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.

See also

• Integration along fibers § Projection formula

References

Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157