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Given a ring $R$ with maximal ideal $\mathfrak{m}$, we can form the localization $R_\mathfrak{m}$, the completion $\hat{R}_\mathfrak{m}$ or the Henselization $\hat{R}^h_\mathfrak{m}$ of $R$ with ...

This is a question related to the definition of Brauer-Manin obstruction. Let $K$ be a number field. $X/K$ be an algebraic variety over $K$. Let $O_{X,P}$ be a local ring of $X$ at $P$. Let $Br(X)=\...

Is the Brauer group $\text{Br}(K)$ of a global field $K$ an $\ell$-divisible group for some prime $\ell$? If so, what $\ell$? Is $\text{Br}(K)[n]$ finite, for $n$ integer? I know from class field ...

If $f : X\to V$ is a smooth proper map of smooth schemes, what are the global sections of $R^if_{fppf, *}\mu_p$ $R^if_{fppf, *}\mathbb{G}_{\rm m}$ I was reading Milne's book "Arithmetic duality", ...