fppf拓扑下无特征约束的相对Brauer群与Kummer序列
Relative Brauer groups and Kummer Sequence without characteristic constraint in $fppf$-topology
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- Indian Institute of Technology Kharagpur（印度理工学院卡拉格普尔分校）
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中文总结 AI 辅助
该研究针对忠实仿射映射构造fppf拓扑下的群同态，消除了相对Kummer正合序列的特征约束，还证明了特定诺特环扩张下相关fppf上同调群的消失性质。
中文摘要 AI 辅助
设$f: X\rightarrow S$为忠实仿射映射，本文构造群同态$\theta_{fppf}: \text{Br}(f) \longrightarrow H^1_{fppf}\\!\bigl(S, (f_*\mathcal{O}_X^\times / \mathcal{O}_S^\times)_{fppf}\bigr)$，拓展了先前在平展 site（étale site）上定义的态射$\theta_{et}$，消除了相对Kummer正合序列中先前存在的特征约束。此外，对诺特环的有限次亚整扩张$A \hookrightarrow B$，证明当$n$不整除$A$的特征时，所有$fppf$上同调群$H^i_{fppf}(\text{Spec}(A),\mu^f_n)$（$i\geq0$）均消失。
英文摘要
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 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, $H^i_{fppf}(Spec(A),μ^f_n); \;\forall i\geq 0$ vanish whenever $n$ does not divide the characteristic of $A$.
