TITLE: Line bundles vs. Cartier divisors on a non-integral scheme
QUESTION [17 upvotes]: It is well-known that if $X$ is an integral scheme, then there is an isomorphism $CaCl(X)\to Pic(X)$ taking $[D]$ to $[\mathcal{O}_X(D)]$. Does anyone know any simple examples where the above map fails to be surjective, i.e., a line bundle on a scheme $X$, not isomorphic to $\mathcal{O}_X(D)$ for any Cartier divisor D?

REPLY [12 votes]: An example is given in this note (it was credited to Kleiman).