  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.
