Zero cycles with modulus and zero cycles on singular varieties

Given a smooth variety X and an effective Cartier divisor D ⊂ 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 CH0(X) and the Chow group of 0-cycles with modulus CH0(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 CH0 (X/D). As a consequence of the above decomposition, we prove the Roitman torsion theorem for the 0-cycles with modulus. We show that CH0 (X/D) is torsion-free and there is an injective cycle class map CH0 (X/D) → K0 (X,D) if X is affine. For a smooth affine surface X, this is strengthened to show that K0 (X,D) is an extension of CH1 (X/D) by CH0 (X/D).