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.
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
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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