110.36 Quasi-coherent Sheaves

Definition 110.36.1. Let $X$ be a scheme. A sheaf $\mathcal{F}$ of $\mathcal{O}_ X$-modules is quasi-coherent if for every affine open $\mathop{\mathrm{Spec}}(R) = U \subset X$ the restriction $\mathcal{F}|_ U$ is of the form $\widetilde M$ for some $R$-module $M$.

It is enough to check this conditions on the members of an affine open covering of $X$. See Schemes, Section 26.24 for more results.

Definition 110.36.2. Let $X$ be a topological space. Let $x, x' \in X$. We say $x$ is a specialization of $x'$ if and only if $x \in \overline{\{ x'\} }$.

Exercise 110.36.3. Let $X$ be a scheme. Let $x, x' \in X$. Let $\mathcal{F}$ be a quasi-coherent sheaf of $\mathcal{O}_ X$-modules. Suppose that (a) $x$ is a specialization of $x'$ and (b) $\mathcal{F}_{x'} \not= 0$. Show that $\mathcal{F}_ x \not= 0$.

Exercise 110.36.4. Find an example of a scheme $X$, points $x, x' \in X$, a sheaf of $\mathcal{O}_ X$-modules $\mathcal{F}$ such that (a) $x$ is a specialization of $x'$ and (b) $\mathcal{F}_{x'} \not= 0$ and $\mathcal{F}_ x = 0$.

Definition 110.36.5. A scheme $X$ is called locally Noetherian if and only if for every point $x \in X$ there exists an affine open $\mathop{\mathrm{Spec}}(R) = U \subset X$ such that $R$ is Noetherian. A scheme is Noetherian if it is locally Noetherian and quasi-compact.

If $X$ is locally Noetherian then any affine open of $X$ is the spectrum of a Noetherian ring, see Properties, Lemma 28.5.2.

Definition 110.36.6. Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a quasi-coherent sheaf of $\mathcal{O}_ X$-modules. We say $\mathcal{F}$ is coherent if for every point $x \in X$ there exists an affine open $\mathop{\mathrm{Spec}}(R) = U \subset X$ such that $\mathcal{F}|_ U$ is isomorphic to $\widetilde M$ for some finite $R$-module $M$.

Exercise 110.36.7. 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 $\mathcal{F}$.

Remark 110.36.8. If $U \to X$ is a quasi-compact immersion then any quasi-coherent sheaf on $U$ is the restriction of a quasi-coherent sheaf on $X$. If $X$ is a Noetherian scheme, and $U \subset X$ is open, then any coherent sheaf on $U$ is the restriction of a coherent sheaf on $X$. Of course the exercise above is easier, and shouldn't use these general facts.

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