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

Abstract

Let $f:X\to S$ be a faithful affine map. In this article we construct a group homomorphism ${\theta }_{fppf}:\text{Br}(f)⟶H_{fppf}^1\left(S,(f_{\ast }{\mathcal{O}}_X^{\times }/{\mathcal{O}}_S^{\times }{)}_{fppf}\right)$, 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, $H_{fppf}^i(Spec(A),{\mu }_n^f);\forall i\ge 0$ vanish whenever $n$ does not divide the characteristic of $A$.

arXiv commentCorrected typos, flatness of the morphism is explicitly added in every theorem of chapter 5, modification done in the proof and statement of the final theorem