  Based on Balmer's tensor triangular Chow group, we propose K-theoretic Chow
groups of derived categories of noetherian schemes and their Milnor variants
for regular schemes and their thickenings. We discuss functoriality and show
that our Chow groups agree with the classical ones for regular schemes.
Moreover, we also define tangent spaces to our Chow groups as usually and
identify them with cohomology groups of differentials. As an application, we
extend Bloch-Quillen identification and soule's variant from regular schemes to
their thickenings. This gives a positive answer to a question by
Green-Griffiths.
