Section 59.35 (03PV): Direct images

59.35 Direct images

Let us define the pushforward of a presheaf.

Definition 59.35.1. Let $f: X\to Y$ be a morphism of schemes. Let $\mathcal{F}$ a presheaf of sets on $X_{\acute{e}tale}$ . The direct image, or pushforward of $\mathcal{F}$ (under $f$ ) is

$f_*\mathcal{F} : Y_{\acute{e}tale}^{opp} \longrightarrow \textit{Sets}, \quad (V/Y) \longmapsto \mathcal{F}(X \times _ Y V/X).$

We sometimes write $f_* = f_{small, *}$ to distinguish from other direct image functors (such as usual Zariski pushforward or $f_{big, *}$ ).

This is a well-defined étale presheaf since the base change of an étale morphism is again étale. A more categorical way of saying this is that $f_*\mathcal{F}$ is the composition of functors $\mathcal{F} \circ u$ where $u$ is as in Equation (59.34.0.1). This makes it clear that the construction is functorial in the presheaf $\mathcal{F}$ and hence we obtain a functor

$f_* = f_{small, *} : \textit{PSh}(X_{\acute{e}tale}) \longrightarrow \textit{PSh}(Y_{\acute{e}tale})$

Note that if $\mathcal{F}$ is a presheaf of abelian groups, then $f_*\mathcal{F}$ is also a presheaf of abelian groups and we obtain

$f_* = f_{small, *} : \textit{PAb}(X_{\acute{e}tale}) \longrightarrow \textit{PAb}(Y_{\acute{e}tale})$

as before (i.e., defined by exactly the same rule).

Remark 59.35.2. We claim that the direct image of a sheaf is a sheaf. Namely, if $\{ V_ j \to V\}$ is an étale covering in $Y_{\acute{e}tale}$ then $\{ X \times _ Y V_ j \to X \times _ Y V\}$ is an étale covering in $X_{\acute{e}tale}$ . Hence the sheaf condition for $\mathcal{F}$ with respect to $\{ X \times _ Y V_ i \to X \times _ Y V\}$ is equivalent to the sheaf condition for $f_*\mathcal{F}$ with respect to $\{ V_ i \to V\}$ . Thus if $\mathcal{F}$ is a sheaf, so is $f_*\mathcal{F}$ .

Definition 59.35.3. Let $f: X\to Y$ be a morphism of schemes. Let $\mathcal{F}$ a sheaf of sets on $X_{\acute{e}tale}$ . The direct image, or pushforward of $\mathcal{F}$ (under $f$ ) is

$f_*\mathcal{F} : Y_{\acute{e}tale}^{opp} \longrightarrow \textit{Sets}, \quad (V/Y) \longmapsto \mathcal{F}(X \times _ Y V/X)$

which is a sheaf by Remark 59.35.2. We sometimes write $f_* = f_{small, *}$ to distinguish from other direct image functors (such as usual Zariski pushforward or $f_{big, *}$ ).

The exact same discussion as above applies and we obtain functors

$f_* = f_{small, *} : \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale})$

and

$f_* = f_{small, *} : \textit{Ab}(X_{\acute{e}tale}) \longrightarrow \textit{Ab}(Y_{\acute{e}tale})$

called direct image again.

The functor $f_*$ on abelian sheaves is left exact. (See Homology, Section 12.7 for what it means for a functor between abelian categories to be left exact.) Namely, if $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3$ is exact on $X_{\acute{e}tale}$ , then for every $U/X \in \mathop{\mathrm{Ob}}\nolimits (X_{\acute{e}tale})$ the sequence of abelian groups $0 \to \mathcal{F}_1(U) \to \mathcal{F}_2(U) \to \mathcal{F}_3(U)$ is exact. Hence for every $V/Y \in \mathop{\mathrm{Ob}}\nolimits (Y_{\acute{e}tale})$ the sequence of abelian groups $0 \to f_*\mathcal{F}_1(V) \to f_*\mathcal{F}_2(V) \to f_*\mathcal{F}_3(V)$ is exact, because this is the previous sequence with $U = X \times _ Y V$ .

Definition 59.35.4. Let $f: X \to Y$ be a morphism of schemes. The right derived functors $\{ R^ pf_*\} _{p \geq 1}$ of $f_* : \textit{Ab}(X_{\acute{e}tale}) \to \textit{Ab}(Y_{\acute{e}tale})$ are called higher direct images.

The higher direct images and their derived category variants are discussed in more detail in (insert future reference here).

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59.35 Direct images

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