Projection formula

In algebraic geometry, the projection formula states the following:[1][2]

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.

See also

• Integration along fibers#Projection formula

References

1. Hartshorne 1977, Ch III, Exercise 8.3
2. http://math.stanford.edu/~vakil/0708-216/216class38.pdf

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