An additive version of higher Chow groups

2001/12/11 by Spencer Bloch, Bloch, Spencer, Hélène Esnault +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AG

paper · pdf · doi:10.48550/arxiv.math/0112101

16 pages

arxiv created 2001/12/11 · openalex publication_date 2001/12/11 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The cosimplicial scheme Delta^bullet = Δ⁰ smallmatrix → smallmatrix Δ¹ smallmatrix to smallmatrix ...; Δⁿ :=\Spec(k[t₀,...c,t_n]/(∑ t_i -t)) was used in B to define higher Chow groups. In this note, we let t tend to 0 and replace Δ^\bullet by a degenerate version Q^\bullet = Q⁰ smallmatrix to smallmatrix Q¹ smallmatrix to smallmatrix ...; Qⁿ := \Spec(k[t₀,...c,t_n]/(∑ t_i)) to define an additive version of the higher Chow groups. For a field k, we show the Chow group of 0-cycles on Qⁿ in this theory is isomorphic to the absolute (n-1)-Kähler forms Ω^n-1_k. An analogous degeneration on the level of de Rham cohomology associated to ``constant modulus'' degenerations of varieties in various contexts is discussed.

Related

• Additive Chow groups with higher modulus and the generalized de Rham-Witt complex
• Additive Chow groups of schemes
• A module structure and a vanishing theorem for cycles with modulus
• Higher Chow groups and not necessarily admissible cycles
• Regulators on additive higher Chow groups