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Brauer group of a curve over non-algebraically closed field

Discussion in 'Mathematics' started by Giulio Bresciani, Oct 8, 2018.

  1. It is a famous consequence of Tsen's theorem that a smooth curve over an algebraically closed field has trivial Brauer group. But what about curves over non algebraically closed fields?

    Let us fix a smooth, projective curve $X$ over some field $k$. If $X$ has a rational point $x\in X(k)$, then the natural map $\text{Br}(k)\to\text{Br}(X)$ has a retraction $\text{Br}(X)\to\text{Br}(k)$, thus it is injective. Moreover, it is widely known that $\text{Br}(X)$ injects in $\text{Br}(k(X))$. But what about closed points? A Brauer class that is trivial on each closed point is globally trivial?

    Precise question: Is the map $$\text{Br}(X)\to\prod_{x\in X^1}\text{Br}(k(x))$$ injective? If not, can we describe its kernel?

    Observe that, for $k$ algebraically closed, the injectivity above is precisely $\text{Br}(X)=0$.

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