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Tag 03DL

Chapter 18: Modules on Sites > Section 18.23: Local types of modules

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 $(\mathcal{C}/U, \mathcal{O}_U)$.

    The code snippet corresponding to this tag is a part of the file sites-modules.tex and is located in lines 2877–2926 (see updates for more information).

    \begin{definition}
    \label{definition-site-local}
    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 \ref{definition-global}.
    \begin{enumerate}
    \item We say $\mathcal{F}$ is {\it 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.
    \item We say $\mathcal{F}$ is {\it 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.
    \item We say $\mathcal{F}$ is {\it 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.
    \item Given $r \geq 0$ we sat $\mathcal{F}$ is {\it 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.
    \item We say $\mathcal{F}$ is {\it 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.
    \item We say $\mathcal{F}$ is {\it 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.
    \item We say $\mathcal{F}$ is {\it 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.
    \item We say $\mathcal{F}$ is {\it 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 $(\mathcal{C}/U, \mathcal{O}_U)$.
    \end{enumerate}
    \end{definition}

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