**Yoshifumi Tsuchimoto**

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

exact

We have thus the associated long exact sequence on cohomology:

0 | ||

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.