It is wellknown 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?
Take the 2minute tour
×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.
An example is given in this note (it was credited to Kleiman). 


From the exact sequence $$1\to O^\*\to K^\*\to K^\* / O^\*\to 1$$ you see that, for as long as $H^1(K^\*)=0$, the map from $H^0( K^\*/ O^\*)$ (i.e. Cartier divisors) to $H^1( O^\*)$ (i.e. line bundles) is surjective. On a Noetherian scheme without embedded primes (for example, reduced), $\mathcal K^\*$ is the direct sum of several constant sheaves on the irreducible components, so it has trivial $H^1$. So the example would have to be a scheme with embedded primes with a tricky nonconstant $K^\*$ (the sheaf of nonzero divisors). I've seen it but can't remember right now. So this is just some general observations to narrow the search. 

