Situation 21.25.1. Let $\mathcal{C}$ be a site. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}$. Let $\mathcal{A} \subset \textit{Mod}(\mathcal{O})$ be a weak Serre subcategory. We assume the following is true: there exists a subset $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ such that

every object of $\mathcal{C}$ has a covering whose members are in $\mathcal{B}$, and

for every $V \in \mathcal{B}$ there exists an integer $d_ V$ and a cofinal system $\text{Cov}_ V$ of coverings of $V$ such that

\[ H^ p(V_ i, \mathcal{F}) = 0 \text{ for } \{ V_ i \to V\} \in \text{Cov}_ V,\ p > d_ V, \text{ and } \mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}) \]

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