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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?

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2 Answers

up vote 8 down vote accepted

An example is given in this note (it was credited to Kleiman).

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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.

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