Lemma 102.14.2. Let $\mathcal{X}$ be an algebraic stack.

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

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,

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

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

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

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

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 $g^{-1} = g^*$.

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

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,

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

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

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

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

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 $g^{-1} = g^*$.

## Comments (0)