Open Access
2013 Albanese varieties with modulus over a perfect field
Henrik Russell
Algebra Number Theory 7(4): 853-892 (2013). DOI: 10.2140/ant.2013.7.853


Let X be a smooth proper variety over a perfect field k of arbitrary characteristic. Let D be an effective divisor on X with multiplicity. We introduce an Albanese variety Alb(X,D) of X of modulus D as a higher-dimensional analogue of the generalized Jacobian of Rosenlicht and Serre with modulus for smooth proper curves. Basing on duality of 1-motives with unipotent part (which are introduced here), we obtain explicit and functorial descriptions of these generalized Albanese varieties and their dual functors.

We define a relative Chow group of zero cycles CH0(X,D) of modulus D and show that Alb(X,D) can be viewed as a universal quotient of CH0(X,D)0.

As an application we can rephrase Lang’s class field theory of function fields of varieties over finite fields in explicit terms.


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Henrik Russell. "Albanese varieties with modulus over a perfect field." Algebra Number Theory 7 (4) 853 - 892, 2013.


Received: 18 February 2011; Revised: 7 April 2012; Accepted: 17 May 2012; Published: 2013
First available in Project Euclid: 20 December 2017

zbMATH: 1282.14078
MathSciNet: MR3095229
Digital Object Identifier: 10.2140/ant.2013.7.853

Primary: 14L10
Secondary: 11G45 , 14C15

Keywords: Albanese with modulus , geometric class field theory , relative Chow group with modulus

Rights: Copyright © 2013 Mathematical Sciences Publishers

Vol.7 • No. 4 • 2013
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