The comparison of the 0-th Suslin homology and Chow groups of 0-cycles with modulus

We show that, for a $K_0$-regular projective normal surface $X$ over a perfect field $k$ of positive characteristic and a reduced effective Cartier divisor $D\hookrightarrow X$, the Chow group of zero cycles on $X$ with modulus $D$ coincides with the 0-th Suslin homology of $X\setminus D$. Moreover, we show that this isomorphism also holds for a projective smooth scheme of any dimension with a reduced effective Cartier divisor.