As an attempt to understand motives over k[x]/(x m ), we define the cubical additive higher Chow groups with modulus for all dimensions extending the works of S. Bloch, H. Esnault and K. Rülling on 0-dimensional cycles. We give an explicit construction of regulator maps on the groups of 1-cycles with an aid of the residue theory of A.

We study additive higher Chow groups with several modulus conditions. Apart from exhibiting the validity of all known results for the additive Chow groups with these modulus conditions, we prove the moving lemma for them: for a smooth projective variety X and a finite collection ᐃ of its locally closed algebraic subsets, every additive higher Chow cycle is congruent to an admissible cycle intersecting properly all members of ᐃ times faces. This is the additive analogue of the moving lemma for the higher Chow groups studied by S. Bloch and M. Levine.As an application, we prove that any morphism from a quasiprojective variety to a smooth projective variety induces a pull-back map of additive higher Chow groups. More important applications of this moving lemma are derived in two separate papers by the authors.

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We construct a graded-commutative differential graded algebra structure on additive higher Chow groups of a smooth projective variety over a perfect field. We show that these groups are equipped with Frobenius and Verschiebung operators, that turn the collection into a Witt-complex.

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