If X is a variety and $Y\subset X$ is a closed subscheme then one can define relative Chow group. The definition is follows: there is subcomplex $\psi_Y\colon z^r_Y(X,*)\hookrightarrow z^r(X,*)$ of cycles which intersects $Y$ properly. One can show that $\psi_Y$ is a quasi-isomorphism. Define $z^r(X,Y, *)=Cone(\psi_Y)$. (See RELATIVE CHOW GROUPS by SE Landsburg).

Is it known that $H^0(z^r(X\times\square^{n}, X\times\partial \square^{n}, *))$ isomorphic to Bloch's higher Chow group $H^n(z^r(X, *))$?

Here $\square^n=\mathbb A^n$ and $\partial \square^{n}$ is given by the equation $x_1(1-x_1)\dots x_n(1-x_n)=0$.

In the paper of M. Levine "Bloch's higher Chow groups revisited" he does something similar, but I don't understand why he gets relative Chow groups. He takes not cohomology of cone but the cycles which do not intersect the divisor $\partial \square^{n}$. Why it is the same thing?

Connected: Chow group of a pair but without any answers(



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