nLab
direct image

Idea

Pairs of adjoint functors between the categories of sheaves appear in varius setups, e.g. geometric morphisms of topoi, abelian categories of quasicoherent sheaves on schemes, bounded derived categories of coherent sheaves on varieties.

For example, the right adjoint part $f_{*}$ of any geometric morphism

$$(f^{*}⊣f_{*}):E_1\overleftarrow{\to }{E}_2$$(f^* \dashv f_*) \;\; : E_1 \stackrel{\leftarrow}{\to} E_2

of toposes is called a direct image.

Specifically for Grothendieck toposes: a morphism of sites $f:X\to Y$ induces a geometric morphism of Grothendieck toposes

$$(p^{*}⊣p_{*}):\mathrm{Sh}(X)\stackrel{}{\stackrel{\stackrel{p_{*}}{\to }}{\stackrel{p^{*}}{\leftarrow }}}\mathrm{Sh}(Y)$$(p^* \dashv p_*) \;\;\; : \;\;\; Sh(X) \stackrel{}{\stackrel{\overset{p_*}{\to}}{\overset{p^*}{\leftarrow}}} Sh(Y)

between the categories of sheaves on the sites, with

• $p_{*}$ the direct image

• and $p^{*}$ its left adjoint: the inverse image.

Definition

Given a morphism of sites $f:X\to Y$ coming from a functor $f^t:S_Y\to S_X$, the direct image operation on presheaves is the functor

$$f_{*}:\mathrm{PSh}(X)\to \mathrm{PSh}(Y)$$f_* : PSh(X) \to PSh(Y)

$$f_{*}F:S_Y^{\mathrm{op}}\stackrel{f^t}{\to }{S}_X^{\mathrm{op}}\stackrel{F}{\to }\mathrm{Set}.$$f_* F : S_Y^{op} \stackrel{f^t}{\to} S_X^{op} \stackrel{F}{\to} Set \,.

The restriction of this operation to sheaves, which respects sheaves, is the direct image of sheaves

$$f_{*}:\mathrm{Sh}(X)\to \mathrm{Sh}(Y).$$f_* : Sh(X) \to Sh(Y) \,.

Examples

Global sections

For $X$ a site with a terminal object, let the morphism of sites be the canonical morphism $p:X\to *$.

• The direct image $p_{*}$ is the global sections functor;

• the inverse image $p^{*}$ is the constant sheaf functor;

• the left adjoint to $p^{*}$ is ${\Pi }_0$, the functor of geometric connected components (see homotopy group of an ∞-stack).

Restriction and extension of sheaves

See

• restriction and extension of sheaves

for the moment.