1. The idea of defining sheaf-cohomology of a ringed space
Let X be a topological space and Sh(X) be the category of abelian sheaves (the sheaf of abelian groups). One can show that the category Sh(X) is an abelian category with enough injectives and the global section functor Γ : Sh(X) → Ab is a left exact functor.
Here Ab is the category of abelian groups. The i-th cohomology functor Hi from Sh(X) to Ab is defined to be the i-th right derived functor RiΓ of the global section functor.
The category ModOX of sheaves of OX-modules on a ringed space (X, OX) is also an abelian category with enough injectives. The global section functor ΓOX from ModOX to Ab is also a left exact functor. We can also consider its right derived functors RiΓOX. Notice that if F is a sheave of OX modules on (X, OX), F can be also considered as an abelian sheaf on the underlying space X of the ringed space (X, OX). Therefore Hi(F ) is defined.
To compute RiΓOX(F ), we choose an injective resolutions (J•) of F in ModOX and RiΓOX(F ) = hi(ΓOX(J•)).
Notice that an injective object in ModOX may not be an injective object in Sh(X). For- tunately, an injective object ModOX is a Γ-acyclic object in Sh(X). Hence an injective resolution of F in ModOX is a Γ-acyclic resolution of F (considered as an abelian sheaf on X) in Sh(X).
Theorem 1.1. Let A be an abelian category with enough injectives and F be a right derived functor from A to an abelian category B. Suppoose that J• is a F -acyclic resolution of an object A of A. Then
RiF (A) ∼= hi(F (J•)).
This theorem tells us that the object RiF (A) can be computed using the complex F (J•) given by the F -acyclic resolution J• of A. This implies that
RiΓOX(F ) ∼= hi(Γ(J•)).
This allows us to define the sheaf cohomology of a sheaf of OX-modules on a ringed space X by considering the derived functors of the global section functor Γ on the category of abelian sheaves Sh(X) on X.
In order to define the sheaf cohomology on a ringed space, we need to prove the followings:
(1) The categories ModOX and Sh(X) are abelian categories with enough injectives.
(2) The functors Γ and ΓOX are left-exact functors.
(3) An injective object in ModOX is a Γ-acyclic object in Sh(X).
(4) Finally, the theorem 1.1.
1
