Algebra & Number Theory

Albanese varieties with modulus over a perfect field

Henrik Russell

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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.

Article information

Algebra Number Theory, Volume 7, Number 4 (2013), 853-892.

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14L10: Group varieties
Secondary: 11G45: Geometric class field theory [See also 11R37, 14C35, 19F05] 14C15: (Equivariant) Chow groups and rings; motives

Albanese with modulus relative Chow group with modulus geometric class field theory


Russell, Henrik. Albanese varieties with modulus over a perfect field. Algebra Number Theory 7 (2013), no. 4, 853--892. doi:10.2140/ant.2013.7.853.

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