On extending Soulé's variant of Bloch-Quillen identification

Abstract: Based on Balmer's tensor triangular Chow group [2], we propose (Milnor)K-theoretic Chow groups of derived categories of schemes. These Milnor K-theoretic Chow groups recover the classical ones [6] for smooth projective varieties and can detect nilpotent, while the classical ones can't do. As an application, we extend Soul\'e's variant of Bloch-Quilled identification from smooth projective varieties to their trivial infinitesimal thickenings. This answers affirmatively a question by Green-Griffiths for trivial deformations, see Question 1.1 below.