Projection formula

In algebraic geometry, the projection formula states the following:

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

• Integration along fibers § Projection formula