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*}

18.23 Local types of modules

According to our general strategy explained in Section 18.18 we first define the local types for sheaves of modules on a ringed site, and then we immediately show that these types are intrinsic, hence make sense for sheaves of modules on ringed topoi.

Definition 18.23.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules. We will freely use the notions defined in Definition 18.17.1.

  1. We say $\mathcal{F}$ is locally free if for every object $U$ of $\mathcal{C}$ there exists a covering $\{ U_ i \to U\} _{i \in I}$ of $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/U_ i}$ is a free $\mathcal{O}_{U_ i}$-module.

  2. We say $\mathcal{F}$ is finite locally free if for every object $U$ of $\mathcal{C}$ there exists a covering $\{ U_ i \to U\} _{i \in I}$ of $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/U_ i}$ is a finite free $\mathcal{O}_{U_ i}$-module.

  3. We say $\mathcal{F}$ is locally generated by sections if for every object $U$ of $\mathcal{C}$ there exists a covering $\{ U_ i \to U\} _{i \in I}$ of $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/U_ i}$ is an $\mathcal{O}_{U_ i}$-module generated by global sections.

  4. Given $r \geq 0$ we sat $\mathcal{F}$ is locally generated by $r$ sections if for every object $U$ of $\mathcal{C}$ there exists a covering $\{ U_ i \to U\} _{i \in I}$ of $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/U_ i}$ is an $\mathcal{O}_{U_ i}$-module generated by $r$ global sections.

  5. We say $\mathcal{F}$ is of finite type if for every object $U$ of $\mathcal{C}$ there exists a covering $\{ U_ i \to U\} _{i \in I}$ of $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/U_ i}$ is an $\mathcal{O}_{U_ i}$-module generated by finitely many global sections.

  6. We say $\mathcal{F}$ is quasi-coherent if for every object $U$ of $\mathcal{C}$ there exists a covering $\{ U_ i \to U\} _{i \in I}$ of $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/U_ i}$ is an $\mathcal{O}_{U_ i}$-module which has a global presentation.

  7. We say $\mathcal{F}$ is of finite presentation if for every object $U$ of $\mathcal{C}$ there exists a covering $\{ U_ i \to U\} _{i \in I}$ of $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/U_ i}$ is an $\mathcal{O}_{U_ i}$-module which has a finite global presentation.

  8. We say $\mathcal{F}$ is coherent if and only if $\mathcal{F}$ is of finite type, and for every object $U$ of $\mathcal{C}$ and any $s_1, \ldots , s_ n \in \mathcal{F}(U)$ the kernel of the map $\bigoplus _{i = 1, \ldots , n} \mathcal{O}_ U \to \mathcal{F}|_ U$ is of finite type on $(\mathcal{C}/U, \mathcal{O}_ U)$.

Proof. Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a special cocontinuous functor. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules on $\mathcal{C}$. Let $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ be the equivalence of topoi associated to $u$. Set $\mathcal{O}' = g_*\mathcal{O}$, and let $g^\sharp : \mathcal{O}' \to g_*\mathcal{O}$ be the identity. Finally, set $\mathcal{F}' = g_*\mathcal{F}$. Let $\mathcal{P}_ l$ be one of the properties (1) – (7) listed in Definition 18.23.1. (We will discuss the coherent case at the end of the proof.) Let $\mathcal{P}_ g$ denote the corresponding property listed in Definition 18.17.1. We have already seen that $\mathcal{P}_ g$ is intrinsic. We have to show that $\mathcal{P}_ l(\mathcal{C}, \mathcal{O}, \mathcal{F})$ holds if and only if $\mathcal{P}_ l(\mathcal{D}, \mathcal{O}', \mathcal{F}')$ holds.

Assume that $\mathcal{F}$ has $\mathcal{P}_ l$. Let $V$ be an object of $\mathcal{D}$. One of the properties of a special cocontinuous functor is that there exists a covering $\{ u(U_ i) \to V\} _{i \in I}$ in the site $\mathcal{D}$. By assumption, for each $i$ there exists a covering $\{ U_{ij} \to U_ i\} _{j \in J_ i}$ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{U_{ij}}$ is $\mathcal{P}_ g$. By Sites, Lemma 7.29.3 we have commutative diagrams of ringed topoi

\[ \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U_{ij}), \mathcal{O}_{U_{ij}}) \ar[r] \ar[d] & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \ar[d] \\ (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}/u(U_{ij})), \mathcal{O}'_{u(U_{ij})}) \ar[r] & (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}') } \]

where the vertical arrows are equivalences. Hence we conclude that $\mathcal{F}'|_{u(U_{ij})}$ has property $\mathcal{P}_ g$ also. And moreover, $\{ u(U_{ij}) \to V\} _{i \in I, j \in J_ i}$ is a covering of the site $\mathcal{D}$. Hence $\mathcal{F}'$ has property $\mathcal{P}_ l$.

Assume that $\mathcal{F}'$ has $\mathcal{P}_ l$. Let $U$ be an object of $\mathcal{C}$. By assumption, there exists a covering $\{ V_ i \to u(U)\} _{i \in I}$ such that $\mathcal{F}'|_{V_ i}$ has property $\mathcal{P}_ g$. Because $u$ is cocontinuous we can refine this covering by a family $\{ u(U_ j) \to u(U)\} _{j \in J}$ where $\{ U_ j \to U\} _{j \in J}$ is a covering in $\mathcal{C}$. Say the refinement is given by $\alpha : J \to I$ and $u(U_ j) \to V_{\alpha (j)}$. Restricting is transitive, i.e., $(\mathcal{F}'|_{V_{\alpha (j)}})|_{u(U_ j)} = \mathcal{F}'|_{u(U_ j)}$. Hence by Lemma 18.17.2 we see that $\mathcal{F}'|_{u(U_ j)}$ has property $\mathcal{P}_ g$. Hence the diagram

\[ \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U_ j), \mathcal{O}_{U_ j}) \ar[r] \ar[d] & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \ar[d] \\ (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}/u(U_ j)), \mathcal{O}'_{u(U_ j)}) \ar[r] & (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}') } \]

where the vertical arrows are equivalences shows that $\mathcal{F}|_{U_ j}$ has property $\mathcal{P}_ g$ also. Thus $\mathcal{F}$ has property $\mathcal{P}_ l$ as desired.

Finally, we prove the lemma in case $\mathcal{P}_ l = coherent$1. Assume $\mathcal{F}$ is coherent. This implies that $\mathcal{F}$ is of finite type and hence $\mathcal{F}'$ is of finite type also by the first part of the proof. Let $V$ be an object of $\mathcal{D}$ and let $s_1, \ldots , s_ n \in \mathcal{F}'(V)$. We have to show that the kernel $\mathcal{K}'$ of $\bigoplus _{j = 1, \ldots , n} \mathcal{O}_ V \to \mathcal{F}'|_ V$ is of finite type on $\mathcal{D}/V$. This means we have to show that for any $V'/V$ there exists a covering $\{ V'_ i \to V'\} $ such that $\mathcal{F}'|_{V'_ i}$ is generated by finitely many sections. Replacing $V$ by $V'$ (and restricting the sections $s_ j$ to $V'$) we reduce to the case where $V' = V$. Since $u$ is a special cocontinuous functor, there exists a covering $\{ u(U_ i) \to V\} _{i \in I}$ in the site $\mathcal{D}$. Using the isomorphism of topoi $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U_ i) = \mathop{\mathit{Sh}}\nolimits (\mathcal{D}/u(U_ i))$ we see that $\mathcal{K}'|_{u(U_ i)}$ corresponds to the kernel $\mathcal{K}_ i$ of a map $\bigoplus _{j = 1, \ldots , n} \mathcal{O}_{U_ i} \to \mathcal{F}|_{U_ i}$. Since $\mathcal{F}$ is coherent we see that $\mathcal{K}_ i$ is of finite type. Hence we conclude (by the first part of the proof again) that $\mathcal{K}|_{u(U_ i)}$ is of finite type. Thus there exist coverings $\{ V_{il} \to u(U_ i)\} $ such that $\mathcal{K}|_{V_{il}}$ is generated by finitely many global sections. Since $\{ V_{il} \to V\} $ is a covering of $\mathcal{D}$ we conclude that $\mathcal{K}$ is of finite type as desired.

Assume $\mathcal{F}'$ is coherent. This implies that $\mathcal{F}'$ is of finite type and hence $\mathcal{F}$ is of finite type also by the first part of the proof. Let $U$ be an object of $\mathcal{C}$, and let $s_1, \ldots , s_ n \in \mathcal{F}(U)$. We have to show that the kernel $\mathcal{K}$ of $\bigoplus _{j = 1, \ldots , n} \mathcal{O}_ U \to \mathcal{F}|_ U$ is of finite type on $\mathcal{C}/U$. Using the isomorphism of topoi $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) = \mathop{\mathit{Sh}}\nolimits (\mathcal{D}/u(U))$ we see that $\mathcal{K}|_ U$ corresponds to the kernel $\mathcal{K}'$ of a map $\bigoplus _{j = 1, \ldots , n} \mathcal{O}_{u(U)} \to \mathcal{F}'|_{u(U)}$. As $\mathcal{F}'$ is coherent, we see that $\mathcal{K}'$ is of finite type. Hence, by the first part of the proof again, we conclude that $\mathcal{K}$ is of finite type. $\square$

Hence from now on we may refer to the properties of $\mathcal{O}$-modules defined in Definition 18.23.1 without specifying a site.

Lemma 18.23.3. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be a ringed topos. Let $\mathcal{F}$ be an $\mathcal{O}$-module. Assume that the site $\mathcal{C}$ has a final object $X$. Then

  1. The following are equivalent

    1. $\mathcal{F}$ is locally free,

    2. there exists a covering $\{ X_ i \to X\} $ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_ i}$ is a locally free $\mathcal{O}_{X_ i}$-module, and

    3. there exists a covering $\{ X_ i \to X\} $ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_ i}$ is a free $\mathcal{O}_{X_ i}$-module.

  2. The following are equivalent

    1. $\mathcal{F}$ is finite locally free,

    2. there exists a covering $\{ X_ i \to X\} $ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_ i}$ is a finite locally free $\mathcal{O}_{X_ i}$-module, and

    3. there exists a covering $\{ X_ i \to X\} $ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_ i}$ is a finite free $\mathcal{O}_{X_ i}$-module.

  3. The following are equivalent

    1. $\mathcal{F}$ is locally generated by sections,

    2. there exists a covering $\{ X_ i \to X\} $ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_ i}$ is an $\mathcal{O}_{X_ i}$-module locally generated by sections, and

    3. there exists a covering $\{ X_ i \to X\} $ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_ i}$ is an $\mathcal{O}_{X_ i}$-module globally generated by sections.

  4. Given $r \geq 0$, the following are equivalent

    1. $\mathcal{F}$ is locally generated by $r$ sections,

    2. there exists a covering $\{ X_ i \to X\} $ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_ i}$ is an $\mathcal{O}_{X_ i}$-module locally generated by $r$ sections, and

    3. there exists a covering $\{ X_ i \to X\} $ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_ i}$ is an $\mathcal{O}_{X_ i}$-module globally generated by $r$ sections.

  5. The following are equivalent

    1. $\mathcal{F}$ is of finite type,

    2. there exists a covering $\{ X_ i \to X\} $ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_ i}$ is an $\mathcal{O}_{X_ i}$-module of finite type, and

    3. there exists a covering $\{ X_ i \to X\} $ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_ i}$ is an $\mathcal{O}_{X_ i}$-module globally generated by finitely many sections.

  6. The following are equivalent

    1. $\mathcal{F}$ is quasi-coherent,

    2. there exists a covering $\{ X_ i \to X\} $ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_ i}$ is a quasi-coherent $\mathcal{O}_{X_ i}$-module, and

    3. there exists a covering $\{ X_ i \to X\} $ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_ i}$ is an $\mathcal{O}_{X_ i}$-module which has a global presentation.

  7. The following are equivalent

    1. $\mathcal{F}$ is of finite presentation,

    2. there exists a covering $\{ X_ i \to X\} $ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_ i}$ is an $\mathcal{O}_{X_ i}$-module of finite presentation, and

    3. there exists a covering $\{ X_ i \to X\} $ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_ i}$ is an $\mathcal{O}_{X_ i}$-module has a finite global presentation.

  8. The following are equivalent

    1. $\mathcal{F}$ is coherent, and

    2. there exists a covering $\{ X_ i \to X\} $ in $\mathcal{C}$ such that each restriction $\mathcal{F}|_{\mathcal{C}/X_ i}$ is a coherent $\mathcal{O}_{X_ i}$-module.

Proof. In each case we have (a) $\Rightarrow (b)$. In each of the cases (1) - (6) condition (b) implies condition (c) by axiom (2) of a site (see Sites, Definition 7.6.2) and the definition of the local types of modules. Suppose $\{ X_ i \to X\} $ is a covering. Then for every object $U$ of $\mathcal{C}$ we get an induced covering $\{ X_ i \times _ X U \to U\} $. Moreover, the global property for $\mathcal{F}|_{\mathcal{C}/X_ i}$ in part (c) implies the corresponding global property for $\mathcal{F}|_{\mathcal{C}/X_ i \times _ X U}$ by Lemma 18.17.2, hence the sheaf has property (a) by definition. We omit the proof of (b) $\Rightarrow $ (a) in case (7). $\square$

Lemma 18.23.4. 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 locally free then $f^*\mathcal{F}$ is locally free.

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

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

  4. If $\mathcal{F}$ is locally generated by $r$ sections then $f^*\mathcal{F}$ is locally generated by $r$ sections.

  5. If $\mathcal{F}$ is of finite type then $f^*\mathcal{F}$ is of finite type.

  6. If $\mathcal{F}$ is quasi-coherent then $f^*\mathcal{F}$ is quasi-coherent.

  7. If $\mathcal{F}$ is of finite presentation then $f^*\mathcal{F}$ is of finite presentation.

Proof. According to the discussion in Section 18.18 we need only check preservation under pullback for a morphism of ringed sites $(f, f^\sharp ) : (\mathcal{C}, \mathcal{O}_\mathcal {C}) \to (\mathcal{D}, \mathcal{O}_\mathcal {D})$ such that $f$ is given by a left exact, continuous functor $u : \mathcal{D} \to \mathcal{C}$ between sites which have all finite limits. Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_\mathcal {D}$-modules which has one of the properties (1) – (6) of Definition 18.23.1. We know $\mathcal{D}$ has a final object $Y$ and $X = u(Y)$ is a final object for $\mathcal{C}$. By assumption we have a covering $\{ Y_ i \to Y\} $ such that $\mathcal{G}|_{\mathcal{D}/Y_ i}$ has the corresponding global property. Set $X_ i = u(Y_ i)$ so that $\{ X_ i \to X\} $ is a covering in $\mathcal{C}$. We get a commutative diagram of morphisms ringed sites

\[ \xymatrix{ (\mathcal{C}/X_ i, \mathcal{O}_\mathcal {C}|_{X_ i}) \ar[r] \ar[d] & (\mathcal{C}, \mathcal{O}_\mathcal {C}) \ar[d] \\ (\mathcal{D}/Y_ i, \mathcal{O}_\mathcal {D}|_{Y_ i}) \ar[r] & (\mathcal{D}, \mathcal{O}_\mathcal {D}) } \]

by Sites, Lemma 7.28.2. Hence by Lemma 18.17.2 that $f^*\mathcal{G}|_{X_ i}$ has the corresponding global property. Hence we conclude that $\mathcal{G}$ has the local property we started out with by Lemma 18.23.3. $\square$

[1] The mechanics of this are a bit awkward, and we suggest the reader skip this part of the proof.

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