The Stacks project

Exercise 111.36.7. 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 $\mathcal{F}$.