Is finitely generated projective module M of rank one over regular commutative notherian ring free? Bass (Illinois Math J, 1963) showed that in case M is nonfinitely generated, it is free. I am wondering about finitely generated case.
Thanks!
Is finitely generated projective module M of rank one over regular commutative notherian ring free? Bass (Illinois Math J, 1963) showed that in case M is nonfinitely generated, it is free. I am wondering about finitely generated case.
Thanks!
No. Take the case of a Dedekind domain $A$ that it is not a principal ideal domain. It is a regular commutative notherian ring (of Krull dimension 1). The group of classes $\mathrm{Pic}(A)$ is not trivial, otherwise it would be a p.i.d. Therefore, there is a rank one projective module, in this case a fractional ideal (i.e. an $A$-submodule of its quotient field), that it's not free.