I am stuck on the following exercise in Hartshorne (Chapter III, 8.1):
$\hspace{3mm} $ "Let $f:X \to Y$ be a continuous map of topological spaces. Let $\mathcal{F}$ be a sheaf of abelian groups on $X$ and assume that $R^if_*(\mathcal{F})=0$ for all $i>0.$ Show that there are natural isomorphisms, for each $i \geq 0,$ $$H^i(X,\mathcal{F}) = H^i(Y, f_*\mathcal{F})."$$
Idea: It seems like Hartshorne would like me to take an injective resolution of $\mathcal{F},$ apply $f_*,$ and argue that this gives me an injective resolution of $f_*\mathcal{F}.$
The vanishing of $R^if_*\mathcal{F}$ ensures that $f_*$ is an exact functor and hence the cohomology groups will agree. The only problem is that I don't see why $f_*$ preserves "invectiveness" of my resolution.
$f_*$ is right adjoint to an exact functor ($f^{-1}$, note that you compute cohomology in the category of sheaves of abelian groups), hence it preserves injectives.