Idele class groups with modulus.

We prove Bloch's formula for the Chow group of 0-cycles with modulus on smooth projective varieties over finite fields. This shows that the K-theoretic idele class group with modulus due to Kato-Saito and the cycle-theoretic idele class group with modulus due to Kerz-Saito coincide for such varieties. The proof relies on new results in ramification theory.