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

RESUMO

Let $f: X\rightarrow S$ be a faithful affine map. In this article we construct a group homomorphism $\theta_{fppf}: \text{Br}(f) \longrightarrow H^1_{fppf}\!\left(S, (f_*\mathcal{O}_X^\times / \mathcal{O}_S^\times)_{fppf}\right)$, extending the formerly defined morphism over an étale site $\theta_{et}$. In doing so, we eliminate the characteristic constraint previously found in the relative version of Kummer's exact sequence. Furthermore, given a finite subintegral extension of noetherian rings $A \hookrightarrow B$, we show that all the $fppf$ cohomology groups, $H^i_{fppf}(Spec(A),\mu^f_n); \;\forall i\geq 0$ vanish whenever $n$ does not divide the characteristic of $A$.