Suppose $\mathcal{L}$ is a line bundle over a smooth projective variety $V$, and suppose $\mathcal{L}$ is trivial on the complement of a prime divisor $X$. Can I conclude $\mathcal{L}$ belongs to the subgroup of line bundles corresponding to the subgroup of Weil divisors generated by $X$? Does this generalize to $\mathcal{L}$ being trivial on a generic Zariski open set?
If the answer were yes, I would really hope there was a proof that did not make use of Cartier divisors, as I am not too familiar with them.