Lemma 21.43.3. In the situation above, suppose that $M$ is an object of $\mathit{QC}(\mathcal{O})$ and $b \in \mathbf{Z}$ such that $H^ i(M) = 0$ for all $i > b$. Then $H^ b(M)$ is a quasi-coherent module on $(\mathcal{C}, \mathcal{O})$ in the sense of Modules on Sites, Definition 18.23.1.
Proof. By Modules on Sites, Lemma 18.24.2 it suffices to show that for every morphism $U \to V$ of $\mathcal{C}$ the map
\[ H^ p(M)(V) \otimes _{\mathcal{O}(V)} \mathcal{O}(U) \to H^ b(M)(U) \]
is an isomorphism. We are given that the map
\[ R\Gamma (V, M) \otimes _{\mathcal{O}(V)}^\mathbf {L} \mathcal{O}(U) \to R\Gamma (U, M) \]
is an isomorphism. Thus the result by the Tor spectral sequence for example. Details omitted. $\square$
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