Direct image functor
|Image functors for sheaves|
|direct image f∗|
|inverse image f∗|
|direct image with compact support f!|
|exceptional inverse image Rf!|
|Base change theorems|
sends a sheaf F on X to its direct image presheaf, which is defined on open subsets U of Y by
which turns out to be a sheaf on Y, also called the pushforward sheaf.
This assignment is functorial, i.e. a morphism of sheaves φ: F → G on X gives rise to a morphism of sheaves f∗(φ): f∗(F) → f∗(G) on Y.
If Y is a point, then the direct image equals the global sections functor. Let f: X → Y be a continuous map of topological spaces or a morphism of schemes. Then the exceptional inverse image is a functor f!: D(Y) → D(X).
Higher direct images
One can show that there is a similar expression as above for higher direct images: for a sheaf F on X, Rq f∗(F) is the sheaf associated to the presheaf
- The direct image functor is right adjoint to the inverse image functor, which means that for any continuous and sheaves respectively on X, Y, there is a natural isomorphism:
- If f is the inclusion of a closed subspace X ⊆ Y then f∗ is exact. Actually, in this case f∗ is an equivalence between sheaves on X and sheaves on Y supported on X. It follows from the fact that the stalk of is if and zero otherwise (here the closedness of X in Y is used).
- Iversen, Birger (1986), Cohomology of sheaves, Universitext, Berlin, New York: Springer-Verlag, ISBN 978-3-540-16389-3, MR 0842190, esp. section II.4