# The Stacks Project

## Tag 029U

Exercise 102.35.7. Let $X = \mathop{\mathrm{Spec}}(R)$ be an affine scheme.

1. Let $f \in R$. Let $\mathcal{G}$ be a quasi-coherent sheaf of $\mathcal{O}_{D(f)}$-modules on the open subscheme $D(f)$. Show that $\mathcal{G} = \mathcal{F}|_{D(f)}$ for some quasi-coherent sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$.
2. Let $I \subset R$ be an ideal. Let $i : Z \to X$ be the closed subscheme of $X$ corresponding to $I$. Let $\mathcal{G}$ be a quasi-coherent sheaf of $\mathcal{O}_Z$-modules on the closed subscheme $Z$. Show that $\mathcal{G} = i^*\mathcal{F}$ for some quasi-coherent sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$. (Why is this silly?)
3. Assume that $R$ is Noetherian. Let $f \in R$. Let $\mathcal{G}$ be a coherent sheaf of $\mathcal{O}_{D(f)}$-modules on the open subscheme $D(f)$. Show that $\mathcal{G} = \mathcal{F}|_{D(f)}$ for some coherent sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$.

The code snippet corresponding to this tag is a part of the file exercises.tex and is located in lines 3321–3346 (see updates for more information).

\begin{exercise}
\label{exercise-extend-quasi-coherent}
Let $X = \Spec(R)$ be an affine scheme.
\begin{enumerate}
\item Let $f \in R$. Let $\mathcal{G}$ be a
quasi-coherent sheaf of $\mathcal{O}_{D(f)}$-modules
on the open subscheme $D(f)$.
Show that $\mathcal{G} = \mathcal{F}|_{D(f)}$ for some
quasi-coherent sheaf of $\mathcal{O}_X$-modules
$\mathcal{F}$.
\item Let $I \subset R$ be an ideal.
Let $i : Z \to X$ be the closed subscheme of $X$ corresponding
to $I$. Let $\mathcal{G}$ be a
quasi-coherent sheaf of $\mathcal{O}_Z$-modules
on the closed subscheme $Z$.
Show that $\mathcal{G} = i^*\mathcal{F}$ for some
quasi-coherent sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$.
(Why is this silly?)
\item Assume that $R$ is Noetherian.
Let $f \in R$. Let $\mathcal{G}$ be a
coherent sheaf of $\mathcal{O}_{D(f)}$-modules
on the open subscheme $D(f)$.
Show that $\mathcal{G} = \mathcal{F}|_{D(f)}$ for some
coherent sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$.
\end{enumerate}
\end{exercise}

## Comments (2)

Comment #2363 by Dominic Wynter on January 30, 2017 a 7:23 pm UTC

I presume that we have set $U:=D(f)$?

Comment #2426 by Johan (site) on February 17, 2017 a 2:23 pm UTC

Thanks, fixed here.

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