A regulator formula for Milnor \(K\)-groups
type: Item
instance of: scholarly article
title: A regulator formula for Milnor \(K\)-groups
zbMATH Open document ID: 1037.19004
published in: \(K\)-Theory
publication date: 2004-02-03T00:00:00Z
review text: Let \(X\) be a projective variety, \(\dim X=d\). In the paper under review the author constructs some new formulas for the regulator maps from Milnor \(K\)-groups \(K_n^M( \mathbb{C}(X))\) to Deligne cohomology \(H^n_\mathcal{D}(\eta_X, \mathbb{Z}(n))\) which allows to make these maps more accessible to computations in many interesting cases. The first part of the paper is devoted to geometric motivation and construction of cycle-class and Abel-Jacobi maps. Let to an element \({f}= \sum m_j f_{1j}\otimes \cdots \otimes f_{nj} \in \otimes^n\mathbb{Z}\{\mathbb{C}(X)\}\) correspond the subvariety \(V_{{f}}:=\bigcup_{ij}| (f_{ij})| \) and the ``graph cycle'' \[ \gamma_{ {f}}=\big[\sum m_j ( \text{id}_X; f_{1j}, \ldots, f_{nj})_*(X\setminus V_{ {f}}))\big]\cap ((X\setminus V_{ {f}}) \times \square^n), \] where \(\square^n:=( \mathbb{P}^1_\mathbb{C}\setminus \{1\})^n\), thus one gets an isomorphism \(K_n^M( \mathbb{C}(X)) \overset{\cong}{\to} CH^n(\eta_X,n)\). On the other hand, the map \( {f}\to \partial {f}\) induces the map \( \text{Tame}: K_n^M( \mathbb{C}(X)) \to \coprod_{x\in X^1}K_n^M( \mathbb{C}(x))\). Further, the author defines the Abel-Jacobi map \({AJ}\) on graphs which takes graphs into functionals on forms \(\omega\in \Gamma_{X^\infty,c}^{2d-n+1}\), and shows that \({AJ}(\gamma_{f})= \int_X R'_{f}\wedge \omega\) for some \(n-1\)-current \(R'_{f}\). It turns out (Proposition 4.4) that if \(\text{Ker}\text{Tame}=0\) or \(n>d\), then the rule \({f} \mapsto R'_{f}\) induces a well defined map \(R':\;K_n^M( \mathbb{C}(X)) \to H^{n-1}(\eta_X, \mathbb{C}/\mathbb{Q}(n))\). The second part of the paper is devoted to employ a local-global spectral sequence to describing the residues of the Milnor currents. Here the author constructs the higher residue maps of the Milnor currents and the maps \(\text{Tame}^i\), generalizing the map \(\text{Tame}\) from Part I, investigates the relationships between these maps and defines the group \(K_n^M(X)=\bigcap_i \text{ker}(\text{…
Mathematics Subject Classification ID: 19D45
Mathematics Subject Classification ID: 19E15
Mathematics Subject Classification ID: 14C30
Mathematics Subject Classification ID: 14C15
zbMATH DE Number: 2037060
zbMATH Keywords: Milnor \(K\)-theory
zbMATH Keywords: regulator
zbMATH Keywords: Abel-Jacobi map
zbMATH Keywords: polylogarithm
zbMATH Keywords: higher Chow group
author: Matt Kerr
reviewed by: Vasyl I. Andriychuk
MaRDI profile type: Publication
Wikidata QID: Q57445990
full work available at URL: b:kthe.0000006920.60109.e8
OpenAlex ID: W2077970398
DOI: 10.1023/B:KTHE.0000006920.60109.E8
description: scientific article; zbMATH DE number 2037060