|
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
Full text:
PDF file (1811 kB)
References:
PDF file
HTML file
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
\Bibitem{Tan03}
\by S.~G.~Tankeev
\paper On the Brauer group of an arithmetic scheme.~II
\jour Izv. RAN. Ser. Mat.
\yr 2003
\vol 67
\issue 5
\pages 155--176
\mathnet{http://mi.mathnet.ru/izv455}
\crossref{https://doi.org/10.4213/im455}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2018744}
\zmath{https://zbmath.org/?q=an:1078.14023}
\elib{http://elibrary.ru/item.asp?id=14229897}
\transl
\jour Izv. Math.
\yr 2003
\vol 67
\issue 5
\pages 1007--1029
\crossref{https://doi.org/10.1070/IM2003v067n05ABEH000455}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000187798600007}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33645399836}
Linking options:
http://mi.mathnet.ru/eng/izv455https://doi.org/10.4213/im455 http://mi.mathnet.ru/eng/izv/v67/i5/p155
Citing articles on Google Scholar:
Russian citations,
English citations
Related articles on Google Scholar:
Russian articles,
English articles
Cycle of papers
This publication is cited in the following articles:
-
S. G. Tankeev, “On the Conjectures of Artin and Shafarevich–Tate”, Proc. Steklov Inst. Math., 241 (2003), 238–248
-
T. V. Zasorina, “On the Brauer group of an algebraic variety over a finite field”, Izv. Math., 69:2 (2005), 331–343
-
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
-
S. G. Tankeev, “On the Finiteness of the Brauer Group of an Arithmetic Scheme”, Math. Notes, 95:1 (2014), 122–133
-
T. V. Prokhorova, “O gruppe Brauera arifmeticheskoi modeli mnogoobraziya nad globalnym polem polozhitelnoi kharakteristiki”, Model. i analiz inform. sistem, 23:2 (2016), 164–172
-
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
|
Number of views: |
This page: | 270 | Full text: | 97 | References: | 31 | First page: | 1 |
|