Projection formula

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

  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


  • v
  • t
  • e