Line bundles vs. Cartier divisors on a non-integral scheme - MathOverflow

Line bundles vs. Cartier divisors on a non-integral scheme

2010-04-21T17:09:04Z

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?

Answer by VA for Line bundles vs. Cartier divisors on a non-integral scheme

2010-04-21T17:22:50Z

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.

Answer by Hailong Dao for Line bundles vs. Cartier divisors on a non-integral scheme

2010-04-21T17:42:37Z

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