# Projection formula

In algebraic geometry, the projection formula states that,[1][2] for a quasi-compact separated morphism of schemes ${\displaystyle f:X\to Y}$, a quasi-coherent sheaf ${\displaystyle {\mathcal {F}}}$ on X, a locally free sheaf ${\displaystyle {\mathcal {E}}}$ on Y, the natural maps of sheaves

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

are isomorphisms.

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