The Stacks project

Lemma 59.36.2. Let $f : X \to Y$ be a morphism of schemes.

  1. The functor $f^{-1} : \textit{Ab}(Y_{\acute{e}tale}) \to \textit{Ab}(X_{\acute{e}tale})$ is exact.

  2. The functor $f^{-1} : \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ is exact, i.e., it commutes with finite limits and colimits, see Categories, Definition 4.23.1.

  3. Let $\overline{x} \to X$ be a geometric point. Let $\mathcal{G}$ be a sheaf on $Y_{\acute{e}tale}$. Then there is a canonical identification

    \[ (f^{-1}\mathcal{G})_{\overline{x}} = \mathcal{G}_{\overline{y}}. \]

    where $\overline{y} = f \circ \overline{x}$.

  4. For any $V \to Y$ étale we have $f^{-1}h_ V = h_{X \times _ Y V}$.

Proof. The exactness of $f^{-1}$ on sheaves of sets is a consequence of Sites, Proposition 7.14.7 applied to our functor $u$ of Equation (59.34.0.1). In fact the exactness of pullback is part of the definition of a morphism of topoi (or sites if you like). Thus we see (2) holds. It implies part (1) since given an abelian sheaf $\mathcal{G}$ on $Y_{\acute{e}tale}$ the underlying sheaf of sets of $f^{-1}\mathcal{F}$ is the same as $f^{-1}$ of the underlying sheaf of sets of $\mathcal{F}$, see Sites, Section 7.44. See also Modules on Sites, Lemma 18.31.2. In the literature (1) and (2) are sometimes deduced from (3) via Theorem 59.29.10.

Part (3) is a general fact about stalks of pullbacks, see Sites, Lemma 7.34.2. We will also prove (3) directly as follows. Note that by Lemma 59.29.9 taking stalks commutes with sheafification. Now recall that $f^{-1}\mathcal{G}$ is the sheaf associated to the presheaf

\[ U \longrightarrow \mathop{\mathrm{colim}}\nolimits _{U \to X \times _ Y V} \mathcal{G}(V), \]

see Equation (59.36.1.1). Thus we have

\begin{align*} (f^{-1}\mathcal{G})_{\overline{x}} & = \mathop{\mathrm{colim}}\nolimits _{(U, \overline{u})} f^{-1}\mathcal{G}(U) \\ & = \mathop{\mathrm{colim}}\nolimits _{(U, \overline{u})} \mathop{\mathrm{colim}}\nolimits _{a : U \to X \times _ Y V} \mathcal{G}(V) \\ & = \mathop{\mathrm{colim}}\nolimits _{(V, \overline{v})} \mathcal{G}(V) \\ & = \mathcal{G}_{\overline{y}} \end{align*}

in the third equality the pair $(U, \overline{u})$ and the map $a : U \to X \times _ Y V$ corresponds to the pair $(V, a \circ \overline{u})$.

Part (4) can be proved in a similar manner by identifying the colimits which define $f^{-1}h_ V$. Or you can use Yoneda's lemma (Categories, Lemma 4.3.5) and the functorial equalities

\[ \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})}(f^{-1}h_ V, \mathcal{F}) = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale})}(h_ V, f_*\mathcal{F}) = f_*\mathcal{F}(V) = \mathcal{F}(X \times _ Y V) \]

combined with the fact that representable presheaves are sheaves. See also Sites, Lemma 7.13.5 for a completely general result. $\square$


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