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Izv. RAN. Ser. Mat., 2001, Volume 65, Issue 2, Pages 155–186 (Mi izv330)  

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

On the Brauer group of an arithmetic scheme

S. G. Tankeev

Vladimir State University

Abstract: For an Enriques surface $V$ over a number field $k$ with a $k$-rational point we prove that the $l$-component of $\operatorname{Br}(V)/{\operatorname{Br}(k)}$ is finite if and only if $l\ne 2$. For a regular projective smooth variety satisfying the Tate conjecture for divisors over a number field, we find a simple criterion for the finiteness of the $l$-component of $\operatorname{Br}'(V)/{\operatorname{Br}(k)}$. Moreover, for an arithmetic model $X$ of $V$ we prove a variant of Artin's conjecture on the finiteness of the Brauer group of $X$. Applications to the finiteness of the $l$-components of Shafarevich–Tate groups are given.

DOI: https://doi.org/10.4213/im330

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English version:
Izvestiya: Mathematics, 2001, 65:2, 357–388

Bibliographic databases:

MSC: 14F22
Received: 01.02.2000

Citation: S. G. Tankeev, “On the Brauer group of an arithmetic scheme”, Izv. RAN. Ser. Mat., 65:2 (2001), 155–186; Izv. Math., 65:2 (2001), 357–388

Citation in format AMSBIB
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\by S.~G.~Tankeev
\paper On the Brauer group of an arithmetic scheme
\jour Izv. RAN. Ser. Mat.
\yr 2001
\vol 65
\issue 2
\pages 155--186
\mathnet{http://mi.mathnet.ru/izv330}
\crossref{https://doi.org/10.4213/im330}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1842843}
\zmath{https://zbmath.org/?q=an:1004.14004}
\elib{http://elibrary.ru/item.asp?id=14229896}
\transl
\jour Izv. Math.
\yr 2001
\vol 65
\issue 2
\pages 357--388
\crossref{https://doi.org/10.1070/im2001v065n02ABEH000330}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33645393994}


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  • https://doi.org/10.4213/im330
  • http://mi.mathnet.ru/eng/izv/v65/i2/p155

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    This publication is cited in the following articles:
    1. S. V. Tikhonov, V. I. Yanchevskii, “The indices of central simple algebras over function fields of projective spaces over $P_{n,r}$-fields”, Sb. Math., 193:11 (2002), 1691–1705  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. S. G. Tankeev, “On the Brauer group of an arithmetic scheme. II”, Izv. Math., 67:5 (2003), 1007–1029  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. S. G. Tankeev, “On the Conjectures of Artin and Shafarevich–Tate”, Proc. Steklov Inst. Math., 241 (2003), 238–248  mathnet  mathscinet  zmath
    4. 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
    5. 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
    6. S. G. Tankeev, “On the Brauer group of an arithmetic model of a hyperkähler variety over a number field”, Izv. Math., 79:3 (2015), 623–644  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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