Cube invariance of higher Chow groups with modulus

The higher Chow groups with modulus generalize both Bloch’s higher Chow groups and the additive higher Chow groups. Our first aim is to formulate and prove a generalization of $\mathbb {A}^1$-homotopy invariance for the higher Chow groups with modulus, called the “cube invariance”. The proof requires a new moving lemma of algebraic cycles. Next, we introduce the obstruction to the $\mathbb {A}^1$-homotopy invariance, called the “nilpotent higher Chow groups”, as analogues of the nilpotent $K$-groups. We prove that the nilpotent higher Chow groups admit module structures over the big Witt ring of the base field. This result implies that the higher Chow groups with modulus with appropriate coefficients satisfy $\mathbb {A}^1$-homotopy invariance. We also prove that $\mathbb {A}^1$-homotopy invariance implies independence from the multiplicity of the modulus divisors.