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 We sometimes write $f_* = f_{small, *}$ to distinguish from other direct image functors (such as usual Zariski pushforward or $f_{big, *}$).
\[ f_*\mathcal{F} : Y_{\acute{e}tale}^{opp} \longrightarrow \textit{Sets}, \quad (V/Y) \longmapsto \mathcal{F}(X \times _ Y V/X). \]
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
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
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 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 $f_{big, *}$).
\[ f_*\mathcal{F} : Y_{\acute{e}tale}^{opp} \longrightarrow \textit{Sets}, \quad (V/Y) \longmapsto \mathcal{F}(X \times _ Y V/X) \]
The exact same discussion as above applies and we obtain functors
and
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).
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)