Direct image functor

In algebraic geometry, the direct image functor generalizes the notion of a section of a sheaf to the relative case.

If $f:X\to Y$ is a continuous map of topological spaces, and if $\mathbf {Sheaves} (X)$ 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 $f_{*}{\mathcal {F}}$ on $Y.$ A morphism of sheaves $g:{\mathcal {F}}\to {\mathcal {G}}$ 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 $X.$

---

Direct image (functor) at PlanetMath.