Projection formula

In algebraic geometry, the projection formula states the following:

For a morphism $$f:X\to Y$$ of ringed spaces, an $$\mathcal{O}_X$$-module $$\mathcal{F}$$ and a locally free $$\mathcal{O}_Y$$-module $$\mathcal{E}$$ of finite rank, the natural maps of sheaves


 * $$R^i f_* \mathcal{F} \otimes \mathcal{E} \to R^i f_* (\mathcal{F} \otimes f^* \mathcal{E})$$

are isomorphisms.

There is yet another projection formula in the setting of étale cohomology.