Direct image functor
|Image functors for sheaves|
|direct image f∗|
|inverse image f∗|
|direct image with compact support f!|
|exceptional inverse image Rf!|
sends a sheaf F on X to its direct image presheaf
which turns out be a sheaf on Y. 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
The direct image functor is left exact, but usually not right exact. Hence one can consider the right derived functors of the direct image. They are called higher direct images and denoted Rq f∗.
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 closeness of X in Y is used).
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