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A higher-dimensional generalization of Lichtenbaum duality in terms of the Albanese map

Part of: Arithmetic problems. Diophantine geometry (Co)homology theory

Published online by Cambridge University Press:  14 July 2016

Wataru Kai
Wataru Kai*
Affiliation:
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan email kaiw@ms.u-tokyo.ac.jp
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Abstract

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In this article, we present a conjectural formula describing the cokernel of the Albanese map of zero-cycles of smooth projective varieties $X$ over $p$-adic fields in terms of the Néron–Severi group and provide a proof under additional assumptions on an integral model of $X$. The proof depends on a non-degeneracy result of Brauer–Manin pairing due to Saito–Sato and on Gabber–de Jong’s comparison result of cohomological and Azumaya–Brauer groups. We will also mention the local–global problem for the Albanese cokernel; the abelian group on the ‘local side’ turns out to be a finite group.

Keywords

Albanese mapBrauer groups

MSC classification

Primary: 14F22: Brauer groups of schemes 14G20: Local ground fields
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 9 , September 2016 , pp. 1915 - 1934
Copyright
© The Author 2016 

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