Additive Chow Groups with Higher Modulus and the Generalized De Rham-witt Complex

Bloch and Esnault defined additive higher Chow groups with modulus (m + 1) on the level of zero cycles over a field k, denoted by TH(k, n;m), n, m ≥ 1. They prove TH(k, n; 1) ∼= Ω k/Z . In this paper we generalize their result and obtain an isomorphism TH(k, n;m) ∼= WmΩ n−1 k , where W−Ω·k is the generalized de Rham-Witt complex of Hesselholt-Madsen, generalizing the p-typical de Rham-Witt complex of Bloch-Deligne-Illusie. Before we can prove this theorem we have to generalize some classical results to the de Rham-Witt complex. We give a construction of the generalized de Rham-Witt complex for Z(p)-algebras analogous to the construction in the ptypical case. We construct a trace Tr : W−Ω·L → W−Ω · k for finite field extensions L ⊃ k and if K is the function field of a smooth projective curve C over k and P ∈ C is a point we define a residue map ResP : W−ΩK → W−(k), which satisfies ∑ P∈C ResP (ω) = 0, for all ω ∈ W−Ω 1 K .