Relative Brauer groups and Kummer Sequence without characteristic constraint in $fppf$-topology

Let $f: X\rightarrow S$ be a faithful affine map. In this article we construct a group homomorphism $θ_{fppf}: \text{Br}(f) \longrightarrow H^1_{fppf}\!\left(S, (f_*\mathcal{O}_X^\times / \mathcal{O}_S^\times)_{fppf}\right)$, extending the formerly defined morphism over an étale site $θ_{et}$. In do