# The Stacks Project

## Tag: 05B0

This tag has label descent-lemma-finite-presentation-descends and it points to

The corresponding content:

Lemma 31.6.2. Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $\{f_i : X_i \to X\}_{i \in I}$ be an fpqc covering such that each $f_i^*\mathcal{F}$ is an $\mathcal{O}_{X_i}$-module of finite presentation. Then $\mathcal{F}$ is an $\mathcal{O}_X$-module of finite presentation.

Proof. Omitted. For the affine case, see Algebra, Lemma 9.80.2. $\square$

\begin{lemma}
\label{lemma-finite-presentation-descends}
Let $X$ be a scheme.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
Let $\{f_i : X_i \to X\}_{i \in I}$ be an fpqc covering such that
each $f_i^*\mathcal{F}$ is an $\mathcal{O}_{X_i}$-module of finite
presentation. Then $\mathcal{F}$ is an $\mathcal{O}_X$-module
of finite presentation.
\end{lemma}

\begin{proof}
Omitted. For the affine case, see
Algebra, Lemma \ref{algebra-lemma-descend-properties-modules}.
\end{proof}


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