17.9 Modules of finite type

Definition 17.9.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules. We say that $\mathcal{F}$ is of finite type if for every $x \in X$ there exists an open neighbourhood $U$ such that $\mathcal{F}|_ U$ is generated by finitely many sections.

Lemma 17.9.2. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. The pullback $f^*\mathcal{G}$ of a finite type $\mathcal{O}_ Y$-module is a finite type $\mathcal{O}_ X$-module.

Proof. Arguing as in the proof of Lemma 17.8.2 we may assume $\mathcal{G}$ is globally generated by finitely many sections. We have seen that $f^*$ commutes with all colimits, and is right exact, see Lemma 17.3.3. Thus if we have a surjection

$\bigoplus \nolimits _{i = 1, \ldots , n} \mathcal{O}_ Y \to \mathcal{G} \to 0$

then upon applying $f^*$ we obtain the surjection

$\bigoplus \nolimits _{i = 1, \ldots , n} \mathcal{O}_ X \to f^*\mathcal{G} \to 0.$

This implies the lemma. $\square$

Lemma 17.9.3. Let $X$ be a ringed space. The image of a morphism of $\mathcal{O}_ X$-modules of finite type is of finite type. Let $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ be a short exact sequence of $\mathcal{O}_ X$-modules. If $\mathcal{F}_1$ and $\mathcal{F}_3$ are of finite type, so is $\mathcal{F}_2$.

Proof. The statement on images is trivial. The statement on short exact sequences comes from the fact that sections of $\mathcal{F}_3$ locally lift to sections of $\mathcal{F}_2$ and the corresponding result in the category of modules over a ring (applied to the stalks for example). $\square$

Lemma 17.9.4. Let $X$ be a ringed space. Let $\varphi : \mathcal{G} \to \mathcal{F}$ be a homomorphism of $\mathcal{O}_ X$-modules. Let $x \in X$. Assume $\mathcal{F}$ of finite type and the map on stalks $\varphi _ x : \mathcal{G}_ x \to \mathcal{F}_ x$ surjective. Then there exists an open neighbourhood $x \in U \subset X$ such that $\varphi |_ U$ is surjective.

Proof. Choose an open neighbourhood $U \subset X$ of $x$ such that $\mathcal{F}$ is generated by $s_1, \ldots , s_ n \in \mathcal{F}(U)$ over $U$. By assumption of surjectivity of $\varphi _ x$, after shrinking $U$ we may assume that $s_ i = \varphi (t_ i)$ for some $t_ i \in \mathcal{G}(U)$. Then $U$ works. $\square$

Lemma 17.9.5. Let $X$ be a ringed space. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module. Let $x \in X$. Assume $\mathcal{F}$ of finite type and $\mathcal{F}_ x = 0$. Then there exists an open neighbourhood $x \in U \subset X$ such that $\mathcal{F}|_ U$ is zero.

Proof. This is a special case of Lemma 17.9.4 applied to the morphism $0 \to \mathcal{F}$. $\square$

Lemma 17.9.6. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules. If $\mathcal{F}$ is of finite type then support of $\mathcal{F}$ is closed.

Proof. This is a reformulation of Lemma 17.9.5. $\square$

Lemma 17.9.7. Let $X$ be a ringed space. Let $I$ be a preordered set and let $(\mathcal{F}_ i, f_{ii'})$ be a system over $I$ consisting of sheaves of $\mathcal{O}_ X$-modules (see Categories, Section 4.21). Let $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$ be the colimit. Assume (a) $I$ is directed, (b) $\mathcal{F}$ is a finite type $\mathcal{O}_ X$-module, and (c) $X$ is quasi-compact. Then there exists an $i$ such that $\mathcal{F}_ i \to \mathcal{F}$ is surjective. If the transition maps $f_{ii'}$ are injective then we conclude that $\mathcal{F} = \mathcal{F}_ i$ for some $i \in I$.

Proof. Let $x \in X$. There exists an open neighbourhood $U \subset X$ of $x$ and finitely many sections $s_ j \in \mathcal{F}(U)$, $j = 1, \ldots , m$ such that $s_1, \ldots , s_ m$ generate $\mathcal{F}$ as $\mathcal{O}_ U$-module. After possibly shrinking $U$ to a smaller open neighbourhood of $x$ we may assume that each $s_ j$ comes from a section of $\mathcal{F}_ i$ for some $i \in I$. Hence, since $X$ is quasi-compact we can find a finite open covering $X = \bigcup _{j = 1, \ldots , m} U_ j$, and for each $j$ an index $i_ j$ and finitely many sections $s_{jl} \in \mathcal{F}_{i_ j}(U_ j)$ whose images generate the restriction of $\mathcal{F}$ to $U_ j$. Clearly, the lemma holds for any index $i \in I$ which is $\geq$ all $i_ j$. $\square$

Lemma 17.9.8. Let $X$ be a ringed space. There exists a set of $\mathcal{O}_ X$-modules $\{ \mathcal{F}_ i\} _{i \in I}$ of finite type such that each finite type $\mathcal{O}_ X$-module on $X$ is isomorphic to exactly one of the $\mathcal{F}_ i$.

Proof. For each open covering $\mathcal{U} : X = \bigcup U_ j$ consider the sheaves of $\mathcal{O}_ X$-modules $\mathcal{F}$ such that each restriction $\mathcal{F}|_{U_ j}$ is a quotient of $\mathcal{O}_{U_ j}^{\oplus r}$ for some $r_ j \geq 0$. These are parametrized by subsheaves $\mathcal{K}_ i \subset \mathcal{O}_{U_ j}^{\oplus r_ j}$ and glueing data

$\varphi _{jj'} : \mathcal{O}_{U_ j \cap U_{j'}}^{\oplus r_ j}/ (\mathcal{K}_ j|_{U_ j \cap U_{j'}}) \longrightarrow \mathcal{O}_{U_ j \cap U_{j'}}^{\oplus r_{j'}}/ (\mathcal{K}_{j'}|_{U_ j \cap U_{j'}})$

see Sheaves, Section 6.33. Note that the collection of all glueing data forms a set. The collection of all coverings $\mathcal{U} : X = \bigcup _{j \in J} U_ i$ where $J \to \mathcal{P}(X)$, $j \mapsto U_ j$ is injective forms a set as well. Hence the collection of all sheaves of $\mathcal{O}_ X$-modules gotten from glueing quotients as above forms a set $\mathcal{I}$. By definition every finite type $\mathcal{O}_ X$-module is isomorphic to an element of $\mathcal{I}$. Choosing an element out of each isomorphism class inside $\mathcal{I}$ gives the desired set of sheaves (uses axiom of choice). $\square$

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