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.

$f : X\to Y$

$\mathbf { Sheaves}(X)$

is the category of sheaves of abelian groups on $X$ (and similarly for $\mathbf { Sheaves}(Y)$ ), then the direct image functor

$f_* : \mathbf { Sheaves}(X)\to \mathbf { Sheaves}(Y)$

sends a sheaf $\mathcal{F}$ on $X$ to its direct image

$f_*\mathcal{F} : U \mapsto \mathcal{F}(f^{-1}(U))$

$g : \mathcal{F}\to\mathcal{G}$

obviously gives rise to a morphism of sheaves

$f_* g : f_*\mathcal{F}\to f_*\mathcal{G}$ , and this determines a functor.

If $\mathcal{F}$ is a sheaf of abelian groups (or anything else), so is $f_*\mathcal{F}$ , so likewise we get direct image functors

$f_* : \mathbf {Ab}(X)\to \mathbf {Ab}(Y)$ ,

where $\mathbf {Ab}(X)$ is the category of sheaves of abelian groups on $X .$

[edit] Properties

The direct image functor is left exact. Hence on can consider the left derived functors of direct image. They are called higher direct images.

The direct image functor is right adjoint to the inverse image functor.

This article incorporates material from Direct image (functor) on PlanetMath, which is licensed under the GFDL.