Zero-cycle groups on algebraic varieties
type: Item
instance of: scholarly article
title: Zero-cycle groups on algebraic varieties
zbMATH Open document ID: 1486.14013
author: F. Binda
author: Amalendu Krishna
publication date: 2022-02-03T00:00:00Z
full work available at URL: https://arxiv.org/abs/2104.07968
review text: On smooth varieties over a field, we have the motivic cohomology theory constructed in a few different way. For instance, Voevodsky's motivic cohomology of smooth varieties is isomorphic to Bloch's higher Chow groups. When the varieties have singularities, there have been a couple of new attempts, such as Levine-Weibel Chow groups \(\mathrm{CH}_0 ^{LW} (X)\) or Chow groups \(\mathrm{CH}_0 (X|D)\) with modulus, for instance, that provided certain successful consequences in a few special cases. This article is one of the recent results by the authors (possibly with other authors, as well), that compare those different attempts. The results in this article present a few omnibus results in a few different special situations in this direction. The inter-relations among those proven are not too strong, but each of them provides certain extensions of previously known results, or clarifications of certain failures of certain expectations. In the first episode of the article, Theorems 1.1 and 1.2 are regarding a natural surjection \[ \lambda_{X|D}: \mathrm{CH}_0 (X|D) \to H_0 ^{\mathrm{Sus}} (U),\] from the Chow group of \(0\)-cycles associated to the modulus pair \((X, D)\) to the \(0\)-th Suslin homology of \(U= X \setminus D\). The first of the theorems shows that when the base field \(k\) is algebraically closed and \(D\) is reduced, this map becomes an isomorphism for \(\mathrm{char} (k) > 0\). Rather interestingly this fails to be an isomorphism in general for \(\mathrm{char} (k) = 0\) case. The second theorem shows that when \(k\) is perfect and the components of \(D_{\mathrm{red}}\) are regular, the map \(\lambda_{X|D}\) becomes an isomorphism after inverting \(p\) for \(\mathrm{char} (k) = p>0\) case, while the map \(\lambda_{X|D}\) becomes an isomorphism mod \(m\) for each integer \(m \not = 0\) for \(\mathrm{char} (k) = 0\). The second episode of this omnibus paper is regarding the torsions of the Suslin homology and the étale cohomology with compact supports. Mor…
Mathematics Subject Classification ID: 14C25
Mathematics Subject Classification ID: 14F42
Mathematics Subject Classification ID: 19E15
zbMATH DE Number: 7468788
zbMATH Keywords: cycles with modulus
zbMATH Keywords: cycles on singular varieties
zbMATH Keywords: motivic cohomology
reviewed by: Jinhyun Park
MaRDI profile type: Publication
OpenAlex ID: W4205572446
arXiv ID: 2104.07968
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DOI: 10.5802/JEP.183
published in: Journal de l’École polytechnique — Mathématiques
description: scientific article