The Stacks project

Remark 37.44.8. As a consequence of Lemma 37.44.7 we obtain a comparison morphism

\[ \epsilon : (\mathit{Sch}/S)_{ph} \longrightarrow (\mathit{Sch}/S)_{fppf} \]

This is the morphism of sites given by the identity functor on underlying categories (with suitable choices of sites as in Topologies, Remark 34.11.1). The functor $\epsilon _*$ is the identity on underlying presheaves and the functor $\epsilon ^{-1}$ associated to an fppf sheaf its ph sheafification. By composition we can in addition compare the ph topology with the syntomic, smooth, ├ętale, and Zariski topologies.

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