Direct Images and Inverse Images of Sheaves

In this post we will be working with the etale topology once again, so we start by formalizing some concepts. We want to first define the category (see Category Theory) called $\text{Et}(X)$ or $\text{Et}/X$ in the literature. The objects of this category are etale morphisms (see Cohomology in Algebraic Geometry) $\varphi:U\rightarrow X$ of schemes to $X$ , while the morphisms are etale morphisms $\psi:U\rightarrow U'$ such that if $\varphi ':U'\rightarrow X$ is another etale morphism to $X$ , then $\varphi' \circ \psi=\varphi$ .

Presheaves of sets, abelian groups, etc. on the category $\text{Et}(X)$ are defined as contravariant functors from the category $\text{Et}(X)$ to sets, abelian groups, etc. They are sheaves if they satisfy the sheaf conditions commonly referred to as local identity and gluing (see Even More Category Theory: The Elementary Topos). We will refer to presheaves (resp. sheaves) on $\text{Et}(X)$ as etale presheaves (resp. etale sheaves) or simply as presheaves (resp. sheaves) on $X$ .

Let $f:X\rightarrow Y$ be a morphism of schemes, and let $\mathcal{F}$ be a sheaf on $X$ . There is a sheaf on $Y$ determined by the $f$ and $\mathcal{F}$ , called the direct image sheaf, written $f_{*}\mathcal{F}$ and defined by

$\displaystyle f_{*}\mathcal{F}(U)=\mathcal{F}(X\times_{Y}U)$

for an etale morphism $U\rightarrow Y$ .

The direct image functor $f_{*}$ is the functor that assigns to a sheaf $\mathcal{F}$ the direct image sheaf $f_{*}\mathcal{F}$ . The derived functor (see More on Chain Complexes and The Hom and Tensor Functors) of $f_{*}$ is called the higher direct image functor and written $R^{n}f_{*}$ .

On the other hand, a morphism $f:X\rightarrow Y$ of schemes and a sheaf $\mathcal{G}$ on $Y$ also determine a sheaf on $X$ called the inverse image sheaf, written $f^{*}\mathcal{F}$ and obtained via the following construction:

Let $U\rightarrow X$ be an etale morphism of schemes. The presheaf $\mathcal{G}'$ is given by

$\mathcal{G}'(U)=\varinjlim \mathcal{G}(V)$

where the direct limit (see Etale Cohomology of Fields and Galois Cohomology) is taken over all $V\rightarrow Y$ such that the morphisms commute (i.e. the composition of morphisms $U\rightarrow V$ and $V\rightarrow Y$ is equal to the composition of morphisms $U\rightarrow X$ and $X\rightarrow Y$ ).

We then define the sheaf $f^{*}\mathcal{G}$ as the sheaf associated to the presheaf $\mathcal{G}'$ (the process of associating a sheaf to a presheaf, also known as sheafification, is left to the references for now).

We now introduce the notions of open subschemes and closed subschemes, and open immersions and closed immersions. We quote directly from the book Algebraic Geometry by Robin Hartshorne:

An open subscheme of a scheme $X$ is a scheme $U$ whose topological space is an open subset of $X$ , and whose structure sheaf $\mathcal{O}_{U}$ is isomorphic to the restriction $\mathcal{O}_{X|U}$ of the structure sheaf of $X$ . An open immersion is a morphism $f:X\rightarrow Y$ which induces an isomorphism of $X$ with an open subscheme of $Y$ .

Note that every open subset of a scheme carries a unique structure of open subscheme.

A closed immersion is a morphism $f: Y\rightarrow X$ of schemes such that induces a homeomorphism of $\text{sp}(Y)$ onto a closed subset of $\text{sp}(X)$ and furthermore the induced map $f^{\#}:\mathcal{O}_{X}\rightarrow f_{*}\mathcal{O}_{Y}$ of sheaves on $X$ is surjective. A closed subscheme of a scheme $X$ is an equivalence class of closed immersions, where we say $f:Y\rightarrow X$ and $f':Y'\rightarrow X$ are equivalent if there is an isomorphism $i: Y'\rightarrow Y$ such that $f'=f\circ i$ .

Now it may happen that given an open immersion $j:U\rightarrow X$ and a sheaf $\mathcal{F}$ on $U$ , the stalks of $j_{*}\mathcal{F}$ may not be zero for points outside $U$ . Therefore we define another sheaf $j_{!}$ on $X$ , given by the following construction:

Given a sheaf $\mathcal{F}$ on $U$ , and an etale morphism $\varphi:V\rightarrow X$ , let

$\displaystyle \mathcal{F}_{!}(V)=\mathcal{F}(V)$ if $\varphi(V)\subseteq U$

$\displaystyle \mathcal{F}_{!}(V)=0$ if $\varphi(V)\nsubseteq U$

Once again, $\mathcal{F}_{!}$ is a presheaf on $X$ , but it need not be a sheaf, therefore we define instead $j_{!}\mathcal{F}$ to be the sheaf associated to the presheaf $\mathcal{F}_{!}$ .

One concept related to this “extension by zero” functor $j_{!}$ is cohomology with compact support, written $H_{c}^{n}(U,\mathcal{F})=H^{r}(X,j_{!}\mathcal{F})$ .

The functors $f_{*}$ , $f^{*}$ , and the generalization of $j_{!}$ , called the direct image with compact support and denoted $f_{!}$ , are part of the so-called “six operations” which play an important role in modern algebraic geometry.