# Direct image functor

In mathematics, in the field of sheaf theory and especially in algebraic geometry, the **direct image functor** generalizes the notion of a section of a sheaf to the relative case.

## DefinitionEdit

Let *f*: *X* → *Y* be a continuous mapping of topological spaces, and Sh(–) denote the category of sheaves of abelian groups on a topological space. The **direct image functor**

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*.

### ExampleEdit

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).

### VariantsEdit

A similar definition applies to sheaves on topoi, such as étale sheaves. Instead of the above preimage *f*^{−1}(*U*) the fiber product of *U* and *X* over *Y* is used.

## Higher direct imagesEdit

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 *R ^{q} f*

_{∗}.

One can show that there is a similar expression as above for higher direct images: for a sheaf *F* on *X*, *R ^{q} f*

_{∗}(

*F*) is the sheaf associated to the presheaf

## PropertiesEdit

- 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).

## See alsoEdit

## ReferencesEdit

- Iversen, Birger (1986),
*Cohomology of sheaves*, Universitext, Berlin, New York: Springer-Verlag, ISBN 978-3-540-16389-3, MR 0842190, esp. section II.4

*This article incorporates material from Direct image (functor) on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*