The Stacks Project


Tag 07B0

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

Lemma 94.12.2. Let $\mathcal{X}$ be an algebraic stack. Notation as in Lemma 94.11.2.

  1. Let $\mathcal{H}$ be a quasi-coherent $\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}}$-module on the lisse-étale site of $\mathcal{X}$. Then $g_!\mathcal{H}$ is a quasi-coherent module on $\mathcal{X}$.
  2. Let $\mathcal{H}$ be a quasi-coherent $\mathcal{O}_{\mathcal{X}_{flat,fppf}}$-module on the flat-fppf site of $\mathcal{X}$. Then $g_!\mathcal{H}$ is a quasi-coherent module on $\mathcal{X}$.

Proof. Pick a scheme $U$ and a surjective smooth morphism $x : U \to \mathcal{X}$. By Modules on Sites, Definition 18.23.1 there exists an étale (resp. fppf) covering $\{U_i \to U\}_{i \in I}$ such that each pullback $f_i^{-1}\mathcal{H}$ has a global presentation (see Modules on Sites, Definition 18.17.1). Here $f_i : U_i \to \mathcal{X}$ is the composition $U_i \to U \to \mathcal{X}$ which is a morphism of algebraic stacks. (Recall that the pullback ''is'' the restriction to $\mathcal{X}/f_i$, see Sheaves on Stacks, Definition 87.9.2 and the discussion following.) Since each $f_i$ is smooth (resp. flat) by Lemma 94.11.6 we see that $f_i^{-1}g_!\mathcal{H} = g_{i, !}(f'_i)^{-1}\mathcal{H}$. Using Lemma 94.12.1 we reduce the statement of the lemma to the case where $\mathcal{H}$ has a global presentation. Say we have $$ \bigoplus\nolimits_{j \in J} \mathcal{O} \longrightarrow \bigoplus\nolimits_{i \in I} \mathcal{O} \longrightarrow \mathcal{H} \longrightarrow 0 $$ of $\mathcal{O}$-modules where $\mathcal{O} = \mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}}$ (resp. $\mathcal{O} = \mathcal{O}_{\mathcal{X}_{flat,fppf}}$). Since $g_!$ commutes with arbitrary colimits (as a left adjoint functor, see Lemma 94.11.3 and Categories, Lemma 4.24.5) we conclude that there exists an exact sequence $$ \bigoplus\nolimits_{j \in J} g_!\mathcal{O} \longrightarrow \bigoplus\nolimits_{i \in I} g_!\mathcal{O} \longrightarrow g_!\mathcal{H} \longrightarrow 0 $$ Finally, Lemma 94.11.4 shows that $g_!\mathcal{O} = \mathcal{O}_\mathcal{X}$ and we win. $\square$

    The code snippet corresponding to this tag is a part of the file stacks-cohomology.tex and is located in lines 2215–2229 (see updates for more information).

    \begin{lemma}
    \label{lemma-shriek-quasi-coherent}
    Let $\mathcal{X}$ be an algebraic stack. Notation as in
    Lemma \ref{lemma-lisse-etale}.
    \begin{enumerate}
    \item Let $\mathcal{H}$ be a quasi-coherent
    $\mathcal{O}_{\mathcal{X}_{lisse,\etale}}$-module 
    on the lisse-\'etale site of $\mathcal{X}$. Then $g_!\mathcal{H}$ is a
    quasi-coherent module on $\mathcal{X}$.
    \item Let $\mathcal{H}$ be a quasi-coherent
    $\mathcal{O}_{\mathcal{X}_{flat,fppf}}$-module 
    on the flat-fppf site of $\mathcal{X}$. Then $g_!\mathcal{H}$ is a
    quasi-coherent module on $\mathcal{X}$.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    Pick a scheme $U$ and a surjective smooth morphism $x : U \to \mathcal{X}$.
    By
    Modules on Sites, Definition \ref{sites-modules-definition-site-local}
    there exists an \'etale (resp.\ fppf) covering
    $\{U_i \to U\}_{i \in I}$ such that each pullback $f_i^{-1}\mathcal{H}$
    has a global presentation (see
    Modules on Sites, Definition \ref{sites-modules-definition-global}).
    Here $f_i : U_i \to \mathcal{X}$ is the composition
    $U_i \to U \to \mathcal{X}$ which is a morphism of algebraic stacks.
    (Recall that the pullback ``is'' the restriction to $\mathcal{X}/f_i$, see
    Sheaves on Stacks, Definition \ref{stacks-sheaves-definition-pullback}
    and the discussion following.) Since each $f_i$ is smooth (resp.\ flat) by
    Lemma \ref{lemma-lisse-etale-functorial}
    we see that $f_i^{-1}g_!\mathcal{H} = g_{i, !}(f'_i)^{-1}\mathcal{H}$.
    Using Lemma \ref{lemma-check-qc-on-etale-covering}
    we reduce the statement of the lemma to the case where $\mathcal{H}$
    has a global presentation. Say we have
    $$
    \bigoplus\nolimits_{j \in J} \mathcal{O} \longrightarrow
    \bigoplus\nolimits_{i \in I} \mathcal{O} \longrightarrow
    \mathcal{H} \longrightarrow 0
    $$
    of $\mathcal{O}$-modules where
    $\mathcal{O} = \mathcal{O}_{\mathcal{X}_{lisse,\etale}}$
    (resp.\ $\mathcal{O} = \mathcal{O}_{\mathcal{X}_{flat,fppf}}$).
    Since $g_!$ commutes with arbitrary colimits (as a left adjoint functor, see
    Lemma \ref{lemma-lisse-etale-modules} and
    Categories, Lemma \ref{categories-lemma-adjoint-exact})
    we conclude that there exists an exact sequence
    $$
    \bigoplus\nolimits_{j \in J} g_!\mathcal{O} \longrightarrow
    \bigoplus\nolimits_{i \in I} g_!\mathcal{O} \longrightarrow
    g_!\mathcal{H} \longrightarrow 0
    $$
    Finally, Lemma \ref{lemma-lisse-etale-structure-sheaf}
    shows that $g_!\mathcal{O} = \mathcal{O}_\mathcal{X}$
    and we win.
    \end{proof}

    Comments (3)

    Comment #3191 by anonymous on February 10, 2018 a 10:09 pm UTC

    The final statement "and we win" is a bit short. The last displayed sequence lives a priori in $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal{X})$. To deduce that $g_!\mathcal{H}$ is a quasi-coherent module in $\textit{Mod}(\mathcal{O}_\mathcal{X})$ we need that this sequence already lives in $\textit{Mod}(\mathcal{O}_\mathcal{X})$ and is exact there. This follows from the results about colimits in tags 0771 and 06WN. It would be good to mention this explicitly.

    Comment #3195 by Johan (site) on February 11, 2018 a 3:19 pm UTC

    In case (1) the sequence is exact as a sequence in modules with the etale topology. Hence the cokernel is a quasi-coherent on the big \'etale topos as the cokernel of a map of free modules. AHA, I see the problem: I should state in Section 06WF explicitly that if you have a sheaf of $\mathcal{O}_\mathcal{X}$-modules $\mathcal{F}$ on $\mathcal{X}$ in the Zariski, respectively \'etale topology and if $\mathcal{F}$ is quasi-coherent in the sense of Definition 03DL on the (ringed) Zariski respectively \'etale topos, then actually it is quasi-coherent in the sense of Definition 06WG. This follows as in my previous comment from the fact that we know this to be true for schemes. OK?

    Busy this week but will add this later. Thanks for the comment.

    Comment #3202 by anonymous on February 14, 2018 a 8:52 am UTC

    This sounds like a very good idea. Many thanks!

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

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