In algebraic geometry, the projection formula states the following:

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

R i f ∗ F ⊗ E → R i f ∗ ( F ⊗ f ∗ E ) {\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.