POTW Projection Formula for Ringed Spaces

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Show that if $f : X \to Y$ is a morphism of ringed spaces, $\mathscr{F}$ is an $\mathcal{O}_X$-module and $\mathscr{E}$ is a locally free $\mathcal{O}_Y$-module of finite rank, then for all $p \ge 0$, there is an isomorphism $$R^pf_*(\mathscr{F}\otimes_{\mathcal{O}_X} f^*\mathscr{E}) \approx R^pf_*(\mathscr{F}) \otimes_{\mathcal{O}_Y} \mathscr{E}$$

 

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Spoiler

When $p = 0$, the result follows from the projection formula $f_*(\mathscr{F}\otimes f^*\mathscr{E}) \approx f_*(\mathscr{F}) \otimes \mathscr{E}$, which may be obtained by reducing to the case $\mathscr{E} = \mathscr{O}_Y^n$. Both sides of the proposed isomorphism are delta functors (applied to $\mathscr{F}$) that are effaceable. Indeed, if $\mathscr{F}$ is injective, then since $f_*$ has an exact left adjoint $f^*$, $f_*(\mathscr{F})$ is injective. Also, by exactness of $\otimes_{\mathscr{O}_X} f^*\mathscr{E}$, the tensor sheaf $\mathscr{F} \otimes_{\mathscr{O}_X} f^*\mathscr{E}$ is injective and hence $f_*(\mathscr{F} \otimes_{\mathscr{O}_X} f^*\mathscr{E})$ is injective. We then deduce $R^pf_*(\mathscr{F}) = 0 = R^p f_*(\mathscr{F} \otimes_{\mathscr{O}_X} \mathscr{E})$, so that the delta functors are effaceable. It follows that these delta functors are universal. Since they agree at $p = 0$, they are naturally isomorphic.