The Stacks project

Lemma 21.43.11. Let $\mathcal{C}$ be a category viewed as a site with the chaotic topology. 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 $\mathit{QC}(\mathcal{O})$ agrees with the subcategory $D_\mathit{QCoh}(\mathcal{O}) \subset D(\mathcal{O})$ of complexes whose cohomology sheaves are quasi-coherent.

Proof. Recall that $\mathit{QCoh}(\mathcal{O}) \subset \textit{Mod}(\mathcal{O})$ is a weak Serre subcategory under our assumptions, see Modules on Sites, Lemma 18.24.3. Thus taking the full subcategory

\[ D_\mathit{QCoh}(\mathcal{O}) = D_{\mathit{QCoh}(\mathcal{O})}(\textit{Mod}(\mathcal{O})) \]

of $D(\mathcal{O})$ makes sense, see Derived Categories, Section 13.17. (Strictly speaking we don't need this in the proof of the lemma.)

Let $M$ be an object of $\mathit{QC}(\mathcal{O})$. Since for every morphism $U \to V$ in $\mathcal{C}$ the restriction map $\mathcal{O}(V) \to \mathcal{O}(U)$ is flat, we see that

\begin{align*} H^ i(M)(U) & = H^ i(R\Gamma (U, M)) \\ & = H^ i(R\Gamma (V, M) \otimes _{\mathcal{O}(V)}^\mathbf {L} \mathcal{O}(U)) \\ & = H^ i(R\Gamma (V, M)) \otimes _{\mathcal{O}(V)} \mathcal{O}(U) \\ & = H^ i(M)(V) \otimes _{\mathcal{O}(V)} \mathcal{O}(U) \end{align*}

and hence $H^ i(M)$ is quasi-coherent by Modules on Sites, Lemma 18.24.2. The first and last equality above follow from the fact that taking sections over an object of $\mathcal{C}$ is an exact functor due to the fact that the topology on $\mathcal{C}$ is chaotic.

Conversely, if $M$ is an object of $D_\mathit{QCoh}(\mathcal{O})$, then due to Modules on Sites, Lemma 18.24.2 we see that the map $R\Gamma (V, M) \to R\Gamma (U, M)$ induces isomorphisms $H^ i(M)(U) \to H^ i(M)(V) \otimes _{\mathcal{O}(V)} \mathcal{O}(U)$. Whence $R\Gamma (V, K) \otimes _{\mathcal{O}(V)}^\mathbf {L} \mathcal{O}(U) \to R\Gamma (U, K)$ is an isomorphism in $D(\mathcal{O}(U))$ by the flatness of $\mathcal{O}(V) \to \mathcal{O}(U)$ and we conclude that $M$ is in $\mathit{QC}(\mathcal{O})$. $\square$

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