The Stacks project

Lemma 103.14.2. Let $\mathcal{X}$ be an algebraic stack.

1. The inclusion functor $\mathcal{X}_{lisse,{\acute{e}tale}} \to \mathcal{X}_{\acute{e}tale}$ is fully faithful, continuous and cocontinuous. It follows that

   1. there is a morphism of topoi

      \[ g : \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{lisse,{\acute{e}tale}}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{\acute{e}tale}) \]

      with $g^{-1}$ given by restriction,

   2. the functor $g^{-1}$ has a left adjoint $g_!^{Sh}$ on sheaves of sets,

   3. the adjunction maps $g^{-1}g_* \to \text{id}$ and $\text{id} \to g^{-1}g_!^{Sh}$ are isomorphisms,

   4. the functor $g^{-1}$ has a left adjoint $g_!$ on abelian sheaves,

   5. the adjunction map $\text{id} \to g^{-1}g_!$ is an isomorphism, and

   6. we have $g^{-1}\mathcal{O}_\mathcal {X} = \mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}}$ hence $g$ induces a flat morphism of ringed topoi such that $g^{-1} = g^*$.

2. The inclusion functor $\mathcal{X}_{flat,fppf} \to \mathcal{X}_{fppf}$ is fully faithful, continuous and cocontinuous. It follows that

   1. there is a morphism of topoi

      \[ g : \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{flat,fppf}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{fppf}) \]

      with $g^{-1}$ given by restriction,

   2. the functor $g^{-1}$ has a left adjoint $g_!^{Sh}$ on sheaves of sets,

   3. the adjunction maps $g^{-1}g_* \to \text{id}$ and $\text{id} \to g^{-1}g_!^{Sh}$ are isomorphisms,

   4. the functor $g^{-1}$ has a left adjoint $g_!$ on abelian sheaves,

   5. the adjunction map $\text{id} \to g^{-1}g_!$ is an isomorphism, and

   6. we have $g^{-1}\mathcal{O}_\mathcal {X} = \mathcal{O}_{\mathcal{X}_{flat,fppf}}$ hence $g$ induces a flat morphism of ringed topoi such that $g^{-1} = g^*$.