The Stacks project

Lemma 102.14.2. 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 $g^{-1} = g^*$.

Proof. In both cases it is immediate that the functor is fully faithful, continuous, and cocontinuous (see Sites, Definitions 7.13.1 and 7.20.1). Hence properties (a), (b), (c) follow from Sites, Lemmas 7.21.5 and 7.21.7. Parts (d), (e) follow from Modules on Sites, Lemmas 18.16.2 and 18.16.4. Part (f) is immediate. $\square$


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