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.

## 17.9 Modules of finite type

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

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

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 $\mathcal{O}_{U_ j}^{\oplus r_ j}$ for some $r_ j \geq 0$. These are parametrized by subsheaves $\mathcal{K}_ j \subset \mathcal{O}_{U_ j}^{\oplus r_ j}$ and glueing data

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). $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (2)

Comment #5863 by Dhivya Prakash.R.V on

Comment #6074 by Johan on