Let be a curve over . A divisor is a formal finite sum of points on the curve . For any such divisor, we may consider a sheaf . We call the sum the degree of . It is also refered to as the degree of .
An -module on is called invertible if it is locally isomorphic to the structure sheaf . Any invertible sheaf is actually isomorphic to a sheaf for some divisor .
A divisor is called effective if for all . For any invertible sheaf over , we have a exact sequence
We have thus the associated long exact sequence on cohomology:
We should also mention the genus of the curve. It is topologically the ``number of holes'' of the surface .
We have an important sheaf on . For any -module on , we may consider the sheaf of -valued -forms .
We also note that for any invertible sheaf on , we have its dual :
We may understand the situation of the two theorems above by using a ``formal version of the Cech cohomology''. Namely, for any point of with a local coordinate such that , We define formal- as a formal ``neighbourhood'' of . may then be covered as
where is the union of such formal ``neighbourhoods'' of 's. One may then mimic the Cech cohomology and obtain a Cech complex. Namely, for any -module on , we have a complex
whose cohomologies are isomorphic to . If is invertible, we have also a residue pairing
which gives rise to the Serre duality.