nLab direct image

This page is about direct images of sheaves and related subjects. For the set-theoretic operation, see image.

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Idea

The right adjoint part $f_*$ of any geometric morphism

$$(f^* \dashv f_*) \;\; : E_1 \leftrightarrows E_2$$

of toposes is called a direct image.

More generally, pairs of adjoint functors between the categories of sheaves appear in various other setups apart from geometric morphisms of topoi, for instance on abelian categories of quasicoherent sheaves, bounded derived categories of coherent sheaves and the term direct image is used for the right adjoint part of these, too.

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

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

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

$$f_* : PSh(X) \to PSh(Y)$$

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

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

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

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.

Direct image with compact supports

Let $f:X\to Y$ be a morphism of locally compact topological spaces. Then there exist a unique subfunctor $f_!: Sh(X)\to Sh(Y)$ of the direct image functor $f_*$ such that for any abelian sheaf $F$ over $X$ the sections of $f_!(F)$ over $U^{open}\subset X$ are those sections $s\in f_*(U)= \Gamma(f^{-1}(U),F)$ for which the restriction $supp(s)|f : supp(s)\hookrightarrow U$ is a proper map.

This is called the direct image with compact support.

It follows that $f_!$ is left exact.

Let $p:X\to {*}$ be the map into the one point space. Then for any $F\in Sh(X)$ the abelian sheaf $p_!F$ is the abelian group consisting of sections $s\in \Gamma(X,F)$ such that $supp(s)$ is compact. One writes $\Gamma_c(X,F) \coloneqq p_! F$ and calls this group a group of sections of $F$ with compact support. If $y\in Y$, then the fiber $(f_! F)_y$ is isomorphic to $\Gamma_c(f^{-1}y,F|_{f^{-1}(y)})$.

Derived direct image

Proposition

Let $f^{-1} \colon Y \to X$ be a morphism of sites. Then the $q$th derived functor $R^q f_\ast$ of the induced direct image functor sends $\mathcal{F} \in Ab(Sh(X_{et}))$ to the sheafification of the presheaf

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

where on the right we have the degree $q$ abelian sheaf cohomology group with coefficients in the given $\mathcal{F}$.

(e.g. Tamme, I (3.7.1), II (1.3.4), Milne, 12.1).

Proof

We have a commuting diagram

$$\array{ Ab(PSh(X)) &\stackrel{(-)\circ f^{-1}}{\longrightarrow}& Ab(PSh(Y)) \\ \uparrow^{\mathrlap{inc}} && \downarrow^{L} \\ Ab(Sh(X)) &\stackrel{f_\ast}{\longrightarrow}& Ab(Sh(Y)) } \,,$$

where the right vertical morphism is sheafification. Because $(-) \circ f^{-1}$ and $L$ are both exact functors it follows that for $I^\bullet \to \mathcal{F}$ an injective resolution that

$$\begin{aligned} R^p f_\ast(\mathcal{F}) & :\simeq H^p( f_\ast I) \\ & = H^p(L I^\bullet(f^{-1}(-))) \\ & = L (H^p(I^\bullet)(f^{-1}(-))) \end{aligned}$$

Related concepts

• derived direct image

References

e.g.

• Günter Tamme, section II 1 of Introduction to Étale Cohomology

• James Milne, section 7 of Lectures on Étale Cohomology