The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

17.11 Modules of finite presentation

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

\[ \bigoplus \nolimits _{j = 1, \ldots , m} \mathcal{O}_ U \longrightarrow \bigoplus \nolimits _{i = 1, \ldots , n} \mathcal{O}_ U \]

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

\[ \bigoplus \nolimits _{j = 1, \ldots , m} \mathcal{O}_ U \longrightarrow \bigoplus \nolimits _{i = 1, \ldots , n} \mathcal{O}_ U \to \mathcal{F}|_ U \to 0. \]

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

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

  2. 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 a $\mathcal{O}_ X$-module of finite presentation.

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

  2. 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

\[ \mathcal{O}_ U^{\oplus m} \xrightarrow {\chi } \mathcal{O}_ U^{\oplus n} \xrightarrow {\varphi } \mathcal{F}|_ U \to 0. \]

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

\[ \mathcal{O}_ U^{\oplus m} \oplus \mathcal{O}_ U^{\oplus r} \xrightarrow {\beta \circ \chi , 1 - \beta \circ \alpha } \mathcal{O}_ U^{\oplus r} \]

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 10.8.12. $\square$

Lemma 17.11.6. Let $X$ be a ringed space. Let $I$ be a preordered set and let $(\mathcal{F}_ i, \varphi _{ii'})$ be a system over $I$ consisting of sheaves of $\mathcal{O}_ X$-modules (see Categories, Section 4.21). Assume

  1. $I$ is directed,

  2. $\mathcal{G}$ is an $\mathcal{O}_ X$-module of finite presentation, and

  3. $X$ has a cofinal system of open coverings $\mathcal{U} : X = \bigcup _{j\in J} U_ j$ with $J$ finite and $U_ j \cap U_{j'}$ quasi-compact for all $j, j' \in J$.

Then we have

\[ \mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Hom}}\nolimits _ X(\mathcal{G}, \mathcal{F}_ i) = \mathop{\mathrm{Hom}}\nolimits _ X(\mathcal{G}, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i). \]

Proof. Let $\alpha $ be an element of the right hand side. For every point $x \in X$ we may choose an open neighbourhood $U \subset X$ and finitely many sections $s_ j \in \mathcal{G}(U)$ which generate $\mathcal{G}$ over $U$ and finitely many relations $\sum f_{kj} s_ j = 0$, $k = 1, \ldots , n$ with $f_{kj} \in \mathcal{O}_ X(U)$ which generate the kernel of $\bigoplus _{j = 1, \ldots , m} \mathcal{O}_ U \to \mathcal{G}$. After possibly shrinking $U$ to a smaller open neighbourhood of $x$ we may assume there exists an index $i \in I$ such that the sections $\alpha (s_ j)$ all come from sections $s_ j' \in \mathcal{F}_ i(U)$. After possibly shrinking $U$ to a smaller open neighbourhood of $x$ and increasing $i$ we may assume the relations $\sum f_{kj} s'_ j = 0$ hold in $\mathcal{F}_ i(U)$. Hence we see that $\alpha |_ U$ lifts to a morphism $\mathcal{G}|_ U \to \mathcal{F}_ i|_ U$ for some index $i \in I$.

By condition (3) and the preceding arguments, we may choose a finite open covering $X = \bigcup _{j = 1, \ldots , m} U_ j$ such that (a) $\mathcal{G}|_{U_ j}$ is generated by finitely many sections $s_{jk} \in \mathcal{G}(U_ j)$, (b) the restriction $\alpha |_{U_ j}$ comes from a morphism $\alpha _ j : \mathcal{G} \to \mathcal{F}_{i_ j}$ for some $i_ j \in I$, and (c) the intersections $U_ j \cap U_{j'}$ are all quasi-compact. For every pair $(j, j') \in \{ 1, \ldots , m\} ^2$ and any $k$ we can find we can find an index $i \geq \max (i_ j, i_{j'})$ such that

\[ \varphi _{i_ ji}(\alpha _ j(s_{jk}|_{U_ j \cap U_{j'}})) = \varphi _{i_{j'}i}(\alpha _{j'}(s_{jk}|_{U_ j \cap U_{j'}})) \]

see Sheaves, Lemma 6.29.1 (2). Since there are finitely many of these pairs $(j, j')$ and finitely many $s_{jk}$ we see that we can find a single $i$ which works for all of them. For this index $i$ all of the maps $\varphi _{i_ ji} \circ \alpha _ j$ agree on the overlaps $U_ j \cap U_{j'}$ as the sections $s_{jk}$ generate $\mathcal{G}$ over this overlap. Hence we get a morphism $\mathcal{G} \to \mathcal{F}_ i$ as desired. $\square$

Remark 17.11.7. In the lemma above some condition beyond the condition that $X$ is quasi-compact is necessary. See Sheaves, Example 6.29.2.

Lemma 17.11.8. 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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