The Stacks project

Proof. An immersion is separated (see Schemes, Lemma 26.23.8) and $j$ is quasi-compact by assumption. Hence for any quasi-coherent sheaf $\mathcal{G}$ on $U$ the sheaf $j_*\mathcal{G}$ is an extension to $X$. See Schemes, Lemma 26.24.1 and Sheaves, Section 6.31.

Assume $\mathcal{F}$, $\mathcal{G}$ are as in (2). Then $j_*\mathcal{G}$ is a quasi-coherent sheaf on $X$ (see above). It is a subsheaf of $j_*j^*\mathcal{F}$. Hence the kernel

is quasi-coherent as well, see Schemes, Section 26.24. It is formal to check that $\mathcal{H} \subset \mathcal{F}$ and that $\mathcal{H}|_ U = \mathcal{G}$ (using the material in Sheaves, Section 6.31 again).

Part (3) is proved in the same manner as (2). Just take $\mathcal{H} = \mathop{\mathrm{Ker}}(\mathcal{F} \oplus j_* \mathcal{G} \to j_*j^*\mathcal{F})$ with its obvious map to $\mathcal{F}$ and its obvious identification with $\mathcal{G}$ over $U$. $\square$

Proof. Let $n$ be the minimal number of affine opens $U_ i \subset X$, $i = 1, \ldots , n$ such that $X = U \cup \bigcup U_ i$. (Here we use that $X$ is quasi-compact.) Suppose we can prove the lemma for the case $n = 1$. Then we can successively extend $\mathcal{G}$ to a $\mathcal{G}_1$ over $U \cup U_1$ to a $\mathcal{G}_2$ over $U \cup U_1 \cup U_2$ to a $\mathcal{G}_3$ over $U \cup U_1 \cup U_2 \cup U_3$, and so on. Thus we reduce to the case $n = 1$.

Thus we may assume that $X = U \cup V$ with $V$ affine. Since $X$ is quasi-separated and $U$, $V$ are quasi-compact open, we see that $U \cap V$ is a quasi-compact open. It suffices to prove the lemma for the system $(V, U \cap V, \mathcal{F}|_ V, \mathcal{G}|_{U \cap V})$ since we can glue the resulting sheaf $\mathcal{G}'$ over $V$ to the given sheaf $\mathcal{G}$ over $U$ along the common value over $U \cap V$. Thus we reduce to the case where $X$ is affine.

Assume $X = \mathop{\mathrm{Spec}}(R)$. Write $\mathcal{F} = \widetilde M$ for some $R$-module $M$. By Lemma 28.22.1 above we may find a quasi-coherent subsheaf $\mathcal{H} \subset \mathcal{F}$ which restricts to $\mathcal{G}$ over $U$. Write $\mathcal{H} = \widetilde N$ for some $R$-module $N$. For every $u \in U$ there exists an $f \in R$ such that $u \in D(f) \subset U$ and such that $N_ f$ is finitely generated, see Lemma 28.16.1. Since $U$ is quasi-compact we can cover it by finitely many $D(f_ i)$ such that $N_{f_ i}$ is generated by finitely many elements, say $x_{i, 1}/f_ i^ N, \ldots , x_{i, r_ i}/f_ i^ N$. Let $N' \subset N$ be the submodule generated by the elements $x_{i, j}$. Then the subsheaf $\mathcal{G}' = \widetilde{N'} \subset \mathcal{H} \subset \mathcal{F}$ works. $\square$

Proof. The colimit is directed because if $\mathcal{G}_1$, $\mathcal{G}_2$ are quasi-coherent subsheaves of finite type, then the image of $\mathcal{G}_1 \oplus \mathcal{G}_2 \to \mathcal{F}$ is a quasi-coherent submodule of finite type. Let $U \subset X$ be any affine open, and let $s \in \Gamma (U, \mathcal{F})$ be any section. Let $\mathcal{G} \subset \mathcal{F}|_ U$ be the subsheaf generated by $s$. Then clearly $\mathcal{G}$ is quasi-coherent and has finite type as an $\mathcal{O}_ U$-module. By Lemma 28.22.2 we see that $\mathcal{G}$ is the restriction of a quasi-coherent subsheaf $\mathcal{G}' \subset \mathcal{F}$ which has finite type. Since $X$ has a basis for the topology consisting of affine opens we conclude that every local section of $\mathcal{F}$ is locally contained in a quasi-coherent submodule of finite type. Thus we win. $\square$

Proof. The beginning of the proof is a repeat of the beginning of the proof of Lemma 28.22.2. We write it out carefuly anyway.

Let $n$ be the minimal number of affine opens $U_ i \subset X$, $i = 1, \ldots , n$ such that $X = U \cup \bigcup U_ i$. (Here we use that $X$ is quasi-compact.) Suppose we can prove the lemma for the case $n = 1$. Then we can successively extend the pair $(\mathcal{G}, \varphi )$ to a pair $(\mathcal{G}_1, \varphi _1)$ over $U \cup U_1$ to a pair $(\mathcal{G}_2, \varphi _2)$ over $U \cup U_1 \cup U_2$ to a pair $(\mathcal{G}_3, \varphi _3)$ over $U \cup U_1 \cup U_2 \cup U_3$, and so on. Thus we reduce to the case $n = 1$.

Thus we may assume that $X = U \cup V$ with $V$ affine. Since $X$ is quasi-separated and $U$ quasi-compact, we see that $U \cap V \subset V$ is quasi-compact. Suppose we prove the lemma for the system $(V, U \cap V, \mathcal{F}|_ V, \mathcal{G}|_{U \cap V}, \varphi |_{U \cap V})$ thereby producing $(\mathcal{G}', \varphi ')$ over $V$. Then we can glue $\mathcal{G}'$ over $V$ to the given sheaf $\mathcal{G}$ over $U$ along the common value over $U \cap V$, and similarly we can glue the map $\varphi '$ to the map $\varphi $ along the common value over $U \cap V$. Thus we reduce to the case where $X$ is affine.

Assume $X = \mathop{\mathrm{Spec}}(R)$. By Lemma 28.22.1 above we may find a quasi-coherent sheaf $\mathcal{H}$ with a map $\psi : \mathcal{H} \to \mathcal{F}$ over $X$ which restricts to $\mathcal{G}$ and $\varphi $ over $U$. By Lemma 28.22.2 we can find a finite type quasi-coherent $\mathcal{O}_ X$-submodule $\mathcal{H}' \subset \mathcal{H}$ such that $\mathcal{H}'|_ U = \mathcal{G}$. Thus after replacing $\mathcal{H}$ by $\mathcal{H}'$ and $\psi $ by the restriction of $\psi $ to $\mathcal{H}'$ we may assume that $\mathcal{H}$ is of finite type. By Lemma 28.16.2 we conclude that $\mathcal{H} = \widetilde{N}$ with $N$ a finitely generated $R$-module. Hence there exists a surjection as in the following short exact sequence of quasi-coherent $\mathcal{O}_ X$-modules

where $\mathcal{K}$ is defined as the kernel. Since $\mathcal{G}$ is of finite presentation and $\mathcal{H}|_ U = \mathcal{G}$ by Modules, Lemma 17.11.3 the restriction $\mathcal{K}|_ U$ is an $\mathcal{O}_ U$-module of finite type. Hence by Lemma 28.22.2 again we see that there exists a finite type quasi-coherent $\mathcal{O}_ X$-submodule $\mathcal{K}' \subset \mathcal{K}$ such that $\mathcal{K}'|_ U = \mathcal{K}|_ U$. The solution to the problem posed in the lemma is to set

which is clearly of finite presentation and restricts to give $\mathcal{G}$ on $U$ with $\varphi '$ equal to the composition

This finishes the proof of the lemma. $\square$

Proof. Part (2) is the special case of Lemma 28.22.4 where $\mathcal{F} = 0$. For part (1) we first write $\mathcal{G} = \mathcal{F}|_ U$ for some quasi-coherent $\mathcal{O}_ X$-module by Lemma 28.22.1 and then we apply Lemma 28.22.2 with $\mathcal{G} = \mathcal{F}|_ U$. $\square$

Proof. Choose a set $I$ and for each $i \in I$ an $\mathcal{O}_ X$-module of finite presentation and a homomorphism of $\mathcal{O}_ X$-modules $\varphi _ i : \mathcal{F}_ i \to \mathcal{F}$ with the following property: For any $\psi : \mathcal{G} \to \mathcal{F}$ with $\mathcal{G}$ of finite presentation there is an $i \in I$ such that there exists an isomorphism $\alpha : \mathcal{F}_ i \to \mathcal{G}$ with $\varphi _ i = \psi \circ \alpha $. It is clear from Modules, Lemma 17.9.8 that such a set exists (see also its proof). We denote $\mathcal{I}$ the category with $\mathop{\mathrm{Ob}}\nolimits (\mathcal{I}) = I$ and given $i, i' \in I$ we set

We claim that $\mathcal{I}$ is a filtered category and that $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i$.

Let $i, i' \in I$. Then we can consider the morphism

which is the direct sum of $\varphi _ i$ and $\varphi _{i'}$. Since a direct sum of finitely presented $\mathcal{O}_ X$-modules is finitely presented we see that there exists some $i'' \in I$ such that $\varphi _{i''} : \mathcal{F}_{i''} \to \mathcal{F}$ is isomorphic to the displayed arrow towards $\mathcal{F}$ above. Since there are commutative diagrams

we see that there are morphisms $i \to i''$ and $i' \to i''$ in $\mathcal{I}$. Next, suppose that we have $i, i' \in I$ and morphisms $\alpha , \beta : i \to i'$ (corresponding to $\mathcal{O}_ X$-module maps $\alpha , \beta : \mathcal{F}_ i \to \mathcal{F}_{i'}$). In this case consider the coequalizer

Note that $\mathcal{G}$ is an $\mathcal{O}_ X$-module of finite presentation. Since by definition of morphisms in the category $\mathcal{I}$ we have $\varphi _{i'} \circ \alpha = \varphi _{i'} \circ \beta $ we see that we get an induced map $\psi : \mathcal{G} \to \mathcal{F}$. Hence again the pair $(\mathcal{G}, \psi )$ is isomorphic to the pair $(\mathcal{F}_{i''}, \varphi _{i''})$ for some $i''$. Hence we see that there exists a morphism $i' \to i''$ in $\mathcal{I}$ which equalizes $\alpha $ and $\beta $. Thus we have shown that the category $\mathcal{I}$ is filtered.

We still have to show that the colimit of the diagram is $\mathcal{F}$. By definition of the colimit, and by our definition of the category $\mathcal{I}$ there is a canonical map

Pick $x \in X$. Let us show that $\varphi _ x$ is an isomorphism. Recall that

see Sheaves, Section 6.29. First we show that the map $\varphi _ x$ is injective. Suppose that $s \in \mathcal{F}_{i, x}$ is an element such that $s$ maps to zero in $\mathcal{F}_ x$. Then there exists a quasi-compact open $U$ such that $s$ comes from $s \in \mathcal{F}_ i(U)$ and such that $\varphi _ i(s) = 0$ in $\mathcal{F}(U)$. By Lemma 28.22.2 we can find a finite type quasi-coherent subsheaf $\mathcal{K} \subset \mathop{\mathrm{Ker}}(\varphi _ i)$ which restricts to the quasi-coherent $\mathcal{O}_ U$-submodule of $\mathcal{F}_ i$ generated by $s$: $\mathcal{K}|_ U = \mathcal{O}_ U\cdot s \subset \mathcal{F}_ i|_ U$. Clearly, $\mathcal{F}_ i/\mathcal{K}$ is of finite presentation and the map $\varphi _ i$ factors through the quotient map $\mathcal{F}_ i \to \mathcal{F}_ i/\mathcal{K}$. Hence we can find an $i' \in I$ and a morphism $\alpha : \mathcal{F}_ i \to \mathcal{F}_{i'}$ in $\mathcal{I}$ which can be identified with the quotient map $\mathcal{F}_ i \to \mathcal{F}_ i/\mathcal{K}$. Then it follows that the section $s$ maps to zero in $\mathcal{F}_{i'}(U)$ and in particular in $(\mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i)_ x = \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_{i, x}$. The injectivity follows. Finally, we show that the map $\varphi _ x$ is surjective. Pick $s \in \mathcal{F}_ x$. Choose a quasi-compact open neighbourhood $U \subset X$ of $x$ such that $s$ corresponds to a section $s \in \mathcal{F}(U)$. Consider the map $s : \mathcal{O}_ U \to \mathcal{F}$ (multiplication by $s$). By Lemma 28.22.4 there exists an $\mathcal{O}_ X$-module $\mathcal{G}$ of finite presentation and an $\mathcal{O}_ X$-module map $\mathcal{G} \to \mathcal{F}$ such that $\mathcal{G}|_ U \to \mathcal{F}|_ U$ is identified with $s : \mathcal{O}_ U \to \mathcal{F}$. Again by definition of $\mathcal{I}$ there exists an $i \in I$ such that $\mathcal{G} \to \mathcal{F}$ is isomorphic to $\varphi _ i : \mathcal{F}_ i \to \mathcal{F}$. Clearly there exists a section $s' \in \mathcal{F}_ i(U)$ mapping to $s \in \mathcal{F}(U)$. This proves surjectivity and the proof of the lemma is complete. $\square$

Proof. This is a direct consequence of Lemma 28.22.6 and Categories, Lemma 4.21.5 (combined with the fact that colimits exist in the category of sheaves of $\mathcal{O}_ X$-modules, see Sheaves, Section 6.29). $\square$

Proof. Write $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{G}_ i$ as a filtered colimit of finitely presented $\mathcal{O}_ X$-modules (Lemma 28.22.7). We claim that $\mathcal{G}_ i \to \mathcal{F}$ is surjective for some $i$. Namely, choose a finite affine open covering $X = U_1 \cup \ldots \cup U_ m$. Choose sections $s_{jl} \in \mathcal{F}(U_ j)$ generating $\mathcal{F}|_{U_ j}$, see Lemma 28.16.1. By Sheaves, Lemma 6.29.1 we see that $s_{jl}$ 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 28.22.3). Then $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{G}_ i/\mathcal{K}_ a$ is a solution to the problem posed by the lemma. $\square$

Proof. Write $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$ as a directed colimit with each $\mathcal{F}_ i$ of finite presentation, see Lemma 28.22.7. Choose a finite affine open covering $X = \bigcup V_ j$ and choose finitely many sections $s_{jl} \in \mathcal{F}(V_ j)$ generating $\mathcal{F}|_{V_ j}$, see Lemma 28.16.1. By Sheaves, Lemma 6.29.1 we see that $s_{jl}$ is in the image of $\mathcal{F}_ i \to \mathcal{F}$ for $i$ large enough. Hence $\mathcal{F}_ i \to \mathcal{F}$ is surjective for $i$ large enough. Choose such an $i$ and let $\mathcal{K} \subset \mathcal{F}_ i$ be the kernel of the map $\mathcal{F}_ i \to \mathcal{F}$. Since $\mathcal{F}_ U$ is of finite presentation, we see that $\mathcal{K}|_ U$ is of finite type, see Modules, Lemma 17.11.3. Hence we can find a finite type quasi-coherent submodule $\mathcal{K}' \subset \mathcal{K}$ with $\mathcal{K}'|_ U = \mathcal{K}|_ U$, see Lemma 28.22.2. Then $\mathcal{G} = \mathcal{F}_ i/\mathcal{K}'$ with the given map $\mathcal{G} \to \mathcal{F}$ is a solution. $\square$

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 28.22.7. 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 28.22.3. 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 $\mathcal{A} = \mathop{\mathrm{colim}}\nolimits _{i, j} \mathcal{A}_{i, j}$. $\square$

Proof. If $\mathcal{A}_1, \mathcal{A}_2 \subset \mathcal{A}$ are quasi-coherent $\mathcal{O}_ X$-subalgebras of finite type, then the image of $\mathcal{A}_1 \otimes _{\mathcal{O}_ X} \mathcal{A}_2 \to \mathcal{A}$ is also a quasi-coherent $\mathcal{O}_ X$-subalgebra of finite type (some details omitted) which contains both $\mathcal{A}_1$ and $\mathcal{A}_2$. In this way we see that the system is directed. To show that $\mathcal{A}$ is the colimit of this system, write $\mathcal{A} = \mathop{\mathrm{colim}}\nolimits _ i \mathcal{A}_ i$ as a directed colimit of finitely presented quasi-coherent $\mathcal{O}_ X$-algebras as in Lemma 28.22.10. Then the images $\mathcal{A}'_ i = \mathop{\mathrm{Im}}(\mathcal{A}_ i \to \mathcal{A})$ are quasi-coherent subalgebras of $\mathcal{A}$ of finite type. Since $\mathcal{A}$ is the colimit of these the result follows. $\square$

Proof. By Lemma 28.22.8 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

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 28.22.3). Set

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 surjections, 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 affine open covering $X = U_1 \cup \ldots \cup U_ m$. Take generators $f_{j, 1}, \ldots , f_{j, N_ j} \in \Gamma (U_ i, \mathcal{F})$. As $\mathcal{A}(U_ j)$ is a finite $\mathcal{O}_ X(U_ j)$-algebra we see that for each $k$ there exists a monic polynomial $P_{j, k} \in \mathcal{O}(U_ j)[T]$ such that $P_{j, k}(f_{j, k})$ is zero in $\mathcal{A}(U_ j)$. Since $\mathcal{A} = \mathop{\mathrm{colim}}\nolimits \mathcal{A}_ i$ by construction, we have $P_{j, k}(f_{j, k}) = 0$ in $\mathcal{A}_ i(U_ j)$ for all sufficiently large $i$. For such $i$ the algebras $\mathcal{A}_ i$ are finite. $\square$

Proof. By Lemma 28.22.11 we have $\mathcal{A} = \mathop{\mathrm{colim}}\nolimits \mathcal{A}_ i$ where $\mathcal{A}_ i \subset \mathcal{A}$ runs through the quasi-coherent $\mathcal{O}_ X$-algebras of finite type. Any finite type quasi-coherent $\mathcal{O}_ X$-subalgebra of $\mathcal{A}$ is finite (apply Algebra, Lemma 10.36.5 to $\mathcal{A}_ i(U) \subset \mathcal{A}(U)$ for affine opens $U$ in $X$). 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 28.22.7. For each $i$, let $\mathcal{J}_ i$ be the kernel of the map

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

commute. This follows from Modules, Lemma 17.22.8. This induces a map

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 28.22.12). Finally, we have

Namely, the first equality was shown in the proof of Lemma 28.22.12 and the second equality because $\mathcal{A}$ is the colimit of the modules $\mathcal{F}_ i$. $\square$