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Tag 07AZ

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

Lemma 94.12.1. Let $\mathcal{X}$ be an algebraic stack.

  1. Let $f_j : \mathcal{X}_j \to \mathcal{X}$ be a family of smooth morphisms of algebraic stacks with $|\mathcal{X}| =\bigcup |f_j|(|\mathcal{X}_j|)$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_\mathcal{X}$-modules on $\mathcal{X}_{\acute{e}tale}$. If each $f_j^{-1}\mathcal{F}$ is quasi-coherent, then so is $\mathcal{F}$.
  2. Let $f_j : \mathcal{X}_j \to \mathcal{X}$ be a family of flat and locally finitely presented morphisms of algebraic stacks with $|\mathcal{X}| =\bigcup |f_j|(|\mathcal{X}_j|)$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_\mathcal{X}$-modules on $\mathcal{X}_{fppf}$. If each $f_j^{-1}\mathcal{F}$ is quasi-coherent, then so is $\mathcal{F}$.

Proof. Proof of (1). We may replace each of the algebraic stacks $\mathcal{X}_j$ by a scheme $U_j$ (using that any algebraic stack has a smooth covering by a scheme and that compositions of smooth morphisms are smooth, see Morphisms of Stacks, Lemma 92.32.2). The pullback of $\mathcal{F}$ to $(\mathit{Sch}/U_j)_{\acute{e}tale}$ is still locally quasi-coherent, see Sheaves on Stacks, Lemma 87.11.2. Then $f = \coprod f_j : U = \coprod U_j \to \mathcal{X}$ is a smooth surjective morphism. Let $x : V \to \mathcal{X}$ be an object of $\mathcal{X}$. By Sheaves on Stacks, Lemma 87.18.10 there exists an étale covering $\{x_i \to x\}_{i \in I}$ such that each $x_i$ lifts to an object $u_i$ of $(\mathit{Sch}/U)_{\acute{e}tale}$. This just means that $x_i$ lives over a scheme $V_i$, that $\{V_i \to V\}$ is an étale covering, and that $x_i$ comes from a morphism $u_i : V_i \to U$. Then $x_i^*\mathcal{F} = u_i^*f^*\mathcal{F}$ is quasi-coherent. This implies that $x^*\mathcal{F}$ on $(\mathit{Sch}/V)_{\acute{e}tale}$ is quasi-coherent, for example by Modules on Sites, Lemma 18.23.3. By Sheaves on Stacks, Lemma 87.11.3 we see that $\mathcal{F}$ is quasi-coherent.

Proof of (2). This is proved using exactly the same argument, which we fully write out here. We may replace each of the algebraic stacks $\mathcal{X}_j$ by a scheme $U_j$ (using that any algebraic stack has a smooth covering by a scheme and that flat and locally finite presented morphisms are preserved by composition, see Morphisms of Stacks, Lemmas 92.24.2 and 92.26.2). The pullback of $\mathcal{F}$ to $(\mathit{Sch}/U_j)_{\acute{e}tale}$ is still locally quasi-coherent, see Sheaves on Stacks, Lemma 87.11.2. Then $f = \coprod f_j : U = \coprod U_j \to \mathcal{X}$ is a surjective, flat, and locally finitely presented morphism. Let $x : V \to \mathcal{X}$ be an object of $\mathcal{X}$. By Sheaves on Stacks, Lemma 87.18.10 there exists an fppf covering $\{x_i \to x\}_{i \in I}$ such that each $x_i$ lifts to an object $u_i$ of $(\mathit{Sch}/U)_{\acute{e}tale}$. This just means that $x_i$ lives over a scheme $V_i$, that $\{V_i \to V\}$ is an fppf covering, and that $x_i$ comes from a morphism $u_i : V_i \to U$. Then $x_i^*\mathcal{F} = u_i^*f^*\mathcal{F}$ is quasi-coherent. This implies that $x^*\mathcal{F}$ on $(\mathit{Sch}/V)_{\acute{e}tale}$ is quasi-coherent, for example by Modules on Sites, Lemma 18.23.3. By Sheaves on Stacks, Lemma 87.11.3 we see that $\mathcal{F}$ is quasi-coherent. $\square$

    The code snippet corresponding to this tag is a part of the file stacks-cohomology.tex and is located in lines 2136–2153 (see updates for more information).

    \begin{lemma}
    \label{lemma-check-qc-on-etale-covering}
    Let $\mathcal{X}$ be an algebraic stack.
    \begin{enumerate}
    \item Let $f_j : \mathcal{X}_j \to \mathcal{X}$ be a family of smooth
    morphisms of algebraic stacks with
    $|\mathcal{X}| =\bigcup |f_j|(|\mathcal{X}_j|)$.
    Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_\mathcal{X}$-modules
    on $\mathcal{X}_\etale$. If each $f_j^{-1}\mathcal{F}$
    is quasi-coherent, then so is $\mathcal{F}$.
    \item Let $f_j : \mathcal{X}_j \to \mathcal{X}$ be a family of flat and
    locally finitely presented morphisms of algebraic stacks with
    $|\mathcal{X}| =\bigcup |f_j|(|\mathcal{X}_j|)$.
    Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_\mathcal{X}$-modules
    on $\mathcal{X}_{fppf}$. If each $f_j^{-1}\mathcal{F}$
    is quasi-coherent, then so is $\mathcal{F}$.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    Proof of (1). We may replace each of the algebraic stacks $\mathcal{X}_j$
    by a scheme $U_j$ (using that any algebraic stack has a smooth covering by
    a scheme and that compositions of smooth morphisms are smooth, see
    Morphisms of Stacks, Lemma \ref{stacks-morphisms-lemma-composition-smooth}).
    The pullback of $\mathcal{F}$ to $(\Sch/U_j)_\etale$ is still
    locally quasi-coherent, see
    Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-pullback-quasi-coherent}.
    Then $f = \coprod f_j : U = \coprod U_j \to \mathcal{X}$ is a smooth surjective
    morphism. Let $x : V \to \mathcal{X}$ be an object of $\mathcal{X}$. By
    Sheaves on Stacks, Lemma
    \ref{stacks-sheaves-lemma-surjective-flat-locally-finite-presentation}
    there exists an \'etale covering $\{x_i \to x\}_{i \in I}$
    such that each $x_i$ lifts to an object $u_i$ of $(\Sch/U)_\etale$.
    This just means that $x_i$ lives over a scheme $V_i$, that
    $\{V_i \to V\}$ is an \'etale covering, and that $x_i$ comes from
    a morphism $u_i : V_i \to U$. Then
    $x_i^*\mathcal{F} = u_i^*f^*\mathcal{F}$ is quasi-coherent.
    This implies that $x^*\mathcal{F}$ on $(\Sch/V)_\etale$
    is quasi-coherent, for example by
    Modules on Sites, Lemma \ref{sites-modules-lemma-local-final-object}.
    By Sheaves on Stacks, Lemma
    \ref{stacks-sheaves-lemma-characterize-quasi-coherent}
    we see that $\mathcal{F}$ is quasi-coherent.
    
    \medskip\noindent
    Proof of (2). This is proved using exactly the same argument, which we fully
    write out here. We may replace each of the algebraic stacks $\mathcal{X}_j$
    by a scheme $U_j$ (using that any algebraic stack has a smooth covering by
    a scheme and that flat and locally finite presented morphisms are preserved
    by composition, see Morphisms of Stacks, Lemmas
    \ref{stacks-morphisms-lemma-composition-flat} and
    \ref{stacks-morphisms-lemma-composition-finite-presentation}).
    The pullback of $\mathcal{F}$ to $(\Sch/U_j)_\etale$ is still
    locally quasi-coherent, see
    Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-pullback-quasi-coherent}.
    Then $f = \coprod f_j : U = \coprod U_j \to \mathcal{X}$ is a surjective,
    flat, and locally finitely presented morphism. Let
    $x : V \to \mathcal{X}$ be an object of $\mathcal{X}$. By
    Sheaves on Stacks, Lemma
    \ref{stacks-sheaves-lemma-surjective-flat-locally-finite-presentation}
    there exists an fppf covering $\{x_i \to x\}_{i \in I}$
    such that each $x_i$ lifts to an object $u_i$ of $(\Sch/U)_\etale$.
    This just means that $x_i$ lives over a scheme $V_i$, that
    $\{V_i \to V\}$ is an fppf covering, and that $x_i$ comes from
    a morphism $u_i : V_i \to U$. Then
    $x_i^*\mathcal{F} = u_i^*f^*\mathcal{F}$ is quasi-coherent.
    This implies that $x^*\mathcal{F}$ on $(\Sch/V)_\etale$
    is quasi-coherent, for example by
    Modules on Sites, Lemma \ref{sites-modules-lemma-local-final-object}.
    By Sheaves on Stacks, Lemma
    \ref{stacks-sheaves-lemma-characterize-quasi-coherent}
    we see that $\mathcal{F}$ is quasi-coherent.
    \end{proof}

    Comments (2)

    Comment #3193 by anonymous on February 11, 2018 a 9:08 am UTC

    Concerning part (1) of the proof: Lemmas 87.11.2 and 87.11.3 speak about fppf-sheaves and not about \'e{}tale sheaves.

    Concerning both parts of the proof: when referring to Lemma 87.11.2 "is still locally quasi-coherent" should be "is still quasi-coherent".

    Comment #3197 by Johan (site) on February 11, 2018 a 3:22 pm UTC

    Please see comment #3195. I will fix the other thing soonish.

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

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