Formal Chow groups, \(p\)-divisible groups and syntomic cohomology.
type: Item
instance of: scholarly article
title: Formal Chow groups, \(p\)-divisible groups and syntomic cohomology.
zbMATH Open document ID: 1114.14017
DOI: 10.1215/S0012-7094-00-10228-1
author: Takao Yamazaki
published in: Duke Mathematical Journal
publication date: 2000-06-07T00:00:00Z
review text: In the paper under review, the author studies the infinitesimal behavior of the Chow groups \(\text{CH}^r(V)\) of a smooth projective variety \(V\) defined over an absolutely unramified complete discrete valuation field \(K\) of characteristic \(p> 0\) with perfect residue field \(k\). This topic is of particular interest in the context of higher algebraic \(K\)-theories and various \(p\)-adic cohomology theories, where both its significance and a first methodological approach to tackling the involved problems were pointed out by \textit{S. Bloch} in the early 1980s [in: Arithmetic and geometry, Pap. dedic. I. R. Shafarevich, Vol. I: Arithmetic, Prog. Math. 35, 13--26 (1983; Zbl 0584.14009)]. In the present work, using the theory of Dieudonné modules over rings of Witt vectors, the theory of \(p\)-divisible groups, \(K\)-cohomological functors, and the framework of the syntomic cohomology theory of Fontaine-Messing and of \textit{K. Kato} [Bull. Soc. Math. Fr. 119, 397--441 (1991; Zbl 0752.14015)], the author develops S. Bloch's approach towards the infinitesimal study of \(p\)-adic Chow groups essentially further, with several deep functorial comparison theorems as breaking new results. More precisely, denote by \({\mathcal K}^M_{r,V}\) the Zariski sheaf of Milnor \(K_r\)-groups on the variety \(V\). Then the so-called Bloch-Quillen formula establishes an isomorphism \[ \text{CH}^r(V)\otimes \mathbb{Z}\Biggl[{1\over (r- 1)!}\Biggr]\cong H^r(V,{\mathcal K}^M_{r,V})\otimes \mathbb{Z}\Biggl[{1\over (r- 1)!}\Biggr], \] which is the starting point ofe the author's refined investigations. Assuming that the variety \(V\) admits a smooth projective model over the valuation ring \(W\) of \(K\), that is, over the ring \(W= W(k)\) of Witt vectors of the residue field \(k\), the category \({\mathcal A}\) of Artinian local \(W\)-algebras with \(k\) as their residue field becomes an appropriate tool for studying the deformation behavior of \(V\) and its Milnor \(K_r\)-sheaves. N…
Mathematics Subject Classification ID: 14G20
zbMATH DE Number: 1436094
zbMATH Keywords: varieties over local fields
zbMATH Keywords: Chow groups
zbMATH Keywords: algebraic \(K\)-theory
zbMATH Keywords: \(p\)-divisible groups
zbMATH Keywords: syntomic cohomology
zbMATH Keywords: Dieudonné modules
zbMATH Keywords: rings of Witt vectors
zbMATH Keywords: Artinian algebras
reviewed by: Werner Kleinert
MaRDI profile type: Publication
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full work available at URL: s0012-7094-00-10228-1
OpenAlex ID: W2026693157
description: scientific article; zbMATH DE number 1436094