# Algebraic geometry and Ring theory

Yoshifumi Tsuchimoto

Note that a sequence

of sheaves of abelian groups is exact if and only if it is exact stalkwise.

PROPOSITION 09.1   Let

be an exact sequence of sheaves of abelian groups on a topological space . Then:
1. For any open subset of , the corresponding sequence

of sections is exact.
2. may not be surjective in general.

In a language of category theory, the global section function

(Sheaf of abelian groups on $X$)   (Ab)

is a left exact functior. (But not exact.) To treat it, we employ derived functors.

LEMMA 09.2
1. For any ring , if an -module is injective, then the associated sheaf on is injective.
2. The category ( -modules) has enough injectives.
3. For any scheme with an affine open covering , for any -quasi coherent sheaf on , we have:

: injective injective for any .

DEFINITION 09.3

THEOREM 09.4 (Serre)   For any affine scheme and for any quasi coherent -module on , we have

for

PROPOSITION 09.5   Let be a scheme with an affine open covering . Then for any quasi coherent -module on , The cohomology may be computed by usin the Cech cohomology with respect to .