On Chow groups and intersection multiplicity of modules. II
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).
Serre's conjecture on intersection multiplicity
scientific article; zbMATH DE number 735305