Here is the definition.
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
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 kth 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
Comments (0)