On Chow groups and intersection multiplicity of modules. II
type: Item
instance of: scholarly article
title: On Chow groups and intersection multiplicity of modules. II
zbMATH Open document ID: 0817.13013
published in: Journal of Algebra
publication date: 1995-08-01T00:00:00Z
review text: This is a continuation of the author's study of the Chow group problem and Serre's conjecture on intersection multiplicity for a complete ramified regular local ring \(R\) [see the author, J. Algebra 161, No. 1, 186-198 (1993; Zbl 0795.13011)]. Assume \(R = S[X_ n]/(f)\) with \(S = V[[X_ 1, \dots, X_{n-1}]]\) where \(V\) is a complete discrete valuation ring and \(f\) an Eisenstein polynomial of degree \(d\). The author proves that if one supposes that the conjectures hold for every Eisenstein extension of degree a proper divisor of \(d\), and if \(0 \neq P\) is a prime ideal of \(R\) such that \(R/P\) is normal but it is not the integral closure of \(S/P \cap S\) then: (1) \([P] = 0\) in the Chow group of \(R\), and (2) \(\chi (R/P, R/Q) > 0\) for every prime ideal \(Q\) of \(R\) such that \(l(R/P + Q) < \infty\) and \(\dim R/P + \dim R/Q = \dim R\). In a final section, the author gives alternative proofs for the vanishing of Chow groups of regular local rings essentially of finite type and smooth over a field (result due to Quillen) or over a discrete valuation ring (result due to Gillet and Levine).
Mathematics Subject Classification ID: 13H15
Mathematics Subject Classification ID: 13H05
Mathematics Subject Classification ID: 14C17
zbMATH DE Number: 735305
zbMATH Keywords: regular local ring \(R\)
zbMATH Keywords: Chow group
zbMATH Keywords: Serre's conjecture on intersection multiplicity
author: S. P. Dutta
reviewed by: Iustin Coandă
MaRDI profile type: Publication
full work available at URL: jabr.1995.1016
OpenAlex ID: W2055010480
DOI: 10.1006/JABR.1995.1016
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description: scientific article; zbMATH DE number 735305