# Algebraic geometry and Ring theory

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 .

THEOREM 10.1 (Riemann-Roch)   Let be a non-singular projective curve over . For any invertible sheaf on , we have

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 :

THEOREM 10.2 (Serre duality)

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.