Additive Chow groups with higher modulus and the generalized de Rham-Witt complex

2005/03/11 by Kay Rülling, Rülling, Kay
Mathematics · #14C15 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #math.AG #msc:14C15

paper · pdf · doi:10.48550/arxiv.math/0503214

74 pages, PhD thesis

arxiv created 2005/03/11 · openalex publication_date 2005/03/11 · arxiv updated 2012/06/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Bloch and Esnault defined additive higher Chow groups with modulus (m+1) on the level of zero cycles over a field k, denoted by THⁿ(k,n;m), n,m >0. They prove that THⁿ(k,n;1) is isomorphic to the group of absolute Kaehler differentials of degree n-1 over k. In this paper we generalize their result and show that THⁿ(k,n;m) is isomorphic to W_mΩ^n-1_k, the group of degree n-1 elements in the generalized de Rham-Witt of length m over k. This complex was defined by Hesselholt and Madsen and generalizes 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 \bbmathZ_(p)-algebras analogous to the construction in the p-typical case. We construct a trace for finite field extensions L / k and if K is the function field of a smooth projective curve C over k and P in C is a point we define a residue map Res_P: W_mΩ¹_K --> W_m(k), which satisfies a "sum-of-residues-equal-zero" theorem.

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