On the Brauer Group of an Arithmetic Model of a Variety over a Global Field of Positive Characteristic
Let V be a smooth projective variety over a global field k = κ(C) of rational functions on a smooth projective curve C over a finite field Fq of characteristic p. Assume that there is a projective flat Fq-morphism π : X → C, where X is a smooth projective variety and the generic scheme fiber of π is...
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| Format: | Article |
| Language: | English |
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Yaroslavl State University
2016-04-01
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| Series: | Моделирование и анализ информационных систем |
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| Online Access: | https://www.mais-journal.ru/jour/article/view/326 |
| _version_ | 1826559003875868672 |
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| author | T. V. Prokhorova |
| author_facet | T. V. Prokhorova |
| author_sort | T. V. Prokhorova |
| collection | DOAJ |
| description | Let V be a smooth projective variety over a global field k = κ(C) of rational functions on a smooth projective curve C over a finite field Fq of characteristic p. Assume that there is a projective flat Fq-morphism π : X → C, where X is a smooth projective variety and the generic scheme fiber of π is isomorphic to a variety V (we call π : X → C an arithmetic model of a variety V ). M. Artin conjectured the finiteness of the Brauer group Br(X) classifying sheaves of Azumaya algebras on X modulo similitude. It is well known that the group Br(X) is contained in the cohomological Brauer group Br′(X)=H2(X,G ). et m By definition, the non−p component of the cohomological Brauer group Br′(X) coincides with the direct sum of the l-primary components of the group Br′(X) for all prime numbers l different from the characteristic p. It is known that the structure of k-variety on V yields the canonical morphism of the groups Br(k) → Br′(V ). The finiteness of the non−p component of the cohomological Brauer group Br′(X) of a variety X has been proved if [Br′(V )/ Im[Br(k) → Br′(V )]](non −p) is finite. In particular, if V is a K 3 surface (in other words, V is a smooth projective simply connected surface over a field k and the canonical class of a surface of V is trivial: Ω2V = OV ) and the characteristic of the ground field p > 2, then, by the Skorobogatov – Zarhin theorem, [Br′(V )/ Im[Br(k) → Br′(V )]](non −p) is finite, so in this case the groups Br′(X)(non−p) and Br(X)(non−p) are finite. |
| first_indexed | 2024-04-10T02:24:29Z |
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| institution | Directory Open Access Journal |
| issn | 1818-1015 2313-5417 |
| language | English |
| last_indexed | 2025-03-14T08:53:29Z |
| publishDate | 2016-04-01 |
| publisher | Yaroslavl State University |
| record_format | Article |
| series | Моделирование и анализ информационных систем |
| spelling | doaj.art-ff02274871fc4f9ca8c10e82530fc91a2025-03-02T12:46:57ZengYaroslavl State UniversityМоделирование и анализ информационных систем1818-10152313-54172016-04-0123216417210.18255/1818-1015-2016-2-164-172288On the Brauer Group of an Arithmetic Model of a Variety over a Global Field of Positive CharacteristicT. V. Prokhorova0A. G. and N. G. Stoletov Vladimir State University, Gorky str., 87, Vladimir, 600000, RussiaLet V be a smooth projective variety over a global field k = κ(C) of rational functions on a smooth projective curve C over a finite field Fq of characteristic p. Assume that there is a projective flat Fq-morphism π : X → C, where X is a smooth projective variety and the generic scheme fiber of π is isomorphic to a variety V (we call π : X → C an arithmetic model of a variety V ). M. Artin conjectured the finiteness of the Brauer group Br(X) classifying sheaves of Azumaya algebras on X modulo similitude. It is well known that the group Br(X) is contained in the cohomological Brauer group Br′(X)=H2(X,G ). et m By definition, the non−p component of the cohomological Brauer group Br′(X) coincides with the direct sum of the l-primary components of the group Br′(X) for all prime numbers l different from the characteristic p. It is known that the structure of k-variety on V yields the canonical morphism of the groups Br(k) → Br′(V ). The finiteness of the non−p component of the cohomological Brauer group Br′(X) of a variety X has been proved if [Br′(V )/ Im[Br(k) → Br′(V )]](non −p) is finite. In particular, if V is a K 3 surface (in other words, V is a smooth projective simply connected surface over a field k and the canonical class of a surface of V is trivial: Ω2V = OV ) and the characteristic of the ground field p > 2, then, by the Skorobogatov – Zarhin theorem, [Br′(V )/ Im[Br(k) → Br′(V )]](non −p) is finite, so in this case the groups Br′(X)(non−p) and Br(X)(non−p) are finite.https://www.mais-journal.ru/jour/article/view/326brauer grouparithmetic modelk 3 surface |
| spellingShingle | T. V. Prokhorova On the Brauer Group of an Arithmetic Model of a Variety over a Global Field of Positive Characteristic Моделирование и анализ информационных систем brauer group arithmetic model k 3 surface |
| title | On the Brauer Group of an Arithmetic Model of a Variety over a Global Field of Positive Characteristic |
| title_full | On the Brauer Group of an Arithmetic Model of a Variety over a Global Field of Positive Characteristic |
| title_fullStr | On the Brauer Group of an Arithmetic Model of a Variety over a Global Field of Positive Characteristic |
| title_full_unstemmed | On the Brauer Group of an Arithmetic Model of a Variety over a Global Field of Positive Characteristic |
| title_short | On the Brauer Group of an Arithmetic Model of a Variety over a Global Field of Positive Characteristic |
| title_sort | on the brauer group of an arithmetic model of a variety over a global field of positive characteristic |
| topic | brauer group arithmetic model k 3 surface |
| url | https://www.mais-journal.ru/jour/article/view/326 |
| work_keys_str_mv | AT tvprokhorova onthebrauergroupofanarithmeticmodelofavarietyoveraglobalfieldofpositivecharacteristic |