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

For a morphism f : X → → Y {\displaystyle f:X\to Y} of ringed spaces, an O X {\displaystyle {\mathcal {O}}_{X}} -module F {\displaystyle {\mathcal {F}}} and a locally free O Y {\displaystyle {\mathcal {O}}_{Y}} -module E {\displaystyle {\mathcal {E}}} of finite rank, the natural maps of sheaves

R i f ∗ ∗ F ⊗ ⊗ E → → R i f ∗ ∗ ( F ⊗ ⊗ f ∗ ∗ E ) {\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

• References

• Integration along fibers § Projection formula

cite-note-11. ↑ citerefhartshorne1977Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157, Ch. III, Exercise 8.3}}

cite-note-22. ↑ citerefvakil2007-2008Vakil, Ravi (2007–2008), Foundations of algebraic geometry class 38 (PDF), Stanford University