Lemma 103.14.4. Let $\mathcal{X}$ be an algebraic stack. Notation as in Lemma 103.14.2.

1. There exists a functor

$g_! : \textit{Mod}(\mathcal{X}_{lisse,{\acute{e}tale}}, \mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}}) \longrightarrow \textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_{\mathcal{X}})$

which is left adjoint to $g^*$. Moreover it agrees with the functor $g_!$ on abelian sheaves and $g^*g_! = \text{id}$.

2. There exists a functor

$g_! : \textit{Mod}(\mathcal{X}_{flat,fppf}, \mathcal{O}_{\mathcal{X}_{flat,fppf}}) \longrightarrow \textit{Mod}(\mathcal{X}_{fppf}, \mathcal{O}_{\mathcal{X}})$

which is left adjoint to $g^*$. Moreover it agrees with the functor $g_!$ on abelian sheaves and $g^*g_! = \text{id}$.

Proof. In both cases, the existence of the functor $g_!$ follows from Modules on Sites, Lemma 18.41.1. To see that $g_!$ agrees with the functor on abelian sheaves we will show the maps Modules on Sites, Equation (18.41.2.1) are isomorphisms.

Lisse-étale case. Let $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_{lisse,{\acute{e}tale}})$ lying over a scheme $U$ with $x : U \to \mathcal{X}$ smooth. Consider the induced fully faithful functor

$g' : \mathcal{X}_{lisse,{\acute{e}tale}}/x \longrightarrow \mathcal{X}_{\acute{e}tale}/x$

The right hand side is identified with $(\mathit{Sch}/U)_{\acute{e}tale}$ and the left hand side with the full subcategory of schemes $U'/U$ such that the composition $U' \to U \to \mathcal{X}$ is smooth. Thus Étale Cohomology, Lemma 59.49.2 applies.

Flat-fppf case. Let $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_{flat,fppf})$ lying over a scheme $U$ with $x : U \to \mathcal{X}$ flat. Consider the induced fully faithful functor

$g' : \mathcal{X}_{flat,fppf}/x \longrightarrow \mathcal{X}_{fppf}/x$

The right hand side is identified with $(\mathit{Sch}/U)_{fppf}$ and the left hand side with the full subcategory of schemes $U'/U$ such that the composition $U' \to U \to \mathcal{X}$ is flat. Thus Étale Cohomology, Lemma 59.49.2 applies.

In both cases the equality $g^*g_! = \text{id}$ follows from $g^* = g^{-1}$ and the equality for abelian sheaves in Lemma 103.14.2. $\square$

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