Misplaced Pages

In algebraic geometry, the projection formula states the following:

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, 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

This algebraic geometry–related article is a stub. You can help Misplaced Pages by expanding it.

• v
• t
• e

• Theorems in algebraic geometry
• Algebraic geometry stubs