 $f \circ h \circ q$ and $f \circ h \circ i\circ q'$ are smooth (with $Bl_{S}X'$ and $E$ quasi-projective in the case $\m V=\m V^{qp}_k$). Let $\mathbb {BL}(\m V/ X \xrightarrow{f}  Y)$ be the free abelian subgroup of $\bM(\m V/X \xrightarrow{f}  Y)$ generated by $$[Bl_{S}X' \xrightarrow{h q}  X] - [E \xrightarrow{hiq'}  X] -  [X'\xrightarrow{h}  X] + [S\xrightarrow{hi}  X]$$ for any such diagram, and define $$\bK_0(\m V/X \xrightarrow{f}  Y) := \frac{\bM(\m V/X \xrightarrow{f}  Y)} {\mathbb {BL}(\m V/X \xrightarrow{f}  Y)}.$$ The corresponding equivalence class of $[V \xrightarrow{p}  X]$ shall be denoted by $\Bigl[[V \xrightarrow{p}  X] \Bigr].$  Note that 