# The Stacks Project

## Tag 01BZ

Lemma 17.12.5. Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. Assume $\mathcal{O}_X$ is a coherent $\mathcal{O}_X$-module. Then $\mathcal{F}$ is coherent if and only if it is of finite presentation.

Proof. Omitted. $\square$

The code snippet corresponding to this tag is a part of the file modules.tex and is located in lines 1823–1830 (see updates for more information).

\begin{lemma}
\label{lemma-coherent-structure-sheaf}
Let $(X, \mathcal{O}_X)$ be a ringed space.
Let $\mathcal{F}$ be an $\mathcal{O}_X$-module.
Assume $\mathcal{O}_X$ is a coherent $\mathcal{O}_X$-module.
Then $\mathcal{F}$ is coherent if and only if it is
of finite presentation.
\end{lemma}

\begin{proof}
Omitted.
\end{proof}

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