Projection formula

In algebraic geometry, the projection formula states that,^[1]^[2] for a quasi-compact separated morphism of schemes $f:X\to Y$ , a quasi-coherent sheaf ${\mathcal {F}}$ on X, a locally free sheaf ${\mathcal {E}}$ on Y, 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 1977, Ch III, Exercise 8.3
↑ 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

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