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.

If

$f:X\to Y$

is a continuous mapping 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

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

on $Y.$ A morphism of sheaves

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

gives rise to a morphism of sheaves

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

Higher direct images

The direct image functor is left exact, but usually not right exact. Hence on can consider the right derived functors of the direct image. They are called higher direct images and denoted R^q f_∗.

One can show that there is a similar expression as above for higher direct images: for a sheaf F on X, R^q f_∗ is the sheaf associated to the presheaves

$U\mapsto H^{q}(f^{-1}(U),F)$

Properties

• The direct image functor is right adjoint to the inverse image functor.
• If f is the inclusion of a closed subspace X ⊂ Y then f_∗ is exact.

Direct image with compact support

There is a variant to the above construction, called the direct image with compact support. For a sheaf F on X it is defined by

f_!(F)(U) := {s ∈ F(f^-1(U)), supp (s) proper over U},

where U is an open subset of Y.

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Direct image (functor) at PlanetMath.