Suslin homology via cycles with modulus and applications

We show that for a smooth projective variety X over a field k and a reduced effective Cartier divisor D subset of X, the Chow group of 0-cycles with modulus CH0(X divide D) coincides with the Suslin homology H0S (X set minus D) under some necessary conditions on k and D. We derive several consequences, and we answer to a question of Barbieri-Viale and Kahn.