Cohomology and homotopy
In higher category theory
This page is about direct images of sheaves and related subjects. For the set-theoretic operation, see image.
The right adjoint part of any geometric morphism
of toposes is called a direct image.
Moe generally, pairs of adjoint functors between the categories of sheaves appear in various other setups apart from geometric morphisms of topoi, for instance on abelian categories of quasicoherent sheaves, bounded derived categories of coherent sheaves and the term direct image is used for the right adjoint part of these, too.
Specifically for Grothendieck toposes: a morphism of sites induces a geometric morphism of Grothendieck toposes
between the categories of sheaves on the sites, with
Given a morphism of sites coming from a functor , the direct image operation on presheaves is the functor
The restriction of this operation to sheaves, which respects sheaves, is the direct image of sheaves
For a site with a terminal object, let the morphism of sites be the canonical morphism .
Restriction and extension of sheaves
for the moment.
Direct image with compact supports
Let be a morphism of locally compact topological spaces. Then there exist a unique subfunctor of the direct image functor such that for any abelian sheaf over the sections of over are those sections for which the restriction is a proper map.
This is called the direct image with compact support.
It follows that is left exact.
Let be the map into the one point space. Then for any the abelian sheaf is the abelian group consisting of sections such that is compact. One writes and calls this group a group of sections of with compact support. If , then the fiber is isomorphic to .
Derived direct image
(e.g. Tamme, I (3.7.1), II (1.3.4), Milne, 12.1).
We have a commuting diagram
where the right vertical morphism is sheafification. Because and are both exact functors it follows that for an injective resolution that