Zero cycles with modulus and zero cycles on singular varieties

Given a smooth variety $X$ and an effective Cartier divisor $D\subset X$, we show that the cohomological Chow group of 0-cycles on the double of $X$ along $D$ has a canonical decomposition in terms of the Chow group of 0-cycles ${\rm CH}_0(X)$andtheChowgroupof0-cycleswithmodulus${\rm CH}_0(X|D)$on$X$. When $X$ is projective, we construct an Albanese variety with modulus and show that this is the universal regular quotient of ${\mathrm{C}\mathrm{H}}_0(X\mathrm{\mid }D)$. As a consequence of the above decomposition, we prove the Roitman torsion theorem for the 0-cycles with modulus. We show that ${\mathrm{C}\mathrm{H}}_0(X\mathrm{\mid }D)$ is torsion-free and there is an injective cycle class map ${\rm CH}_0(X|D) \hookrightarrow K_0(X,D)$if$X$isaffine\mathrm{.}Forasmoothaffinesurface$X$, this is strengthened to show that $K_0(X,D)$ is an extension of ${\rm CH}_1(X|D)$by${\rm CH}_0(X|D)$.