Definition 17.11.1. 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 presentation* if for every point $x \in X$ there exists an open neighbourhood $x\in U \subset X$, and $n, m \in \mathbf{N}$ such that $\mathcal{F}|_ U$ is isomorphic to the cokernel of a map

## 17.11 Modules of finite presentation

Here is the definition.

This means that $X$ is covered by open sets $U$ such that $\mathcal{F}|_ U$ has a *presentation* of the form

Here presentation signifies that the displayed sequence is exact. In other words

for every point $x$ of $X$ there exists an open neighbourhood such that $\mathcal{F}|_ U$ is generated by finitely many global sections, and

for a suitable choice of these sections the kernel of the associated surjection is also generated by finitely many global sections.

Lemma 17.11.2. Let $(X, \mathcal{O}_ X)$ be a ringed space. Any $\mathcal{O}_ X$-module of finite presentation is quasi-coherent.

**Proof.**
Immediate from definitions.
$\square$

Lemma 17.11.3. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module of finite presentation.

If $\psi : \mathcal{O}_ X^{\oplus r} \to \mathcal{F}$ is a surjection, then $\mathop{\mathrm{Ker}}(\psi )$ is of finite type.

If $\theta : \mathcal{G} \to \mathcal{F}$ is surjective with $\mathcal{G}$ of finite type, then $\mathop{\mathrm{Ker}}(\theta )$ is of finite type.

**Proof.**
Proof of (1). Let $x \in X$. Choose an open neighbourhood $U \subset X$ of $x$ such that there exists a presentation

Let $e_ k$ be the section generating the $k$th factor of $\mathcal{O}_ X^{\oplus r}$. For every $k = 1, \ldots , r$ we can, after shrinking $U$ to a small neighbourhood of $x$, lift $\psi (e_ k)$ to a section $\tilde e_ k$ of $\mathcal{O}_ U^{\oplus n}$ over $U$. This gives a morphism of sheaves $\alpha : \mathcal{O}_ U^{\oplus r} \to \mathcal{O}_ U^{\oplus n}$ such that $\varphi \circ \alpha = \psi $. Similarly, after shrinking $U$, we can find a morphism $\beta : \mathcal{O}_ U^{\oplus n} \to \mathcal{O}_ U^{\oplus r}$ such that $\psi \circ \beta = \varphi $. Then the map

is a surjection onto the kernel of $\psi $.

To prove (2) we may locally choose a surjection $\eta : \mathcal{O}_ X^{\oplus r} \to \mathcal{G}$. By part (1) we see $\mathop{\mathrm{Ker}}(\theta \circ \eta )$ is of finite type. Since $\mathop{\mathrm{Ker}}(\theta ) = \eta (\mathop{\mathrm{Ker}}(\theta \circ \eta ))$ we win. $\square$

Lemma 17.11.4. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. The pullback $f^*\mathcal{G}$ of a module of finite presentation is of finite presentation.

**Proof.**
Exactly the same as the proof of Lemma 17.10.4 but with finite index sets.
$\square$

Lemma 17.11.5. Let $(X, \mathcal{O}_ X)$ be a ringed space. Set $R = \Gamma (X, \mathcal{O}_ X)$. Let $M$ be an $R$-module. The $\mathcal{O}_ X$-module $\mathcal{F}_ M$ associated to $M$ is a directed colimit of finitely presented $\mathcal{O}_ X$-modules.

**Proof.**
This follows immediately from Lemma 17.10.5 and the fact that any module is a directed colimit of finitely presented modules, see Algebra, Lemma 10.11.3.
$\square$

Lemma 17.11.6. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$ be a finitely presented $\mathcal{O}_ X$-module. Let $x \in X$ such that $\mathcal{F}_ x \cong \mathcal{O}_{X, x}^{\oplus r}$. Then there exists an open neighbourhood $U$ of $x$ such that $\mathcal{F}|_ U \cong \mathcal{O}_ U^{\oplus r}$.

**Proof.**
Choose $s_1, \ldots , s_ r \in \mathcal{F}_ x$ mapping to a basis of $\mathcal{O}_{X, x}^{\oplus r}$ by the isomorphism. Choose an open neighbourhood $U$ of $x$ such that $s_ i$ lifts to $s_ i \in \mathcal{F}(U)$. After shrinking $U$ we see that the induced map $\psi : \mathcal{O}_ U^{\oplus r} \to \mathcal{F}|_ U$ is surjective (Lemma 17.9.4). By Lemma 17.11.3 we see that $\mathop{\mathrm{Ker}}(\psi )$ is of finite type. Then $\mathop{\mathrm{Ker}}(\psi )_ x = 0$ implies that $\mathop{\mathrm{Ker}}(\psi )$ becomes zero after shrinking $U$ once more (Lemma 17.9.5).
$\square$

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