd be interested in that as well.

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What is the simplest example of a domain $R$ which is regular (in particular Noetherian) and factorial which admits a finitely generated projective module that is not free?

In fact I'll be at least somewhat happy with any example, since I can't think of one at the moment.

Some brief comments: $R$ needs to have Krull dimension greater than one or else it is a PID. The module in question needs to have rank greater than one, because the hypotheses force the Picard group to be equal to the divisor class group and the divisor class group to trivial. And famously, by work of Quillen and Suslin, one cannot take $R$ to be a polynomial ring over a field. Oh yes, and of course $R$ can't be local (or even semilocal, I suppose). I'm already out of ideas...

P.S.: If you can get an easier example by removing the hypothesis of finite generation, I'd be interested in that as well.

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# Nonfree projective module over a regular UFD?

What is the simplest example of a domain $R$ which is regular (in particular Noetherian) and factorial which admits a finitely generated projective module that is not free?

In fact I'll be at least somewhat happy with any example, since I can't think of one at the moment.

Some brief comments: $R$ needs to have Krull dimension greater than one or else it is a PID. The module in question needs to have rank greater than one, because the hypotheses force the Picard group to be equal to the divisor class group and the divisor class group to trivial. And famously, by work of Quillen and Suslin, one cannot take $R$ to be a polynomial ring over a field. Oh yes, and of course $R$ can't be local (or even semilocal, I suppose). I'm already out of ideas...