# Invertible sheaves on affine varieties

Let $X:=\rm{Spec}(A)$ be an integral, noetherian, affine variety, and let $L$ be an invertible sheaf on $X$, I try to find an example where $L$ is not isomorphic to the structure sheaf of $X$.

In the language of commutative algebra, this simply means to find an $A$-module $M$ which is locally free, but $M$ is not isomorphic to $A$ as a module.

Because when $A$ is UFD, the Weil class group is trivial, such example should not exists in this case. Also, I know for affine toric variety, the Picard group is also trivial.

• To add to this, assuming that $A$ is normal, then such a line bundle will exist if and only if $A$ is not a UFD. Since, in this case $\text{Pic}(A)=\text{Cl}(A)$, and $\text{Cl}(A)=0$ if and only if $A$ is a UFD> – Alex Youcis Jul 25 '14 at 1:24
• @LiYutong I was talking about for $A$ a dimension one domain, and Noetherian, as in Bruno's example. In that case, normality is the same as regularity. – Alex Youcis Jul 26 '14 at 1:22