Higher Chow groups and not necessarily admissible cycles

2023/11/13 by Bolbachan, Vasily
#Algebraic Geometry (math.AG) #FOS: Mathematics #K-Theory and Homology (math.KT)

paper · doi:10.48550/arxiv.2311.07567

Abstract

We construct some analog of cubical Bloch's higher Chow groups. Instead of considering cycles in X×\mathbb Aⁿ we consider varieties Y over X together with a distinguished element in the n-th exterior power of the multiplicative group of the field of fraction on Y. This definition allows us to make sense of a cycle in X×\mathbb Aⁿ intersecting faces improperly as an element in this complex. We prove that this complex is well-defined and study its basic properties: flat pullback, the localization sequence etc. As an application we prove that the cohomology of this complex in degrees m-1, m and weight m isomorphic to the cohomology of polylogarithmic complex.

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