On 0-cycles with modulus

Abstract

Given a nonsingular surface $X$ over a field and an effective Cartier divisor $D$, we provide an exact sequence connecting CH${}_0\left(X,D\right)$ and the relative $K$-group $K_0\left(X,D\right)$. We use this exact sequence to answer a question of Kerz and Saito whenever $X$ is a resolution of singularities of a normal surface. This exact sequence and two vanishing theorems are used to show that the localization sequence for ordinary Chow groups does not extend to Chow groups with modulus. This in turn shows that the additive Chow groups of 0-cycles on smooth projective schemes cannot always be represented as reciprocity functors.