Asian Journal of Mathematics

Volume 23 (2019)

Number 1

On extending Soulé’s variant of Bloch–Quillen identification

Pages: 49 – 70



Sen Yang (School of Mathematics, Southeast University, Nanjing, Jiangsu, China; and Yau Mathematical Sciences Center, Tsinghua University, Beijing, China)


Based on Balmer’s tensor triangular Chow group, we propose (Milnor) $K$-theoretic Chow groups of derived categories of schemes. These Milnor $K$-theoretic Chow groups recover the classical ones for smooth projective varieties and can detect nilpotent, while the classical ones can’t do.

As an application, we extend Soulé’s variant of Bloch–Quillen identification from smooth projective varieties to their trivial infinitesimal thickenings. This answers affirmatively a question by Green–Griffiths for trivial deformations.


Chow groups, deformation, $K$-theory, Bloch formula, Chern character, negative cyclic homology, derived category

2010 Mathematics Subject Classification


Received 25 April 2016

Received revised 14 July 2017

Published 3 May 2019