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Higher Chow groups and not necessarily admissible cycles

Vasily Bolbachan

Abstract

We construct some analog of cubical Bloch's higher Chow groups. Instead of considering cycles in $X\times\mathbb A^n$ 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 $X\times\mathbb A^n$ 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.

Higher Chow groups and not necessarily admissible cycles

Abstract

We construct some analog of cubical Bloch's higher Chow groups. Instead of considering cycles in we consider varieties over together with a distinguished element in the -th exterior power of the multiplicative group of the field of fraction on . This definition allows us to make sense of a cycle in 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 and weight isomorphic to the cohomology of polylogarithmic complex.
Paper Structure (26 sections, 47 theorems, 113 equations)

This paper contains 26 sections, 47 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. The statement of the main results
  3. Preliminary definitions
  4. Definition of the complex $\Lambda(X, m)$
  5. The outline of the paper
  6. Acknowledgements
  7. The complex $\Lambda(X,m)$ is well-defined
  8. Some vanishing statements
  9. The differential is well-defined
  10. The square of the differential is zero
  11. Properties of the complex $\Lambda(X, m)$
  12. Proper pushforward
  13. Flat pullback
  14. Localization
  15. The complex $\Lambda(X,m)$ and Bloch's higher Chow group.
  16. ...and 11 more sections

Key Result

Theorem 1.2

The complex $\Lambda(X,m)$ is well-defined. It satisfies the following properties:

Theorems & Definitions (92)

  • Definition 1.1
  • Theorem 1.2
  • Conjecture 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 1.6
  • Lemma 1.7
  • Lemma 1.8
  • proof
  • Lemma 1.9
  • ...and 82 more