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I've been reading Ravi Vakil's Albegraic Geometry notes , and he has you use the FHHF Theorem to prove the projection formula for $R^i\pi_i$ (ex 20.7.E in the Jan. 2011 version), when we are dealing with schemes and sheaves of modules.

The FHHF theorem says that if you have a functor $F: A \rightarrow B$ and a complex $C^\bullet$ in $A$, and $H$ is right exact, you have a map $FH(C^\bullet) \rightarrow HF(C^{\bullet})$ (where $H$ means take cohomology). If instead $H$ is left exact, the map goes the other way, and if it is exact, there is an isomorphism.

I'm not familiar with the other examples, but I would suspect that they would be use the FHHF theorem as well.

Based on the statements of these exercises in the notes, I think you need more hypotheses to get an isomorphism in example 1, rather than just a map one way. In the scheme and sheaves of modules case, having $\mathcal G$ locally free will do it.
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I've been reading Ravi Vakil's Albegraic Geometry notes , and he has you use the FHHF Theorem to prove the projection formula for $R^i\pi_i$ (ex 20.7.E in the Jan. 2011 version), when we are dealing with schemes and sheaves of modules.

The FHHF theorem says that if you have a functor $F: A \rightarrow B$ and a complex $C^\bullet$ in $A$, and $H$ is right exact, you have a map $FH(C^\bullet) \rightarrow HF(C^{\bullet})$ (where $H$ means take cohomology). If instead $H$ is left exact, the map goes the other way, and if it is exact, there is an isomorphism.

I'm not familiar with the other examples, but I would suspect that they would be use the FHHF theorem as well.

Based on the statements of these exercises in the notes, I think you need more hypotheses to get an isomorphism in example 1, rather than just a map one way. In the scheme and sheaves of modules case, having $\mathcal G$ locally free will do it.