The Stacks project

Remark 46.10.1. Let $S$ be a scheme. Let $\tau , \tau ' \in \{ Zar, {\acute{e}tale}, smooth, syntomic, fppf\}$. Denote $\mathcal{O}_\tau$, resp. $\mathcal{O}_{\tau '}$ the structure sheaf $\mathcal{O}$ viewed as a sheaf on $(\mathit{Sch}/S)_\tau$, resp. $(\mathit{Sch}/S)_{\tau '}$. Then $D_{\textit{Adeq}}(\mathcal{O}_\tau )$ and $D_{\textit{Adeq}}(\mathcal{O}_{\tau '})$ are canonically isomorphic. This follows from Cohomology on Sites, Lemma 21.29.1. Namely, assume $\tau$ is stronger than the topology $\tau '$, let $\mathcal{C} = (\mathit{Sch}/S)_{fppf}$, and let $\mathcal{B}$ the collection of affine schemes over $S$. Assumptions (1) and (2) we've seen above. Assumption (3) is clear and assumption (4) follows from Lemma 46.5.8.