The Stacks project

Lemma 102.16.3. Let $\mathcal{X}$ be an algebraic stack.

  1. With $g$ as in Lemma 102.14.2 for the lisse-étale site we have

    1. the functors $g^{-1}$ and $g_!$ define mutually inverse functors

      \[ \xymatrix{ \mathit{QCoh}(\mathcal{O}_\mathcal {X}) \ar@<1ex>[r]^-{g^{-1}} & \mathit{QCoh}(\mathcal{X}_{lisse,{\acute{e}tale}}, \mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}}) \ar@<1ex>[l]^-{g_!} } \]
    2. if $\mathcal{F}$ is in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ then $g^{-1}\mathcal{F}$ is in $\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}})$ and

    3. $Q(\mathcal{F}) = g_!g^{-1}\mathcal{F}$ where $Q$ is as in Lemma 102.10.1.

  2. With $g$ as in Lemma 102.14.2 for the flat-fppf site we have

    1. the functors $g^{-1}$ and $g_!$ define mutually inverse functors

      \[ \xymatrix{ \mathit{QCoh}(\mathcal{O}_\mathcal {X}) \ar@<1ex>[r]^-{g^{-1}} & \mathit{QCoh}(\mathcal{X}_{flat,fppf}, \mathcal{O}_{\mathcal{X}_{flat,fppf}}) \ar@<1ex>[l]^-{g_!} } \]
    2. if $\mathcal{F}$ is in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ then $g^{-1}\mathcal{F}$ is in $\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{flat,fppf}})$ and

    3. $Q(\mathcal{F}) = g_!g^{-1}\mathcal{F}$ where $Q$ is as in Lemma 102.10.1.

Proof. Pullback by any morphism of ringed topoi preserves categories of quasi-coherent modules, see Modules on Sites, Lemma 18.23.4. Hence $g^{-1}$ preserves the categories of quasi-coherent modules; here we use that $\mathit{QCoh}(\mathcal{O}_\mathcal {X}) = \mathit{QCoh}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ by Sheaves on Stacks, Lemma 95.11.4. The same is true for $g_!$ by Lemma 102.16.2. We know that $\mathcal{H} \to g^{-1}g_!\mathcal{H}$ is an isomorphism by Lemma 102.14.2. Conversely, if $\mathcal{F}$ is in $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ then the map $g_!g^{-1}\mathcal{F} \to \mathcal{F}$ is a map of quasi-coherent modules on $\mathcal{X}$ whose restriction to any scheme smooth over $\mathcal{X}$ is an isomorphism. Then the discussion in Sheaves on Stacks, Sections 95.14 and 95.15 (comparing with quasi-coherent modules on presentations) shows it is an isomorphism. This proves (1)(a) and (2)(a).

Let $\mathcal{F}$ be an object of $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$. By Lemma 102.10.2 the kernel and cokernel of the map $Q(\mathcal{F}) \to \mathcal{F}$ are parasitic. Hence by Lemma 102.14.6 and since $g^* = g^{-1}$ is exact, we conclude $g^*Q(\mathcal{F}) \to g^*\mathcal{F}$ is an isomorphism. Thus $g^*\mathcal{F}$ is quasi-coherent. This proves (1)(b) and (2)(b). Finally, (1)(c) and (2)(c) follow because $g_!g^*Q(\mathcal{F}) \to Q(\mathcal{F})$ is an isomorphism by our arguments above. $\square$


Comments (3)

Comment #3189 by anonymous on

Concerning the first two sentences of the proof: The functor has source and not . What is the relation between quasi-coherent -modules and quasi-coherent -modules?

There are also:

  • 2 comment(s) on Section 102.16: Quasi-coherent modules and the lisse-étale and flat-fppf sites

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