Projection formula

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

$R^i{f}_{*}{ℱ}\otimes {ℰ}\to R^i{f}_{*}({ℱ}\otimes f^{*}{ℰ})$

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