= Projection formula
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= Projection formula for sheaves
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For a morphism of <ringed space>[ringed spaces], there is a natural morphism
$$
f_*\mathcal F\otimes_{\mathcal O_Y}\mathcal E\longrightarrow f_*\bigl(\mathcal F\otimes_{\mathcal O_X}f^*\mathcal E\bigr).
$$
It is an isomorphism when $\mathcal E$ is a <locally free sheaf> of finite rank, because the claim is local and then reduces to distributivity over a finite direct sum.
The same formula holds for higher direct images: $R^if_*\mathcal F\otimes\mathcal E\cong R^if_*(\mathcal F\otimes f^*\mathcal E)$ when $\mathcal E$ is locally free of finite rank.
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