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Izv. RAN. Ser. Mat., 2003, Volume 67, Issue 5, Pages 155–176 (Mi izv455)  

This article is cited in 6 scientific papers (total in 6 papers)

On the Brauer group of an arithmetic scheme. II

S. G. Tankeev

Vladimir State University

Abstract: Let $\pi\colon X\to\operatorname{Spec}A$ be an arithmetic model of a regular smooth projective variety $V$ over a number field $k$. We prove the finiteness of $H^1(\operatorname{Spec} A,R^1\pi_\ast\operatorname{G}_m)$ under the assumption that $\pi_\ast\operatorname{G}_m=\operatorname{G}_m$ for the étale topology. (This assumption holds automatically if all geometric fibres of $\pi$ are reduced and connected.) If a prime $l$ does not divide $\operatorname{Card}([\operatorname{NS}(V\otimes \bar k)]_{\mathrm{tors}})$, $V(k)\ne\varnothing$, and the Tate conjecture holds for divisors on $V$, then the $l$-primary component $\operatorname{Br}'(X)(l)$ is finite. We also study finiteness properties of the Brauer group of a Calabi–Yau variety $V$ of dimension $\geqslant 2$ over a number field.

DOI: https://doi.org/10.4213/im455

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English version:
Izvestiya: Mathematics, 2003, 67:5, 1007–1029

Bibliographic databases:

UDC: 512.6
MSC: 14F22
Received: 24.04.2002

Citation: S. G. Tankeev, “On the Brauer group of an arithmetic scheme. II”, Izv. RAN. Ser. Mat., 67:5 (2003), 155–176; Izv. Math., 67:5 (2003), 1007–1029

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. S. G. Tankeev, “On the Conjectures of Artin and Shafarevich–Tate”, Proc. Steklov Inst. Math., 241 (2003), 238–248  mathnet  mathscinet  zmath
    2. T. V. Zasorina, “On the Brauer group of an algebraic variety over a finite field”, Izv. Math., 69:2 (2005), 331–343  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. Skorobogatov A.N., Zarhin Yu.G., “A finiteness theorem for the Brauer group of abelian varieties and $K3$ surfaces”, J. Algebraic Geom., 17:3 (2008), 481–502  crossref  mathscinet  zmath  isi  elib  scopus
    4. S. G. Tankeev, “On the Finiteness of the Brauer Group of an Arithmetic Scheme”, Math. Notes, 95:1 (2014), 122–133  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    5. T. V. Prokhorova, “O gruppe Brauera arifmeticheskoi modeli mnogoobraziya nad globalnym polem polozhitelnoi kharakteristiki”, Model. i analiz inform. sistem, 23:2 (2016), 164–172  mathnet  crossref  mathscinet  elib
    6. T. V. Prokhorova, “O gipotezakh Teita dlya divizorov na rassloennom mnogoobrazii i ego obschem skhemnom sloe v sluchae konechnoi kharakteristiki”, Model. i analiz inform. sistem, 24:2 (2017), 205–214  mathnet  crossref  mathscinet  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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