The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 95.12.3. 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 95.11.2 for the lisse-├ętale 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 95.9.1.

  2. With $g$ as in Lemma 95.11.2 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 95.9.1.

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; 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 88.11.4. The same is true for $g_!$ by Lemma 95.12.2. We know that $\mathcal{H} \to g^{-1}g_!\mathcal{H}$ is an isomorphism by Lemma 95.11.2. 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 88.13 and 88.14 (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 95.9.2 the kernel and cokernel of the map $Q(\mathcal{F}) \to \mathcal{F}$ are parasitic. Hence by Lemma 95.11.5 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. $\square$


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 95.12: Quasi-coherent modules, II

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 07B1. Beware of the difference between the letter 'O' and the digit '0'.