The Stacks project

68.9 Applications

The following lemma can also be deduced directly from Decent Spaces, Lemma 66.8.6 without passing through absolute Noetherian approximation.

Lemma 68.9.1. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Every quasi-coherent $\mathcal{O}_ X$-module is a filtered colimit of finitely presented $\mathcal{O}_ X$-modules.

Proof. We may view as an algebraic space over $\mathop{\mathrm{Spec}}(\mathbf{Z})$, see Spaces, Definition 63.16.2 and Properties of Spaces, Definition 64.3.1. Thus we may apply Proposition 68.8.1 and write $X = \mathop{\mathrm{lim}}\nolimits X_ i$ with $X_ i$ of finite presentation over $\mathbf{Z}$. Thus $X_ i$ is a Noetherian algebraic space, see Morphisms of Spaces, Lemma 65.28.6. The morphism $X \to X_ i$ is affine, see Lemma 68.4.1. Conclusion by Cohomology of Spaces, Lemma 67.15.2. $\square$

The rest of this section consists of straightforward applications of Lemma 68.9.1.

Lemma 68.9.2. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Then $\mathcal{F}$ is the directed colimit of its finite type quasi-coherent submodules.

Proof. If $\mathcal{G}, \mathcal{H} \subset \mathcal{F}$ are finite type quasi-coherent $\mathcal{O}_ X$-submodules then the image of $\mathcal{G} \oplus \mathcal{H} \to \mathcal{F}$ is another finite type quasi-coherent $\mathcal{O}_ X$-submodule which contains both of them. In this way we see that the system is directed. To show that $\mathcal{F}$ is the colimit of this system, write $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i$ as a directed colimit of finitely presented quasi-coherent sheaves as in Lemma 68.9.1. Then the images $\mathcal{G}_ i = \mathop{\mathrm{Im}}(\mathcal{F}_ i \to \mathcal{F})$ are finite type quasi-coherent subsheaves of $\mathcal{F}$. Since $\mathcal{F}$ is the colimit of these the result follows. $\square$

Lemma 68.9.3. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module. Then we can write $\mathcal{F} = \mathop{\mathrm{lim}}\nolimits \mathcal{F}_ i$ where each $\mathcal{F}_ i$ is an $\mathcal{O}_ X$-module of finite presentation and all transition maps $\mathcal{F}_ i \to \mathcal{F}_{i'}$ surjective.

Proof. Write $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{G}_ i$ as a filtered colimit of finitely presented $\mathcal{O}_ X$-modules (Lemma 68.9.1). We claim that $\mathcal{G}_ i \to \mathcal{F}$ is surjective for some $i$. Namely, choose an étale surjection $U \to X$ where $U$ is an affine scheme. Choose finitely many sections $s_ k \in \mathcal{F}(U)$ generating $\mathcal{F}|_ U$. Since $U$ is affine we see that $s_ k$ is in the image of $\mathcal{G}_ i \to \mathcal{F}$ for $i$ large enough. Hence $\mathcal{G}_ i \to \mathcal{F}$ is surjective for $i$ large enough. Choose such an $i$ and let $\mathcal{K} \subset \mathcal{G}_ i$ be the kernel of the map $\mathcal{G}_ i \to \mathcal{F}$. Write $\mathcal{K} = \mathop{\mathrm{colim}}\nolimits \mathcal{K}_ a$ as the filtered colimit of its finite type quasi-coherent submodules (Lemma 68.9.2). Then $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{G}_ i/\mathcal{K}_ a$ is a solution to the problem posed by the lemma. $\square$

Let $X$ be an algebraic space. In the following lemma we use the notion of a finitely presented quasi-coherent $\mathcal{O}_ X$-algebra $\mathcal{A}$. This means that for every affine $U = \mathop{\mathrm{Spec}}(R)$ étale over $X$ we have $\mathcal{A}|_ U = \widetilde{A}$ where $A$ is a (commutative) $R$-algebra which is of finite presentation as an $R$-algebra.

Lemma 68.9.4. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\mathcal{A}$ be a quasi-coherent $\mathcal{O}_ X$-algebra. Then $\mathcal{A}$ is a directed colimit of finitely presented quasi-coherent $\mathcal{O}_ X$-algebras.

Proof. First we write $\mathcal{A} = \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i$ as a directed colimit of finitely presented quasi-coherent sheaves as in Lemma 68.9.1. For each $i$ let $\mathcal{B}_ i = \text{Sym}(\mathcal{F}_ i)$ be the symmetric algebra on $\mathcal{F}_ i$ over $\mathcal{O}_ X$. Write $\mathcal{I}_ i = \mathop{\mathrm{Ker}}(\mathcal{B}_ i \to \mathcal{A})$. Write $\mathcal{I}_ i = \mathop{\mathrm{colim}}\nolimits _ j \mathcal{F}_{i, j}$ where $\mathcal{F}_{i, j}$ is a finite type quasi-coherent submodule of $\mathcal{I}_ i$, see Lemma 68.9.2. Set $\mathcal{I}_{i, j} \subset \mathcal{I}_ i$ equal to the $\mathcal{B}_ i$-ideal generated by $\mathcal{F}_{i, j}$. Set $\mathcal{A}_{i, j} = \mathcal{B}_ i/\mathcal{I}_{i, j}$. Then $\mathcal{A}_{i, j}$ is a quasi-coherent finitely presented $\mathcal{O}_ X$-algebra. Define $(i, j) \leq (i', j')$ if $i \leq i'$ and the map $\mathcal{B}_ i \to \mathcal{B}_{i'}$ maps the ideal $\mathcal{I}_{i, j}$ into the ideal $\mathcal{I}_{i', j'}$. Then it is clear that $\mathcal{A} = \mathop{\mathrm{colim}}\nolimits _{i, j} \mathcal{A}_{i, j}$. $\square$

Let $X$ be an algebraic space. In the following lemma we use the notion of a quasi-coherent $\mathcal{O}_ X$-algebra $\mathcal{A}$ of finite type. This means that for every affine $U = \mathop{\mathrm{Spec}}(R)$ étale over $X$ we have $\mathcal{A}|_ U = \widetilde{A}$ where $A$ is a (commutative) $R$-algebra which is of finite type as an $R$-algebra.

Lemma 68.9.5. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\mathcal{A}$ be a quasi-coherent $\mathcal{O}_ X$-algebra. Then $\mathcal{A}$ is the directed colimit of its finite type quasi-coherent $\mathcal{O}_ X$-subalgebras.

Proof. Omitted. Hint: Compare with the proof of Lemma 68.9.2. $\square$

Let $X$ be an algebraic space. In the following lemma we use the notion of a finite (resp. integral) quasi-coherent $\mathcal{O}_ X$-algebra $\mathcal{A}$. This means that for every affine $U = \mathop{\mathrm{Spec}}(R)$ étale over $X$ we have $\mathcal{A}|_ U = \widetilde{A}$ where $A$ is a (commutative) $R$-algebra which is finite (resp. integral) as an $R$-algebra.

Lemma 68.9.6. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\mathcal{A}$ be a finite quasi-coherent $\mathcal{O}_ X$-algebra. Then $\mathcal{A} = \mathop{\mathrm{colim}}\nolimits \mathcal{A}_ i$ is a directed colimit of finite and finitely presented quasi-coherent $\mathcal{O}_ X$-algebras with surjective transition maps.

Proof. By Lemma 68.9.3 there exists a finitely presented $\mathcal{O}_ X$-module $\mathcal{F}$ and a surjection $\mathcal{F} \to \mathcal{A}$. Using the algebra structure we obtain a surjection

\[ \text{Sym}^*_{\mathcal{O}_ X}(\mathcal{F}) \longrightarrow \mathcal{A} \]

Denote $\mathcal{J}$ the kernel. Write $\mathcal{J} = \mathop{\mathrm{colim}}\nolimits \mathcal{E}_ i$ as a filtered colimit of finite type $\mathcal{O}_ X$-submodules $\mathcal{E}_ i$ (Lemma 68.9.2). Set

\[ \mathcal{A}_ i = \text{Sym}^*_{\mathcal{O}_ X}(\mathcal{F})/(\mathcal{E}_ i) \]

where $(\mathcal{E}_ i)$ indicates the ideal sheaf generated by the image of $\mathcal{E}_ i \to \text{Sym}^*_{\mathcal{O}_ X}(\mathcal{F})$. Then each $\mathcal{A}_ i$ is a finitely presented $\mathcal{O}_ X$-algebra, the transition maps are surjective, and $\mathcal{A} = \mathop{\mathrm{colim}}\nolimits \mathcal{A}_ i$. To finish the proof we still have to show that $\mathcal{A}_ i$ is a finite $\mathcal{O}_ X$-algebra for $i$ sufficiently large. To do this we choose an étale surjective map $U \to X$ where $U$ is an affine scheme. Take generators $f_1, \ldots , f_ m \in \Gamma (U, \mathcal{F})$. As $\mathcal{A}(U)$ is a finite $\mathcal{O}_ X(U)$-algebra we see that for each $j$ there exists a monic polynomial $P_ j \in \mathcal{O}(U)[T]$ such that $P_ j(f_ j)$ is zero in $\mathcal{A}(U)$. Since $\mathcal{A} = \mathop{\mathrm{colim}}\nolimits \mathcal{A}_ i$ by construction, we have $P_ j(f_ j) = 0$ in $\mathcal{A}_ i(U)$ for all sufficiently large $i$. For such $i$ the algebras $\mathcal{A}_ i$ are finite. $\square$

Lemma 68.9.7. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\mathcal{A}$ be an integral quasi-coherent $\mathcal{O}_ X$-algebra. Then

  1. $\mathcal{A}$ is the directed colimit of its finite quasi-coherent $\mathcal{O}_ X$-subalgebras, and

  2. $\mathcal{A}$ is a directed colimit of finite and finitely presented $\mathcal{O}_ X$-algebras.

Proof. By Lemma 68.9.5 we have $\mathcal{A} = \mathop{\mathrm{colim}}\nolimits \mathcal{A}_ i$ where $\mathcal{A}_ i \subset \mathcal{A}$ runs through the quasi-coherent $\mathcal{O}_ X$-sub algebras of finite type. Any finite type quasi-coherent $\mathcal{O}_ X$-subalgebra of $\mathcal{A}$ is finite (use Algebra, Lemma 10.35.5 on affine schemes étale over $X$). This proves (1).

To prove (2), write $\mathcal{A} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$ as a colimit of finitely presented $\mathcal{O}_ X$-modules using Lemma 68.9.1. For each $i$, let $\mathcal{J}_ i$ be the kernel of the map

\[ \text{Sym}^*_{\mathcal{O}_ X}(\mathcal{F}_ i) \longrightarrow \mathcal{A} \]

For $i' \geq i$ there is an induced map $\mathcal{J}_ i \to \mathcal{J}_{i'}$ and we have $\mathcal{A} = \mathop{\mathrm{colim}}\nolimits \text{Sym}^*_{\mathcal{O}_ X}(\mathcal{F}_ i)/\mathcal{J}_ i$. Moreover, the quasi-coherent $\mathcal{O}_ X$-algebras $\text{Sym}^*_{\mathcal{O}_ X}(\mathcal{F}_ i)/\mathcal{J}_ i$ are finite (see above). Write $\mathcal{J}_ i = \mathop{\mathrm{colim}}\nolimits \mathcal{E}_{ik}$ as a colimit of finitely presented $\mathcal{O}_ X$-modules. Given $i' \geq i$ and $k$ there exists a $k'$ such that we have a map $\mathcal{E}_{ik} \to \mathcal{E}_{i'k'}$ making

\[ \xymatrix{ \mathcal{J}_ i \ar[r] & \mathcal{J}_{i'} \\ \mathcal{E}_{ik} \ar[u] \ar[r] & \mathcal{E}_{i'k'} \ar[u] } \]

commute. This follows from Cohomology of Spaces, Lemma 67.5.3. This induces a map

\[ \mathcal{A}_{ik} = \text{Sym}^*_{\mathcal{O}_ X}(\mathcal{F}_ i)/(\mathcal{E}_{ik}) \longrightarrow \text{Sym}^*_{\mathcal{O}_ X}(\mathcal{F}_{i'})/(\mathcal{E}_{i'k'}) = \mathcal{A}_{i'k'} \]

where $(\mathcal{E}_{ik})$ denotes the ideal generated by $\mathcal{E}_{ik}$. The quasi-coherent $\mathcal{O}_ X$-algebras $\mathcal{A}_{ki}$ are of finite presentation and finite for $k$ large enough (see proof of Lemma 68.9.6). Finally, we have

\[ \mathop{\mathrm{colim}}\nolimits \mathcal{A}_{ik} = \mathop{\mathrm{colim}}\nolimits \mathcal{A}_ i = \mathcal{A} \]

Namely, the first equality was shown in the proof of Lemma 68.9.6 and the second equality because $\mathcal{A}$ is the colimit of the modules $\mathcal{F}_ i$. $\square$

Lemma 68.9.8. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $U \subset X$ be a quasi-compact open. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $\mathcal{G} \subset \mathcal{F}|_ U$ be a quasi-coherent $\mathcal{O}_ U$-submodule which is of finite type. Then there exists a quasi-coherent submodule $\mathcal{G}' \subset \mathcal{F}$ which is of finite type such that $\mathcal{G}'|_ U = \mathcal{G}$.

Proof. Denote $j : U \to X$ the inclusion morphism. As $X$ is quasi-separated and $U$ quasi-compact, the morphism $j$ is quasi-compact. Hence $j_*\mathcal{G} \subset j_*\mathcal{F}|_ U$ are quasi-coherent modules on $X$ (Morphisms of Spaces, Lemma 65.11.2). Let $\mathcal{H} = \mathop{\mathrm{Ker}}(j_*\mathcal{G} \oplus \mathcal{F} \to j_*\mathcal{F}|_ U)$. Then $\mathcal{H}|_ U = \mathcal{G}$. By Lemma 68.9.2 we can find a finite type quasi-coherent submodule $\mathcal{H}' \subset \mathcal{H}$ such that $\mathcal{H}'|_ U = \mathcal{H}|_ U = \mathcal{G}$. Set $\mathcal{G}' = \mathop{\mathrm{Im}}(\mathcal{H}' \to \mathcal{F})$ to conclude. $\square$


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