Lemma 20.7.3. Let $f : X \to Y$ be a morphism of ringed spaces. Let $\mathcal{F}$ be a $\mathcal{O}_ X$-module. The sheaves $R^ if_*\mathcal{F}$ are the sheaves associated to the presheaves

\[ V \longmapsto H^ i(f^{-1}(V), \mathcal{F}) \]

with restriction mappings as in Equation (20.7.1.1). There is a similar statement for $R^ if_*$ applied to a bounded below complex $\mathcal{F}^\bullet $.

**Proof.**
Let $\mathcal{F} \to \mathcal{I}^\bullet $ be an injective resolution. Then $R^ if_*\mathcal{F}$ is by definition the $i$th cohomology sheaf of the complex

\[ f_*\mathcal{I}^0 \to f_*\mathcal{I}^1 \to f_*\mathcal{I}^2 \to \ldots \]

By definition of the abelian category structure on $\mathcal{O}_ Y$-modules this cohomology sheaf is the sheaf associated to the presheaf

\[ V \longmapsto \frac{\mathop{\mathrm{Ker}}(f_*\mathcal{I}^ i(V) \to f_*\mathcal{I}^{i + 1}(V))}{\mathop{\mathrm{Im}}(f_*\mathcal{I}^{i - 1}(V) \to f_*\mathcal{I}^ i(V))} \]

and this is obviously equal to

\[ \frac{\mathop{\mathrm{Ker}}(\mathcal{I}^ i(f^{-1}(V)) \to \mathcal{I}^{i + 1}(f^{-1}(V)))}{\mathop{\mathrm{Im}}(\mathcal{I}^{i - 1}(f^{-1}(V)) \to \mathcal{I}^ i(f^{-1}(V)))} \]

which is equal to $H^ i(f^{-1}(V), \mathcal{F})$ and we win.
$\square$

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