The Stacks project

Lemma 96.17.6. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Let $\tau \in \{ Zar, {\acute{e}tale}, smooth, syntomic, fppf\} $. The functor $f^* : \textit{Mod}(\mathcal{Y}_\tau , \mathcal{O}_\mathcal {Y}) \to \textit{Mod}(\mathcal{X}_\tau , \mathcal{O}_\mathcal {X})$ has a left adjoint $f_! : \textit{Mod}(\mathcal{X}_\tau , \mathcal{O}_\mathcal {X}) \to \textit{Mod}(\mathcal{Y}_\tau , \mathcal{O}_\mathcal {Y})$ which agrees with the functor $f_!$ of Lemma 96.17.5 on underlying abelian sheaves. If $f$ is faithful and $\mathcal{X}$ has equalizers, then

  1. $f_!$ is exact, and

  2. $f^{-1}\mathcal{I}$ is injective in $\textit{Mod}(\mathcal{X}_\tau , \mathcal{O}_\mathcal {X})$ for $\mathcal{I}$ injective in $\textit{Mod}(\mathcal{Y}_\tau , \mathcal{O}_\mathcal {X})$.

Proof. Recall that $f$ is a continuous and cocontinuous functor of sites and that $f^{-1}\mathcal{O}_\mathcal {Y} = \mathcal{O}_\mathcal {X}$. Hence Modules on Sites, Lemma 18.41.1 implies $f^*$ has a left adjoint $f_!^{Mod}$. Let $x$ be an object of $\mathcal{X}$ lying over the scheme $U$. Then $f$ induces an equivalence of ringed sites

\[ \mathcal{X}/x \longrightarrow \mathcal{Y}/f(x) \]

as both sides are equivalent to $(\mathit{Sch}/U)_\tau $, see Lemma 96.9.4. Modules on Sites, Remark 18.41.2 shows that $f_!$ agrees with the functor on abelian sheaves.

Assume now that $\mathcal{X}$ has equalizers and that $f$ is faithful. Lemma 96.17.5 tells us that $f_!$ is exact. Finally, Homology, Lemma 12.29.1 implies the statement on pullbacks of injective modules. $\square$

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