next up previous
Next: About this document ...

Algebraic geometry and Ring theory

Yoshifumi Tsuchimoto

\fbox{curves (over $\mathbb {C}$)}

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 % latex2html id marker 708
$ n_i\geq 0$ for all $ i$ . For any invertible sheaf $ \mathcal F$ over $ C$ , we have a exact sequence

% latex2html id marker 716
$\displaystyle 0 \to \mathcal F \to \mathcal F(D) \to \mathcal F(D)/\mathcal F \to 0.
\quad :$exact

We have thus the associated long exact sequence on cohomology:

    0 $\displaystyle \to H^0( \mathcal F) \to H^0(\mathcal F(D)) \to H^0( \mathcal F(D)/\mathcal F)$
    % latex2html id marker 718
$\displaystyle \quad$ $\displaystyle \to H^1( \mathcal F) \to H^1(\mathcal F(D)) \to 0 .$

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

$\displaystyle \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$ :

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

THEOREM 10.2 (Serre duality)  

$\displaystyle 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

$\displaystyle 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

$\displaystyle \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

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

which gives rise to the Serre duality.

next up previous
Next: About this document ...