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*}

18.17 Global types of modules

Definition 18.17.1. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be a ringed topos. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules.

  1. We say $\mathcal{F}$ is a free $\mathcal{O}$-module if $\mathcal{F}$ is isomorphic as an $\mathcal{O}$-module to a sheaf of the form $\bigoplus _{i \in I} \mathcal{O}$.

  2. We say $\mathcal{F}$ is finite free if $\mathcal{F}$ is isomorphic as an $\mathcal{O}$-module to a sheaf of the form $\bigoplus _{i \in I} \mathcal{O}$ with a finite index set $I$.

  3. We say $\mathcal{F}$ is generated by global sections if there exists a surjection

    \[ \bigoplus \nolimits _{i \in I} \mathcal{O} \longrightarrow \mathcal{F} \]

    from a free $\mathcal{O}$-module onto $\mathcal{F}$.

  4. Given $r \geq 0$ we say $\mathcal{F}$ is generated by $r$ global sections if there exists a surjection $\mathcal{O}^{\oplus r} \to \mathcal{F}$.

  5. We say $\mathcal{F}$ is generated by finitely many global sections if it is generated by $r$ global sections for some $r \geq 0$.

  6. We say $\mathcal{F}$ has a global presentation if there exists an exact sequence

    \[ \bigoplus \nolimits _{j \in J} \mathcal{O} \longrightarrow \bigoplus \nolimits _{i \in I} \mathcal{O} \longrightarrow \mathcal{F} \longrightarrow 0 \]

    of $\mathcal{O}$-modules.

  7. We say $\mathcal{F}$ has a global finite presentation if there exists an exact sequence

    \[ \bigoplus \nolimits _{j \in J} \mathcal{O} \longrightarrow \bigoplus \nolimits _{i \in I} \mathcal{O} \longrightarrow \mathcal{F} \longrightarrow 0 \]

    of $\mathcal{O}$-modules with $I$ and $J$ finite sets.

Note that for any set $I$ the direct sum $\bigoplus _{i \in I} \mathcal{O}$ exists (Lemma 18.14.2) and is the sheafification of the presheaf $U \mapsto \bigoplus _{i \in I} \mathcal{O}(U)$. This module is called the free $\mathcal{O}$-module on the set $I$.

Lemma 18.17.2. Let $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. Let $\mathcal{F}$ be an $\mathcal{O}_\mathcal {D}$-module.

  1. If $\mathcal{F}$ is free then $f^*\mathcal{F}$ is free.

  2. If $\mathcal{F}$ is finite free then $f^*\mathcal{F}$ is finite free.

  3. If $\mathcal{F}$ is generated by global sections then $f^*\mathcal{F}$ is generated by global sections.

  4. Given $r \geq 0$ if $\mathcal{F}$ is generated by $r$ global sections, then $f^*\mathcal{F}$ is generated by $r$ global sections.

  5. If $\mathcal{F}$ is generated by finitely many global sections then $f^*\mathcal{F}$ is generated by finitely many global sections.

  6. If $\mathcal{F}$ has a global presentation then $f^*\mathcal{F}$ has a global presentation.

  7. If $\mathcal{F}$ has a finite global presentation then $f^*\mathcal{F}$ has a finite global presentation.

Proof. This is true because $f^*$ commutes with arbitrary colimits (Lemma 18.14.3) and $f^*\mathcal{O}_\mathcal {D} = \mathcal{O}_\mathcal {C}$. $\square$


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