The Stacks project

Lemma 96.17.5. 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 $f^{-1} : \textit{Ab}(\mathcal{Y}_\tau ) \to \textit{Ab}(\mathcal{X}_\tau )$ has a left adjoint $f_! : \textit{Ab}(\mathcal{X}_\tau ) \to \textit{Ab}(\mathcal{Y}_\tau )$. If $f$ is faithful and $\mathcal{X}$ has equalizers, then

  1. $f_!$ is exact, and

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

Proof. By Stacks, Lemma 8.10.3 the functor $f$ is continuous and cocontinuous. Hence by Modules on Sites, Lemma 18.16.2 the functor $f^{-1} : \textit{Ab}(\mathcal{Y}_\tau ) \to \textit{Ab}(\mathcal{X}_\tau )$ has a left adjoint $f_! : \textit{Ab}(\mathcal{X}_\tau ) \to \textit{Ab}(\mathcal{Y}_\tau )$. To see (1) we apply Modules on Sites, Lemma 18.16.3 and to see that the hypotheses of that lemma are satisfied use Lemmas 96.17.2 and 96.17.3 above. Part (2) follows from this formally, see Homology, Lemma 12.29.1. $\square$

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