Cube invariance of higher Chow groups with modulus

Hiroyasu Miyazaki

Published 2016-04-21Version 1

We introduce a generalized notion of higher Chow groups with modulus. We generalize $\mathbb{A}^1$-homotopy invariance of Bloch's higher Chow groups to an invariance property of higher Chow groups with modulus, called cube-invariance. For the proof of cube-invariance, we need a moving lemma of algebraic cycles which takes modulus conditions into account. As an application of cube-invariance, we prove that over a field of positive characteristic $p$, the higher Chow group with modulus, after $p$ inverted, is $\mathbb{A}^1$-homotopy invariant and is independent of the multiplicity of modulus divisors. We can obtain a similar independence result on relative motivic cohomology groups.