Algebraic geometry and Ring theory

Yoshifumi Tsuchimoto

Let $C$ be a curve over $\mathbb{C}$ . A divisor $D$ is a formal finite sum $\sum n_i P_i$ of points $P_i$ on the curve $C$ . For any such divisor, we may consider a sheaf $\mathcal O(D)$ . We call the sum $\sum_i n_i$ the degree of $D$ . It is also refered to as the degree of $\mathcal O(D)$ .

An $\mathcal O$ -module $\mathcal F$ on $C$ is called invertible if it is locally isomorphic to the structure sheaf $\mathcal O$ . Any invertible sheaf is actually isomorphic to a sheaf $\mathcal O(D)$ for some divisor $D$ .

A divisor $D=\sum n_i P_i$ is called effective if $n_i\geq 0$ for all $i$ . For any invertible sheaf $\mathcal F$ over $C$ , we have a exact sequence

$$0 \to \mathcal F \to \mathcal F(D) \to \mathcal F(D)/\mathcal F \to 0. \quad :$$exact

We have thus the associated long exact sequence on cohomology:

$$\to H^0( \mathcal F) \to H^0(\mathcal F(D)) \to H^0( \mathcal F(D)/\mathcal F)$$ 0

$$\quad \to H^1( \mathcal F) \to H^1(\mathcal F(D)) \to 0 .$$

We should also mention the genus $g(C)$ of the curve. It is topologically the ``number of holes'' of the surface $C(\mathbb{C})$ .

THEOREM 10.1 (Riemann-Roch)   Let $C$ be a non-singular projective curve over $\mathbb{C}$ . For any invertible sheaf $\mathcal F$ on $C$ , we have

$$\dim H^0 (\mathcal F) -\dim H^1 (\mathcal F) = 1-g+ \deg(\mathcal F)$$

We have an important sheaf $\omega=\Omega^1$ on $C$ . For any $\mathcal O$ -module $\mathcal F$ on $C$ , we may consider the sheaf of $\mathcal F$ -valued $1$ -forms $\mathcal F\otimes \omega$ .

We also note that for any invertible sheaf $\mathcal F$ on $C$ , we have its dual $\mathcal F^\vee$ :

$$\mathcal O(D)^\vee=\mathcal O(-D).$$

THEOREM 10.2 (Serre duality)  

$$H^i(\mathcal F)^{\vee}\cong H^{1-i}(\mathcal F^{\vee}\otimes \omega)$$

We may understand the situation of the two theorems above by using a ``formal version of the Cech cohomology''. Namely, for any point $P$ of $C$ with a local coordinate $t$ such that $t(P)=0$ , We define formal- $\operatorname{Spec}(\mathbb{C}[[t]])$ as a formal ``neighbourhood'' of $P$ . $C$ may then be covered as

$$C= C\setminus \{P_1,\dots,P_n\} \cup U= \dot C\cup U$$

where $U$ is the union of such formal ``neighbourhoods'' of $P_i$ 's. One may then mimic the Cech cohomology and obtain a Cech complex. Namely, for any $\mathcal O$ -module $\mathcal F$ on $C$ , we have a complex

$$\mathcal F(\dot C) \oplus \mathcal F(U) \to \mathcal F(\dot U)$$

whose cohomologies are isomorphic to $H^\bullet (X; \mathcal F)$ . If $\mathcal F$ is invertible, we have also a residue pairing

$$F(\dot U) \times \mathcal F^\vee (\dot U) \to \mathbb{C}$$

which gives rise to the Serre duality.