Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Cube invariance of higher Chow groups with modulus


Author: Hiroyasu Miyazaki
Journal: J. Algebraic Geom. 28 (2019), 339-390
DOI: https://doi.org/10.1090/jag/726
Published electronically: January 24, 2019
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Abstract: The higher Chow groups with modulus generalize both Bloch's higher Chow groups and the additive higher Chow groups. Our first aim is to formulate and prove a generalization of $ \mathbb{A}^1$-homotopy invariance for the higher Chow groups with modulus, called the ``cube invariance''. The proof requires a new moving lemma of algebraic cycles. Next, we introduce the obstruction to the $ \mathbb{A}^1$-homotopy invariance, called the ``nilpotent higher Chow groups'', as analogues of the nilpotent $ K$-groups. We prove that the nilpotent higher Chow groups admit module structures over the big Witt ring of the base field. This result implies that the higher Chow groups with modulus with appropriate coefficients satisfy $ \mathbb{A}^1$-homotopy invariance. We also prove that $ \mathbb{A}^1$-homotopy invariance implies independence from the multiplicity of the modulus divisors.


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Hiroyasu Miyazaki
Affiliation: Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG), Sorbonne Université - Campus Pierre et Marie Curie 4, place Jussieu, 75005 Paris, France – and – RIKEN iTHEMS, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan – and – Department of Mathematical Sciences, the University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8914 Japan
Address at time of publication: RIKEN iTHEMS, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
Email: hiroyasu.miyazaki@riken.jp, hiroyasumyzk@gmail.com

DOI: https://doi.org/10.1090/jag/726
Received by editor(s): August 11, 2017
Received by editor(s) in revised form: May 20, 2018, and July 24, 2018
Published electronically: January 24, 2019
Additional Notes: This work was supported by a Grant-in-Aid for JSPS Fellows (Grant Number 15J08833) and the Program for Leading Graduate Schools, MEXT, Japan. This work was also supported by RIKEN iTHEMS Program and by Fondation Sciences Mathématiques de Paris.
Article copyright: © Copyright 2019 University Press, Inc.