Duality of Albanese and Picard 1-motives
type: Item
instance of: scholarly article
title: Duality of Albanese and Picard 1-motives
zbMATH Open document ID: 0983.14003
DOI: 10.1023/A:1011100812900
author: Niranjan Ramachandran
published in: \(K\)-Theory
publication date: 2001-11-11T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/9804042
review text: Let \(k\) be a perfect field. If \(X\) is a smooth projective variety over \(k\), then it is well-known that the associated Albanese variety \(\text{Alb} (X)\) and the Picard variety \(\text{Pic} (X)\) are naturally dual to each other. This result is extended in this paper to arbitrary schemes by the introduction of Albanese and Picard 1-motives for smooth simplicial schemes over \(k\). For smooth proper schemes these are the classical Albanese and Picard varieties, and for a curve they are the homological 1-motive of Lichtenbaum and the motivic \(H^1\) of Deligne. This paper proves a conjecture of Deligne about providing an algebraic description, via 1-motives, of the first homology and cohomology groups of a complex algebraic variety, which has also been proved independently by Barbieri-Viale and Srinivas. It also contains a purely algebraic proof of Lichtenbaum's conjecture that the Albanese and Picard 1-motives of a simplicial scheme are dual to each other. This provides a new proof of an unpublished theorem of Lichtenbaum that Deligne's 1-motive of a curve is dual to Lichtenbaum's 1-motive.
reviewed by: Min Ho Lee
Mathematics Subject Classification ID: 14C25
Mathematics Subject Classification ID: 14F42
Mathematics Subject Classification ID: 14C30
zbMATH DE Number: 1624882
zbMATH Keywords: 1-motive
zbMATH Keywords: Albanese varieties
zbMATH Keywords: Picard varieties
MaRDI profile type: Publication
arXiv ID: math/9804042
OpenAlex ID: W2955310887
description: scientific article; zbMATH DE number 1624882