Lemma 103.3.1. Let $\mathcal{X}$ be an algebraic stack. Notation as in Cohomology of Stacks, Lemmas 102.14.2 and 102.14.4.

The functor $g_! : \textit{Ab}(\mathcal{X}_{lisse,{\acute{e}tale}}) \to \textit{Ab}(\mathcal{X}_{\acute{e}tale})$ has a left derived functor

\[ Lg_! : D(\mathcal{X}_{lisse,{\acute{e}tale}}) \longrightarrow D(\mathcal{X}_{\acute{e}tale}) \]which is left adjoint to $g^{-1}$ and such that $g^{-1}Lg_! = \text{id}$.

The functor $g_! : \textit{Mod}(\mathcal{X}_{lisse,{\acute{e}tale}}, \mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}}) \to \textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_{\mathcal{X}})$ has a left derived functor

\[ Lg_! : D(\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}}) \longrightarrow D(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X}) \]which is left adjoint to $g^*$ and such that $g^*Lg_! = \text{id}$.

The functor $g_! : \textit{Ab}(\mathcal{X}_{flat,fppf}) \to \textit{Ab}(\mathcal{X}_{fppf})$ has a left derived functor

\[ Lg_! : D(\mathcal{X}_{flat, fppf}) \longrightarrow D(\mathcal{X}_{fppf}) \]which is left adjoint to $g^{-1}$ and such that $g^{-1}Lg_! = \text{id}$.

The functor $g_! : \textit{Mod}(\mathcal{X}_{flat,fppf}, \mathcal{O}_{\mathcal{X}_{flat,fppf}}) \to \textit{Mod}(\mathcal{X}_{fppf}, \mathcal{O}_{\mathcal{X}})$ has a left derived functor

\[ Lg_! : D(\mathcal{O}_{\mathcal{X}_{flat, fppf}}) \longrightarrow D(\mathcal{O}_\mathcal {X}) \]which is left adjoint to $g^*$ and such that $g^*Lg_! = \text{id}$.

Warning: It is not clear (a priori) that $Lg_!$ on modules agrees with $Lg_!$ on abelian sheaves, see Cohomology on Sites, Remark 21.37.3.

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