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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