On the additive dilogarithm
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 \[ \Gamma_k(n)_{\mathbb Q}:\mathcal{B}_n(k)\to \mathcal{B}_{n-1}(k)\otimes (k\otimes{\mathbb Q})^\times\to \mathcal{B}_{n-2}(k)\otimes \wedge^2(k\otimes{\mathbb Q})^\times\to\ldots\to\wedge^n(k\otimes{\mathbb Q})^\times \] involving the so-called Bloch groups and supposed to compute motivic cohomology. For \(k={\mathbb C}\), integration over the polylogarithmic configurations gives a map \(\mathcal{B}_n({\mathbb C})\to{\mathbb R}\) supposed to induce the regulator \(K_{2n-1}^{(n)}({\mathbb C})\otimes{\mathbb Q}\to\mathcal{B}_n({\mathbb C})\to{\mathbb R}\) [Goncharov, loc. cit]. For general fields \(k\), Goncharov again expects an infinitesimal variant of this map, an infinitesimal polylogarithmic function, that acts like a regulator map \(K_{2n-1}^{(n)}(k[\varepsilon]/(\varepsilon^2),(\varepsilon))\to k\). Assuming the existence of the complex \(\Gamma(n)\) over \(k[\varepsilon]/(\varepsilon^2)\), this expectation translates into the existence of a map \(\mathcal{B}_n(k[\varepsilon]/(\varepsilon)^2)/\mathcal{B}_n(k)\to k\) giving an isomorphism if it is composed with \(K_{2n-1}^{(n)}(k[\varepsilon]/(\varepsilon^2),(\varepsilon))\to \mathcal{B}_n(k[\varepsilon]/(\varepsilon)^2)/\mathcal{B}_n(k)\). In this paper, the attention is drawn to weight \(2\). The author proves that there is a natural map \(Li_{2,n}:\mathcal{B}_2(k[\varepsilon]/(\varepsilon^n))\to k^{\oplus(n-1)}\), which composed with the inclusion \(K_3^{(2)}(k[\varepsilon]/(\varepsilon^n),(\varepsilon))\hookrightarrow\mathcal{B}_2(k[\varepsilon]/(\varepsilon^n))\) gives an isomorphism \(K_3^{(2)}(k[\varepsilon]/(\varepsilon^n),(\varepsilon))\simeq k^{\oplus(n-1)}\). This construction is compared to the additive dilogarithm by \textit{S. Bloch} and \textit{H. Esnault} [Doc. Math., J. DMV Extra Vol., 131--155 (2003; Zbl 1052.11048)], which was further investigated and applied by \textit{K. Rülling} [J. Algebr. Geom. 16, No. 1, 109--169 (2007; Zbl 1122.14006)] and by \textit{J. Park} [Int. Math. Res. Not. 2007, No. 18, Article ID rnm067, 19 p. (2007; Zbl 1124.14013)]. The construction of the map mentioned above makes use of Goodwillie's description of relative \(K\)-theory via cyclic homology [\textit{T. G. Goodwillie}, Ann. Math. (2) 124, 347--402 (1986; Zbl 0627.18004)] and the description of rational \(K\)-theory in terms of the primitive part of the group homology of the general linear group. Further, thanks to \textit{J.-L. Cathelineau} [\(K\)-Theory 4, No. 6, 591--606 (1991; Zbl 0735.19005)] there is a basis of the cyclic homology groups of interest available.
scientific article; zbMATH DE number 5566602