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Tag 07B1

Chapter 94: Cohomology of Algebraic Stacks > Section 94.12: Quasi-coherent modules, II

Lemma 94.12.3. Let $\mathcal{X}$ be an algebraic stack. Let $\mathcal{M}_\mathcal{X}$ be the category of locally quasi-coherent $\mathcal{O}_\mathcal{X}$-modules with the flat base change property.

  1. With $g$ as in Lemma 94.11.2 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 $\mathcal{M}_\mathcal{X}$ then $g^{-1}\mathcal{F}$ is in $\mathit{QCoh}(\mathcal{X}_{lisse,{\acute{e}tale}}, \mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}})$ and
    3. $Q(\mathcal{F}) = g_!g^{-1}\mathcal{F}$ where $Q$ is as in Lemma 94.9.1.
  2. With $g$ as in Lemma 94.11.2 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 $\mathcal{M}_\mathcal{X}$ then $g^{-1}\mathcal{F}$ is in $\mathit{QCoh}(\mathcal{X}_{flat,fppf}, \mathcal{O}_{\mathcal{X}_{flat,fppf}})$ and
    3. $Q(\mathcal{F}) = g_!g^{-1}\mathcal{F}$ where $Q$ is as in Lemma 94.9.1.

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. The same is true for $g_!$ by Lemma 94.12.2. We know that $\mathcal{H} \to g^{-1}g_!\mathcal{H}$ is an isomorphism by Lemma 94.11.2. 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 87.13 and 87.14 (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 $\mathcal{M}_\mathcal{X}$. By Lemma 94.9.2 the kernel and cokernel of the map $Q(\mathcal{F}) \to \mathcal{F}$ are parasitic. Hence by Lemma 94.11.5 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. $\square$

    The code snippet corresponding to this tag is a part of the file stacks-cohomology.tex and is located in lines 2271–2314 (see updates for more information).

    \begin{lemma}
    \label{lemma-quasi-coherent}
    Let $\mathcal{X}$ be an algebraic stack. Let $\mathcal{M}_\mathcal{X}$
    be the category of locally quasi-coherent $\mathcal{O}_\mathcal{X}$-modules
    with the flat base change property.
    \begin{enumerate}
    \item With $g$ as in Lemma \ref{lemma-lisse-etale}
    for the lisse-\'etale site we have
    \begin{enumerate}
    \item the functors $g^{-1}$ and $g_!$ define mutually inverse functors
    $$
    \xymatrix{
    \QCoh(\mathcal{O}_\mathcal{X}) \ar@<1ex>[r]^-{g^{-1}} &
    \QCoh(\mathcal{X}_{lisse,\etale},
    \mathcal{O}_{\mathcal{X}_{lisse,\etale}}) \ar@<1ex>[l]^-{g_!}
    }
    $$
    \item if $\mathcal{F}$ is in $\mathcal{M}_\mathcal{X}$
    then $g^{-1}\mathcal{F}$ is in
    $\QCoh(\mathcal{X}_{lisse,\etale},
    \mathcal{O}_{\mathcal{X}_{lisse,\etale}})$ and
    \item $Q(\mathcal{F}) = g_!g^{-1}\mathcal{F}$ where $Q$ is as in
    Lemma \ref{lemma-adjoint}.
    \end{enumerate}
    \item With $g$ as in Lemma \ref{lemma-lisse-etale}
    for the flat-fppf site we have
    \begin{enumerate}
    \item the functors $g^{-1}$ and $g_!$ define mutually inverse functors
    $$
    \xymatrix{
    \QCoh(\mathcal{O}_\mathcal{X}) \ar@<1ex>[r]^-{g^{-1}} &
    \QCoh(\mathcal{X}_{flat,fppf},
    \mathcal{O}_{\mathcal{X}_{flat,fppf}}) \ar@<1ex>[l]^-{g_!}
    }
    $$
    \item if $\mathcal{F}$ is in $\mathcal{M}_\mathcal{X}$
    then $g^{-1}\mathcal{F}$ is in
    $\QCoh(\mathcal{X}_{flat,fppf}, \mathcal{O}_{\mathcal{X}_{flat,fppf}})$
    and
    \item $Q(\mathcal{F}) = g_!g^{-1}\mathcal{F}$ where $Q$ is as in
    Lemma \ref{lemma-adjoint}.
    \end{enumerate}
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    Pullback by any morphism of ringed topoi preserves categories of quasi-coherent
    modules, see
    Modules on Sites, Lemma \ref{sites-modules-lemma-local-pullback}.
    Hence $g^{-1}$ preserves the categories of quasi-coherent modules.
    The same is true for $g_!$ by
    Lemma \ref{lemma-shriek-quasi-coherent}.
    We know that $\mathcal{H} \to g^{-1}g_!\mathcal{H}$ is an isomorphism by
    Lemma \ref{lemma-lisse-etale}.
    Conversely, if $\mathcal{F}$ is in $\QCoh(\mathcal{O}_\mathcal{X})$
    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
    \ref{stacks-sheaves-section-quasi-coherent-presentation} and
    \ref{stacks-sheaves-section-quasi-coherent-algebraic-stacks}
    (comparing with quasi-coherent modules on presentations)
    shows it is an isomorphism. This proves (1)(a) and (2)(a).
    
    \medskip\noindent
    Let $\mathcal{F}$ be an object of $\mathcal{M}_\mathcal{X}$. By
    Lemma \ref{lemma-adjoint-kernel-parasitic}
    the kernel and cokernel of the map
    $Q(\mathcal{F}) \to \mathcal{F}$ are parasitic. Hence by
    Lemma \ref{lemma-parasitic-in-terms-flat-fppf}
    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.
    \end{proof}

    Comments (2)

    Comment #3192 by anonymous on February 11, 2018 a 8:48 am UTC

    Concerning the first two sentences of the proof: The functor $g^{-1}$ has source $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_{\mathcal{X}})$ and not $\textit{Mod}(\mathcal{O}_\mathcal{X})$. What is the relation between quasi-coherent $\mathcal{O}_\mathcal{X}$-modules and quasi-coherent $\mathcal{O}_{\mathcal{X}_{{\acute{e}tale}}}$-modules?

    Comment #3196 by Johan (site) on February 11, 2018 a 3:20 pm UTC

    Please see comment #3195

    There are also 2 comments on Section 94.12: Cohomology of Algebraic Stacks.

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