Projection formula

In algebraic geometry, the projection formula states the following: For a morphism [math]\displaystyle{ f:X\to Y }[/math] of ringed spaces, an [math]\displaystyle{ \mathcal{O}_X }[/math]-module [math]\displaystyle{ \mathcal{F} }[/math] and a locally free [math]\displaystyle{ \mathcal{O}_Y }[/math]-module [math]\displaystyle{ \mathcal{E} }[/math] of finite rank, the natural maps of sheaves

[math]\displaystyle{ R^i f_* \mathcal{F} \otimes \mathcal{E} \to R^i f_* (\mathcal{F} \otimes f^* \mathcal{E}) }[/math]

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, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9