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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 
$\text{CH}_{0}(X)$ and the Chow group of 0-cycles with modulus 
$\text{CH}_{0}(X|D)$ on 
$X$. When 
$X$ is projective, we construct an Albanese variety with modulus and show that this is the universal regular quotient of 
$\text{CH}_{0}(X|D)$. As a consequence of the above decomposition, we prove the Roitman torsion theorem for the 0-cycles with modulus. We show that 
$\text{CH}_{0}(X|D)$ is torsion-free and there is an injective cycle class map 
$\text{CH}_{0}(X|D){\hookrightarrow}K_{0}(X,D)$ if 
$X$ is affine. For a smooth affine surface 
$X$, this is strengthened to show that 
$K_{0}(X,D)$ is an extension of 
$\text{CH}_{1}(X|D)$ by 
$\text{CH}_{0}(X|D)$.
