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.

Definition

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∗: f∗ → f∗ on Y.

Example

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 → D.

Variants

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

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∗ is the sheaf associated to the presheaf

Properties