The generalized de Rham-Witt complex over a field is a complex of zero-cycles

Abstract

Bloch and Esnault defined additive higher Chow groups with modulus m m on the level of zero cycles over a field k k denoted by CH n ( ( A k 1 , ( m + 1 ) { 0 } ) , n − 1 ) \text {CH}^n((\mathbb {A}^1_k,(m+1)\{0\}),n-1) , n , m ≥ 1 n,m\ge 1 . Bloch and Esnault prove CH n ( ( A k 1 , 2 { 0 } ) , n − 1 ) ≅ Ω k / Z n − 1 \text {CH}^n((\mathbb {A}^1_k,2\{0\}),n-1)\cong \Omega ^{n-1}_{k/\mathbb {Z}} . In this paper we generalize their result and prove that the additive Chow groups with higher modulus form a generalized Witt complex over k k and are as such isomorphic to the generalized de Rham-Witt complex of Bloch-Deligne-Hesselholt-Illusie-Madsen.