This is my first post, so I’ll just write about something I’ve been thinking about as a pilot for my blog. Let’s talk about the relationship between (Cartier) divisors and the Picard group of line bundles up to isomorphism.
Let be an integral scheme with structure sheaf ; for all intents and purposes, one can imagine to be a variety in this post. Let be the sheaf of rational functions on , and let be the subsheaf of invertible rational functions. By examining some sheaf-theoretic properties of , we will show that the divisor class group is isomorphic to .
First, we have an exact sequence of sheaves of abelian groups on : which induces a long exact sequence on cohomology. Looking at a portion of this sequence, we have Note that the cokernel of the map on global sections is by definition the group of Cartier divisors on , while , as is well-known.
What about the last term? It turns out that the sheaf is flasque, i.e. for any open subset , the map is surjective. An important consequence of this is – all higher cohomology vanishes. Now, is flasque (hence, acyclic) because , is, implying that !
We conclude that , which is to say that line bundles (up to isomorphism) correspond to Cartier divisors (up to linear equivalence). Interpretations and consequences of this isomorphism to come.