  The notion of modulus is a striking feature of Rosenlicht-Serre's theory of
generalized Jacobian varieties of curves. It was carried over to algebraic
cycles on general varieties by Bloch-Esnault, Park, R\"ulling, Krishna-Levine.
Recently, Kerz-Saito introduced a notion of Chow group of $0$-cycles with
modulus in connection with geometric class field theory with wild ramification
for varieties over finite fields. We study the non-homotopy invariant part of
the Chow group of $0$-cycles with modulus and show their torsion and
divisibility properties. Modulus is being brought to sheaf theory by
Kahn-Saito-Yamazaki in their attempt to construct a generalization of
Voevodsky-Suslin-Friedlander's theory of homotopy invariant presheaves with
transfers. We prove parallel results about torsion and divisibility properties
for them.
