The Stacks project

Definition 59.36.1. Let $f: X\to Y$ be a morphism of schemes. The inverse image, or pullback1 functors are the functors

\[ f^{-1} = f_{small}^{-1} : \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \]

and

\[ f^{-1} = f_{small}^{-1} : \textit{Ab}(Y_{\acute{e}tale}) \longrightarrow \textit{Ab}(X_{\acute{e}tale}) \]

which are left adjoint to $f_* = f_{small, *}$. Thus $f^{-1}$ is characterized by the fact that

\[ \mathop{\mathrm{Hom}}\nolimits _{{\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})}} (f^{-1}\mathcal{G}, \mathcal{F}) = \mathop{\mathrm{Hom}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale})} (\mathcal{G}, f_*\mathcal{F}) \]

functorially, for any $\mathcal{F} \in \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ and $\mathcal{G} \in \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale})$. We similarly have

\[ \mathop{\mathrm{Hom}}\nolimits _{{\textit{Ab}(X_{\acute{e}tale})}} (f^{-1}\mathcal{G}, \mathcal{F}) = \mathop{\mathrm{Hom}}\nolimits _{\textit{Ab}(Y_{\acute{e}tale})} (\mathcal{G}, f_*\mathcal{F}) \]

for $\mathcal{F} \in \textit{Ab}(X_{\acute{e}tale})$ and $\mathcal{G} \in \textit{Ab}(Y_{\acute{e}tale})$.

[1] We use the notation $f^{-1}$ for pullbacks of sheaves of sets or sheaves of abelian groups, and we reserve $f^*$ for pullbacks of sheaves of modules via a morphism of ringed sites/topoi.

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