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## Tag: 03DL

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Definition 17.23.1. 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 17.17.1.
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 $(\mathcal{C}/U, \mathcal{O}_U)$.

\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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