Illinois Journal of Mathematics

On Chow groups of complete regular local rings

Sichang Lee

Full-text: Open access

Abstract

In this paper, we establish the validity of the Chow group problem for complete regular local rings $R$ of dimension up to 4. For dimension $n$ ($>4$) over ramified regular local ring $R$, we have two results: (1) When $I$ is an ideal of height 3 such that $R/I$ is a Gorenstein ring, then $[I]=0$ in $A_{n-3}(R)$. (2) We reduce any prime ideal of height $i$ to an almost complete intersection ideal of height $i$ and in some special cases of almost complete intersection ideal of height $i$, we show that all Chow groups except the top one vanish. A necessary and sufficient condition for the vanishing of Chow groups is also derived using Eisenstein extension.

Article information

Source
Illinois J. Math., Volume 56, Number 4 (2012), 1085-1093.

Dates
First available in Project Euclid: 6 May 2014

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1399395823

Digital Object Identifier
doi:10.1215/ijm/1399395823

Mathematical Reviews number (MathSciNet)
MR3231474

Zentralblatt MATH identifier
1295.13033

Subjects
Primary: 13H05: Regular local rings 13D45: Local cohomology [See also 14B15]

Citation

Lee, Sichang. On Chow groups of complete regular local rings. Illinois J. Math. 56 (2012), no. 4, 1085--1093. doi:10.1215/ijm/1399395823. https://projecteuclid.org/euclid.ijm/1399395823


Export citation

References

  • D. A. Buchsbaum and D. Eisenbud, Lifting modules and a theorem on finite free resolutions, Ring theory (R. Gorden, ed.), Acdemic Press, New York, 1987.
  • L. Claborn and R. Fossum, Generalization of the notion of class group, Illinois J. Math. 12 (1968), 228–253.
  • S. P. Dutta, A note on Chow groups and intersection multiplicity of modules, J. Algebra 161 (1993), 186–198.
  • S. P. Dutta, On Chow groups and intersection multiplicity of modules II, J. Algebra 171 (1995), 370–382.
  • H. Gillet and M. Levine, The relative form of Gersten's conjecture over a discrete valuation ring: The smooth case, J. Pure Appl. Algebra 46 (1987), no. 1, 59–71.
  • I. Kaplansky, Commutative rings, Allyn and Bacon, Boston, 1970.
  • S. Lee, Chow groups of complete regular local rings III, Commun. Korean Math. Soc. 17 (2002), no. 2, 221–227.
  • D. Quillen, Higher algebraic $K$-theory. I, Algebraic $K$-theory, I–-Higher $K$-theories, Lecture Notes in Math., vol. 341, Springer-Verlag, Berlin, 1973, pp. 85–147.
  • W. Smoke, Perfect modules over Cohen–Macaulay local rings, J. Algebra 106 (1987), 367–375.