Weil divisors v.s. Cartier divisors
Let’s continue the discussion of the last post. In this post, we shall establish a correspondence between Weil divisors and Cartier divisors for special schemes.
First let’s repeat what this correspondence is:
Suppose that is an integral, separated noetherian scheme such that all of its local rings are UFDs. Then the group of Weil divisors on is isomorphic to the group of Cartier divisors. Besides, the principal Weil divisors correspond to the principal Cartier divisors.
This correspondence and the following proof are taken from Hartshorne’s book, ‘algebraic geometry’.
We shall not give a complete and rigorous proof to this statement. We only give a rough idea of what this correspondence is like. First note that is integral(reduced and irreducible), thus the sheaf is the constant sheaf of on where is the function field on , which is also the local ring with the generic point of . So, a Cartier divisor is given by a family where is an open cover of and . We now construct a Weil divisor: for each prime divisor of , by definition, this means that is an integral closed subscheme of , now for each , we set the coefficient of to be . Note that for any two such that , then by definition, , that is to say, is invertible on , thus , so the coefficient of is well-defined. And we set the Weil divisor to be . Note here we should guarantee that this is a finite sum. This can be deduced easily from the assumption that is noetherian(we will say about this later).
Now conversely, if is a Weil divisor on , then for each point , gives a divisor on . Now by assumption, is a UFD, we can show that is a principal divisor(we shall not prove this point for the present), that is, for some . This means that there is an open subset of such that and agree on . In this way, varying , we get an open cover of with such that and agree on . And we set to be the corresponding Cartier divisor. So, we finish the construction of the correspondence between Weil divisors and Cartier divisors.
It is not too hard to see that these two construction are inverse one to another.
Relation to invertible sheaves
Next we want to relate divisors to invertible sheaves.
Suppose that is a scheme and its structure sheaf. Then we define an invertible sheaf to be an -module locally free of rank .
Now let be a Cartier divisor on given by , then we define the sheaf to be the submodule of generated by on each for all . We call to be the sheaf associated to .
The important result is that, this gives a one-to-one correspondence between the set of Cartier divisors and the set of invertible submodules of . This can be seen like this: given a invertible submodule of , for an open subset of , is a free module of of rank , which is also a submodule of . So, we take a generator of (it is not zero since is of rank , not ), then we get an open covering of such that on each there is an . We can verify that these can glue together, and thus we can set to be the Cartier divisor. It can be verified that this process is converse to the above one, thus we showed that there is a correspondence between the set of Cartier divisors and the set of invertible submodules of .
In fact, this is more than a bijection, this is a group homomorphism. Recall that we give a group structure to the set of invertible sheaves on by tensor product: suppose that are two invertible sheaves on , then by definition, is again an invertible sheaf on . Besides, is also an invertible sheaf on , and . We call this group the Picard group of ,
and denote it by . So the set of invertible submodules of is a subgroup of .
It is not hard to see that , so this gives a group homomorphism between the group of Cartier divisors and the group of invertible submodules of .
In general, the group of Cartier divisors is not equal to the Picard group of . Yet, under some mild condition, we can show that, these two groups are isomorphic under the above morphism:
Suppose that is an integral scheme, then the group of Cartier divisors on is isomorphic to .
We will establish this isomorphism in the next post.