projection formula

In algebraic geometry, the projection formula states the following:{{Hartshorne AG}}, Ch. III, Exercise 8.3}}{{citation|url=http://math.stanford.edu/~vakil/0708-216/216class38.pdf|title=Foundations of algebraic geometry class 38|first=Ravi|last=Vakil|publisher=Stanford University|year=2007–2008}}

For a morphism $f:X\backslash to\; Y$ of ringed spaces, an $\backslash mathcal\{O\}\_X$-module $\backslash mathcal\{F\}$ and a locally free $\backslash mathcal\{O\}\_Y$-module $\backslash mathcal\{E\}$ of finite rank, the natural maps of sheaves

:$R^i\; f\_*\; \backslash mathcal\{F\}\; \backslash otimes\; \backslash mathcal\{E\}\; \backslash to\; R^i\; f\_*\; (\backslash mathcal\{F\}\; \backslash otimes\; f^*\; \backslash mathcal\{E\})$

are isomorphisms.

There is yet another projection formula in the setting of étale cohomology.