Lemma 20.13.3. Let X be a ringed space. Let \mathcal{F} be an \mathcal{O}_ X-module.
The cohomology groups H^ i(U, \mathcal{F}) for U \subset X open of \mathcal{F} computed as an \mathcal{O}_ X-module, or computed as an abelian sheaf are identical.
Let f : X \to Y be a morphism of ringed spaces. The higher direct images R^ if_*\mathcal{F} of \mathcal{F} computed as an \mathcal{O}_ X-module, or computed as an abelian sheaf are identical.
There are similar statements in the case of bounded below complexes of \mathcal{O}_ X-modules.
Proof.
Consider the morphism of ringed spaces (X, \mathcal{O}_ X) \to (X, \underline{\mathbf{Z}}_ X) given by the identity on the underlying topological space and by the unique map of sheaves of rings \underline{\mathbf{Z}}_ X \to \mathcal{O}_ X. Let \mathcal{F} be an \mathcal{O}_ X-module. Denote \mathcal{F}_{ab} the same sheaf seen as an \underline{\mathbf{Z}}_ X-module, i.e., seen as a sheaf of abelian groups. Let \mathcal{F} \to \mathcal{I}^\bullet be an injective resolution. By Remark 20.13.2 we see that \Gamma (X, \mathcal{I}^\bullet ) computes both R\Gamma (X, \mathcal{F}) and R\Gamma (X, \mathcal{F}_{ab}). This proves (1).
To prove (2) we use (1) and Lemma 20.7.3. The result follows immediately.
\square
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