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.

If

$f\; :\; X\backslash to\; Y$

is a continuous mapping of topological spaces, and if

$\backslash mathbf\; \{\; Sheaves\}(X)$

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

$f\_*\; :\; \backslash mathbf\; \{\; Sheaves\}(X)\backslash to\; \backslash mathbf\; \{\; Sheaves\}(Y)$

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

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

on $Y.$ A morphism of sheaves

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

obviously gives rise to a morphism of sheaves

$f\_*\; g\; :\; f\_*\backslash mathcal\{F\}\backslash to\; f\_*\backslash mathcal\{G\}$, and this determines a functor.

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

$f\_*\; :\; \backslash mathbf\; \{Ab\}(X)\backslash to\; \backslash mathbf\; \{Ab\}(Y)$,

where $\backslash mathbf\; \{Ab\}(X)$ is the category of sheaves of abelian groups on $X.$ Derived functors of direct image are called higher direct images.

Algebraic geometry | Sheaf theory