On the additive dilogarithm
type: Item
instance of: scholarly article
title: On the additive dilogarithm
zbMATH Open document ID: 1194.11073
author: Sinan Ünver
published in: Algebra \& Number Theory
publication date: 2009-06-19T00:00:00Z
review text: Let \(S\) be any scheme, and \(k\) be any field. According to [cf. \textit{A. Beilinson}, Current trends in arithmetical algebraic geometry, Proc. Summer Res. Conf., Arcata 1985, Comtemp. Math. 67, 1--24 (1987; Zbl 0624.14005)] there should be an abelian tensor category \(\mathcal{M}_S\) of motivic (perverse) sheaves over \(S\) with certain distinguished objects \({\mathbb Z}_{\mathcal{M}}(n), n\in {\mathbb Z}\), called Tate sheaves, with which one can define the motivic cohomology of \(S\) as \(H^i(S,{\mathbb Z}_{\mathcal{M}}(n)):=\text{Ext}_{\mathcal{M}_S}^i({\mathbb Z}_{\mathcal{M}}(0),{\mathbb Z}_{\mathcal{M}}(n))\). If we let \(K_{2n-1}^{(n)}(S)\otimes{\mathbb Q}\to H^i(S,{\mathbb Q}_{\mathcal{M}}(n))\) be the Chern character map from the \(n\)-th graded piece of Quillen's rational \(K\)-theory, the image of the composition \(K_{2n-1}^{(n)}(S)\otimes{\mathbb Q}\to H^i(S,{\mathbb Q}_{\mathcal{M}}(n))\to H_{\text{real}}^i(S,{\mathbb Q}_{\mathcal{M}}(n))\) mapping into one of the realizations of motivic cohomology should contain interesting arithmetic information. The present paper focuses on the category \(\mathcal{MTM}_S\subset\mathcal{M}_S\) of the Tate objects. One expects this subcategory to be constructed using only relative cohomologies of hyperplane arrangements [cf. \textit{P. Deligne} and \textit{A. B. Goncharov}, Ann. Sci. Éc. Norm. Supér. (4) 38, No. 1, 1--56 (2005; Zbl 1084.14024)]. Furthermore there are special degenerate configurations as building blocks for all configurations, the so-called polylogarithmic configurations [\textit{A. A. Beilinson}, \textit{A. B. Goncharov}, \textit{V. V. Schechtman} and \textit{A. N. Varchenko}, The Grothendieck Festschrift, Vol. I. Prog. Math. 86, 135--172 (1990; Zbl 0737.14003)], [\textit{A. B. Goncharov}, Adv. Math. 114, 197--318 (1995; Zbl 0863.19004)]. Using the relations satisfied by polylogarithmic configurations, Goncharov in [loc. cit] defines a complex \[ \Gamma_k(n)_{\mathbb Q}:\mathcal{B}_n(k)\to \mathca…
reviewed by: Oliver Petras
Mathematics Subject Classification ID: 11G55
Mathematics Subject Classification ID: 14F42
Mathematics Subject Classification ID: 19D45
Mathematics Subject Classification ID: 19F27
zbMATH DE Number: 5566602
zbMATH Keywords: polylogarithms
zbMATH Keywords: additive polylogarithms
zbMATH Keywords: mixed Tate motives
zbMATH Keywords: Bloch group
MaRDI profile type: Publication
full work available at URL: ant.2009.3.1
OpenAlex ID: W2072354960
DOI: 10.2140/ANT.2009.3.1
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description: scientific article; zbMATH DE number 5566602