The Stacks project

Lemma 20.13.3. Let $X$ be a ringed space. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module.

  1. 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.

  2. 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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