Lemma 18.24.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\theta : \mathcal{G} \to \mathcal{F}$ be a surjective $\mathcal{O}$-module map with $\mathcal{F}$ of finite presentation and $\mathcal{G}$ of finite type. Then $\mathop{\mathrm{Ker}}(\theta )$ is of finite type.
18.24 Basic results on local types of modules
Basic lemmas related to the definitions made above.
Proof. Omitted. Hint: See Modules, Lemma 17.11.3. $\square$
Lemma 18.24.2. Let $\mathcal{C}$ be a category viewed as a site with the chaotic topology, see Sites, Example 7.6.6. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}$ and let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules. Then $\mathcal{F}$ is quasi-coherent if and only if for all $U \to V$ in $\mathcal{C}$ the canonical map is an isomorphism.
\[ \mathcal{F}(V) \otimes _{\mathcal{O}(V)} \mathcal{O}(U) \longrightarrow \mathcal{F}(U) \]
Proof. Assume $\mathcal{F}$ is quasi-coherent and let $U \to V$ be a morphism of $\mathcal{C}$. Since every covering of $V$ is given by an isomorphism we conclude from Definition 18.23.1 that there exists a presentation
\[ \bigoplus \nolimits _{j \in J} \mathcal{O}_ V \longrightarrow \bigoplus \nolimits _{i \in I} \mathcal{O}_ V \longrightarrow \mathcal{F}|_{\mathcal{C}/V} \longrightarrow 0 \]
Since the topology on $\mathcal{C}$ is chaotic, taking sections over any object of $\mathcal{C}$ is exact. We conclude that we obtain a presentation
\[ \bigoplus \nolimits _{j \in J} \mathcal{O}(V) \longrightarrow \bigoplus \nolimits _{i \in I} \mathcal{O}(V) \longrightarrow \mathcal{F}(V) \longrightarrow 0 \]
of $\mathcal{F}(V)$ as an $\mathcal{O}(V)$-module and similarly for $\mathcal{F}(U)$. This easily shows that the displayed map in the statement of the lemma is an isomorphism.
Assume the displayed map in the statement of the lemma is an isomorphism for every morphism $U \to V$ in $\mathcal{C}$. Fix $V$ and choose a presentation
\[ \bigoplus \nolimits _{j \in J} \mathcal{O}(V) \longrightarrow \bigoplus \nolimits _{i \in I} \mathcal{O}(V) \longrightarrow \mathcal{F}(V) \longrightarrow 0 \]
of $\mathcal{F}(V)$ as an $\mathcal{O}(V)$-module. Then the assumption on $\mathcal{F}$ exactly means that the corresponding sequence
\[ \bigoplus \nolimits _{j \in J} \mathcal{O}_ V \longrightarrow \bigoplus \nolimits _{i \in I} \mathcal{O}_ V \longrightarrow \mathcal{F}|_{\mathcal{C}/V} \longrightarrow 0 \]
is exact and we conclude that $\mathcal{F}$ is quasi-coherent. $\square$
Lemma 18.24.3. Let $\mathcal{C}$ be a category viewed as a site with the chaotic topology, see Sites, Example 7.6.6. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}$. Assume for all $U \to V$ in $\mathcal{C}$ the restriction map $\mathcal{O}(V) \to \mathcal{O}(U)$ is a flat ring map. Then the category of quasi-coherent $\mathcal{O}$-modules is a weak Serre subcategory of $\textit{Mod}(\mathcal{O})$.
Proof. We will check the definition of a weak Serre subcategory, see Homology, Definition 12.10.1. To do this we will use the characterization of quasi-coherent modules given in Lemma 18.24.2. Consider an exact sequence
\[ \mathcal{F}_0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to \mathcal{F}_4 \]
in $\textit{Mod}(\mathcal{O})$ with $\mathcal{F}_0$, $\mathcal{F}_1$, $\mathcal{F}_3$, and $\mathcal{F}_4$ quasi-coherent. Let $U \to V$ be a morphism of $\mathcal{C}$ and consider the commutative diagram
\[ \xymatrix{ \mathcal{F}_0(V) \otimes _{\mathcal{O}(V)} \mathcal{O}(U) \ar[r] \ar[d] & \mathcal{F}_1(V) \otimes _{\mathcal{O}(V)} \mathcal{O}(U) \ar[r] \ar[d] & \mathcal{F}_2(V) \otimes _{\mathcal{O}(V)} \mathcal{O}(U) \ar[r] \ar[d] & \mathcal{F}_3(V) \otimes _{\mathcal{O}(V)} \mathcal{O}(U) \ar[r] \ar[d] & \mathcal{F}_4(V) \otimes _{\mathcal{O}(V)} \mathcal{O}(U) \ar[d] \\ \mathcal{F}_0(U) \ar[r] & \mathcal{F}_1(U) \ar[r] & \mathcal{F}_2(U) \ar[r] & \mathcal{F}_3(U) \ar[r] & \mathcal{F}_4(U) } \]
By assumption the vertical arrows with indices $0$, $1$, $3$, $4$ are isomorphisms. Since the topology on $\mathcal{C}$ is chaotic taking sections over an object of $\mathcal{C}$ is exact and hence the lower row is exact. Since $\mathcal{O}(V) \to \mathcal{O}(U)$ is flat also the upper row is exact. Thus we conclude that the middle arrow is an isomorphism by the $5$ lemma (Homology, Lemma 12.5.20). $\square$
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