# The Stacks Project

## Tag: 01BE

This tag has label modules-definition-quasi-coherent and it points to

The corresponding content:

Definition 16.10.1. Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. We say that $\mathcal{F}$ is a {\it quasi-coherent sheaf of $\mathcal{O}_X$-modules} if for every point $x \in X$ there exists an open neighbourhood $x\in U \subset X$ such that $\mathcal{F}|_U$ is isomorphic to the cokernel of a map $$\bigoplus\nolimits_{j \in J} \mathcal{O}_U \longrightarrow \bigoplus\nolimits_{i \in I} \mathcal{O}_U$$ The category of quasi-coherent $\mathcal{O}_X$-modules is denoted $\textit{QCoh}(\mathcal{O}_X)$.

\begin{definition}
\label{definition-quasi-coherent}
Let $(X, \mathcal{O}_X)$ be a ringed space.
Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules.
We say that $\mathcal{F}$ is a {\it quasi-coherent
sheaf of $\mathcal{O}_X$-modules} if for every
point $x \in X$ there exists an open neighbourhood
$x\in U \subset X$ such that $\mathcal{F}|_U$
is isomorphic to the cokernel of a map
$$\bigoplus\nolimits_{j \in J} \mathcal{O}_U \longrightarrow \bigoplus\nolimits_{i \in I} \mathcal{O}_U$$
The category of quasi-coherent $\mathcal{O}_X$-modules
is denoted $\textit{QCoh}(\mathcal{O}_X)$.
\end{definition}


To cite this tag (see How to reference tags), use:

\cite[\href{http://stacks.math.columbia.edu/tag/01BE}{Tag 01BE}]{stacks-project}


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 lower-right corner).