On the additive dilogarithm

Ünver, Sinan

doi:10.2140/ant.2009.3.1

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On the additive dilogarithm

Sinan Ünver

Vol. 3 (2009), No. 1, 1–34

DOI: 10.2140/ant.2009.3.1

Abstract

Let $k$ be a field of characteristic zero, and let $k {[ε]}_{n} : = k [ε] ∕ (ε^{n})$ . We construct an additive dilogarithm ${Li}_{2, n} : B_{2} (k {[ε]}_{n}) \to k^{\oplus (n - 1)}$ , where $B_{2}$ is the Bloch group which is crucial in studying weight two motivic cohomology. We use this construction to show that the Bloch complex of $k {[ε]}_{n}$ has cohomology groups expressed in terms of the K-groups $K_{(\cdot)} (k {[ε]}_{n})$ as expected. Finally we compare this construction to the construction of the additive dilogarithm by Bloch and Esnault defined on the complex $T_{n} ℚ (2) (k)$ .

Keywords

polylogarithms, additive polylogarithms, mixed Tate motives, Hilbert's 3rd problem

Mathematical Subject Classification 2000

Primary: 11G55

Milestones

Received: 14 September 2007

Revised: 1 November 2008

Accepted: 11 November 2008

Published: 1 February 2009

Authors

Sinan Ünver
Koç University Department of Mathematics Rumelifeneri Yolu 34450 Sarıyer-İstanbul Turkey

Vol. 3, No. 1, 2009