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


Contents

Idea

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

(f *f *):E 1E 2 (f^* \dashv f_*) \;\; : E_1 \leftrightarrows E_2

of toposes is called a direct image.

Moe 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:XYf : X \to Y induces a geometric morphism of Grothendieck toposes

(p *p *):Sh(X)p *p *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

Definition

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

f *:PSh(X)PSh(Y) f_* : PSh(X) \to PSh(Y)
f *F:S Y opf tS X opFSet. 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)Sh(Y). f_* : Sh(X) \to Sh(Y) \,.

Examples

Global sections

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

Restriction and extension of sheaves

See

for the moment.

Direct image with compact supports

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

This is called the direct image with compact support.

It follows that f !f_! is left exact.

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

Derived direct image

Proposition

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

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

where on the right we have the degree qq 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

Ab(PSh(X)) ()f 1 Ab(PSh(Y)) inc L Ab(Sh(X)) f * Ab(Sh(Y)), \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 ()f 1(-) \circ f^{-1} and LL are both exact functors it follows that for I I^\bullet \to \mathcal{F} an injective resolution that

R pf *() :H p(f *I) =H p(LI (f 1())) =L(H p(I )(f 1())) \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}

References

e.g.

Revised on November 24, 2013 11:55:37 by Urs Schreiber (89.204.139.68)