The Stacks project

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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