  We show that for certain singular projective varieties the Levine-Weibel Chow
group of 0-cycles coincides with the corresponding Friedlander-Voevodsky
motivic cohomology, and that the Chow group of 0-cycles with modulus on a
smooth projective variety over a perfect field coincides with the Suslin
homology of the complement of the divisor if the latter has normal crossings.
Among the applications, we prove several generalizations of the finiteness
theorem of Saito and Sato for the Chow group of 0-cycles over $p$-adic fields.
We also prove a torsion theorem for Suslin homology which extends a result of
Bloch to open varieties.
