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) .