The Stacks project

Remark 63.6.6. The material in this section can be generalized to sheaves of pointed sets. Namely, for a site $\mathcal{C}$ denote $\mathop{\mathit{Sh}}\nolimits ^*(\mathcal{C})$ the category of sheaves of pointed sets. The constructions in this and the preceding section apply, mutatis mutandis, to sheaves of pointed sets. Thus given a locally quasi-finite morphism $f : X \to Y$ of schemes we obtain an adjoint pair of functors

\[ f_! : \mathop{\mathit{Sh}}\nolimits ^*(X_{\acute{e}tale}) \longrightarrow \mathop{\mathit{Sh}}\nolimits ^*(Y_{\acute{e}tale}) \quad \text{and}\quad f^! : \mathop{\mathit{Sh}}\nolimits ^*(Y_{\acute{e}tale}) \longrightarrow \mathop{\mathit{Sh}}\nolimits ^*(X_{\acute{e}tale}) \]

such that for every geometric point $\overline{y}$ of $Y$ there are isomorphisms

\[ (f_!\mathcal{F})_{\overline{y}} = \coprod \nolimits _{f(\overline{x}) = \overline{y}} \mathcal{F}_{\overline{x}} \]

(coproduct taken in the category of pointed sets) functorial in $\mathcal{F} \in \mathop{\mathit{Sh}}\nolimits ^*(X_{\acute{e}tale})$ and isomorphisms

\[ f^!(\overline{y}_*S) = \prod \nolimits _{f(\overline{x}) = \overline{y}} \overline{x}_*S \]

functorial in the pointed set $S$. If $F : \textit{Ab}(X_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits ^*(X_{\acute{e}tale})$ and $F : \textit{Ab}(Y_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits ^*(Y_{\acute{e}tale})$ denote the forgetful functors, compatibility between the constructions will guarantee the existence of canonical maps

\[ f_!F(\mathcal{F}) \longrightarrow F(f_!\mathcal{F}) \]

functorial in $\mathcal{F} \in \textit{Ab}(X_{\acute{e}tale})$ and

\[ F(f^!\mathcal{G}) \longrightarrow f^!F(\mathcal{G}) \]

functorial in $\mathcal{G} \in \textit{Ab}(Y_{\acute{e}tale})$ which produce the obvious maps on stalks, resp. skyscraper sheaves. In fact, the transformation $F \circ f^! \to f^! \circ F$ is an isomorphism (because $f^!$ commutes with products).

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