A regulator map for 1-cycles with modulus
type: Item
instance of: scholarly article
title: A regulator map for 1-cycles with modulus
zbMATH Open document ID: 1400.14023
author: Mirai Onoda
published in: Kodai Mathematical Journal
publication date: 2018-08-01T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.kmj/1521424831
full work available at URL: http://t2r2.star.titech.ac.jp/cgi-bin/publicationinfo.cgi
review text: In [Am. J. Math. 131, No. 1, 257--276 (2009; Zbl 1176.14001)] \textit{J. Park} defined cubical additive higher Chow groups \(\mathrm{ACH}_p(X,n;m)\) with modulus \(m\) for any irreducible scheme \(X\) over a field \(k\) of characteristic zero. For the case \(p=1\) and \(X = \mathrm{Spec} \;k \;\) he constructed a regulator map \(R_{2,m} : \mathrm{ACH}_1(k,n;m) \rightarrow \Omega^{n-2}_{k/\mathbb{Z}}\). Park asked the question whether \(R_{2,2}\) is an isomorphism and he stated his belief that the additive Chow groups are related to the Hochschild cohomology. The paper under review contributes to this line of investigation. The author remarks that there are two different definitions for the additive Chow groups of a field, he writes them \(\mathrm{CH}_d(A_k(m),n)_{\mathrm{ssup}}\) and \(\mathrm{CH}_d(A_k(m),n)_{\sup}\), a natural map goes from the first to the second. Park's regulator's domain was \(\mathrm{CH}_1(A_k(m),n)_{\sup}\), but it appears that \(R_{2,m}\) is not well defined for this choice. Let \( \Omega^{n-2}_{k / \mathbb{Z} } \langle w \rangle \) be the vector space of absolute Kähler differentials with the twisted \(k^*\) action of weight \(w\): Oneda's first main result says that for \( m \leq c < 2m \) there are regulator like weight preserving maps \(L^n_c : \mathrm{CH}_1(A_k(m),n)_{\mathrm{ssup}} \to \Omega_k^{n-2} \langle c \rangle \) such that \( L^n_{m+1}\) is equal to the composition of the projection to \(\mathrm{CH}_1(A_k(m),n)_{\sup}\) with Park's \(R_{2,m}\). Moreover if \(k\) is algebraically closed then the map \(L^n \;: \;\mathrm{CH}_1(A_k(m),n)_{\mathrm{ssup}} \to \bigoplus_{m \leq c < 2m } \;\Omega_k^{n-2} \langle c \rangle \) is surjective. Elements in \(\mathrm{CH}_1\) are given by curves, when \(n=2\) parametric curves are used to provide a map \(\Phi ^{\prime} : x^m \;k[ x ] \to \;\mathrm{CH}_1(A_k(m),2)_{\mathrm{ssup}} \). The author then shows that the composition of \(\Phi ^{\prime} \) with the arrow \(\mathrm{CH}_1(A_k(m),2)_{\ma…
reviewed by: Alberto Collino
Mathematics Subject Classification ID: 14C15
Mathematics Subject Classification ID: 14C25
Mathematics Subject Classification ID: 19D55
zbMATH DE Number: 6912408
zbMATH Keywords: algebraic cycle
zbMATH Keywords: Chow group
zbMATH Keywords: regulator map
MaRDI profile type: Publication
OpenAlex ID: W2792415494
Wikidata QID: Q130103110
DOI: 10.2996/KMJ/1521424831
description: scientific article; zbMATH DE number 6912408