# On geometric Brauer groups and Tate-Shafarevich groups.

@article{Qin2020OnGB, title={On geometric Brauer groups and Tate-Shafarevich groups.}, author={Yanshuai Qin}, journal={arXiv: Algebraic Geometry}, year={2020} }

Let $X$ be a smooth projective variety over a finitely generated field $K$ of characteristic $p>0$. We proved that the finiteness of the $\ell$-primary part of $\mathrm{Br}(X_{K^s})^{G_K}$ for a single prime $\ell\neq p$ will imply the finiteness of the prime-to-$p$ part of $\mathrm{Br}(X_{K^s})^{G_K}$, generalizing a theorem of Tate and Lichtenbaum for varieties over finite fields. For an abelian variety $A$ over $K$, we proved a similar result for the Tate-Shafarevich group of $A…

## One Citation

On the Brauer groups of fibrations II

- Mathematics
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Let C be the spectrum of the ring of integers in a number field or a smooth projective geometrically connected curve over a finite field with function field K. Let X be a 2dimensional regular scheme…

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