Projection formula

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

For a morphism $f:X\to Y$ of ringed spaces, an ${\mathcal {O}}_{X}$ -module ${\mathcal {F}}$ and a locally free ${\mathcal {O}}_{Y}$ -module ${\mathcal {E}}$ of finite rank, 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.

References

^ Hartshorne, 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}}
^ Vakil, Ravi (2007–2008), Foundations of algebraic geometry class 38 (PDF), Stanford University

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[1] Hartshorne, 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}}

[2] Vakil, Ravi (2007–2008), Foundations of algebraic geometry class 38 (PDF), Stanford University

[1]

[2]

Projection formula

See also

References