Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Rees algebras and mixed multiplicities

Author: J. K. Verma
Journal: Proc. Amer. Math. Soc. 104 (1988), 1036-1044
MSC: Primary 13H15; Secondary 13H10
MathSciNet review: 929432
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ (R,m)$ be a local ring of positive dimension $ d$ and $ I$ and $ J$ two $ m$-primary ideals of $ R$. Let $ T$ denote the Rees algebra $ R[Jt]$ localized at the maximal homogeneous ideal $ (m,Jt)$. It is proved that

$\displaystyle e((I,Jt)T = {e_0}(I\vert J) + {e_1}(I\vert J) + \cdots + {e_{d - 1}}(I\vert J),$

where $ {e_i}(I\vert J),i = 0,1, \ldots ,d - 1$ are the first $ d$ mixed multiplicities of $ I$ and $ J$. A formula due to Huneke and Sally concerning the multiplicity of the Rees algebra (of a complete zero-dimensional ideal of a two dimensional regular local ring) at its maximal homogeneous ideal is recovered.

References [Enhancements On Off] (What's this?)

  • [B] P. B. Bhattacharya, The Hilbert function of two ideals, Proc. Cambridge Philos. Soc. 53 (1957), 568-575. MR 0089835 (19:727b)
  • [H] C. Huneke, Complete ideals in two dimensional regular local rings (after Zariski and Lipman), Proc. "Microprogram in Commutative Algebra" (MSRI, Berkeley, Calif., 1987) (to appear). MR 1015525 (90i:13020)
  • [HS] C. Huneke and J. Sally, Birational extensions in dimension two and integrally closed ideals, J. Algebra 115 (1988), 481-500. MR 943272 (89e:13025)
  • [L] J. Lipman, On complete ideals in regular local rings, preprint, 1987. MR 977761 (90g:14003)
  • [NR] D. G. Northcott and D. Rees, Reductions of ideals in local rings, Proc. Cambridge Philos. Soc. 50 (1954), 145-158. MR 0059889 (15:596a)
  • [R1] D. Rees, $ \mathcal{A}$ -transforms of local rings and a theorem on multiplicities of ideals, Proc. Cambridge Philos. Soc. 57 (1961), 8-17. MR 0118750 (22:9521)
  • [R2] -, Hilbert functions and pseudo-rational local rings of dimension two, J. London Math. Soc. 24 (1981), 467-479. MR 635878 (83d:13032)
  • [R3] -, The general extension of a local ring and mixed multiplicities, Algebra, Algebraic Topology and Their Interactions (Proc., Stockholm, 1983), Lecture Notes in Math., vol. 1183, Springer-Verlag, Berlin, 1986, pp. 339-360. MR 846458 (87m:13036)
  • [R4] -, Generalizations of reductions and mixed multiplicities, J. London Math. Soc. 29 (1984), 397-414. MR 754926 (86e:13023)
  • [Ra] H. Rademacher, Topics in analytic number theory, Springer-Verlag, New York, 1973. MR 0364103 (51:358)
  • [Ri] J. Riordan, Combinatorial identities, Wiley, New York, 1968. MR 0231725 (38:53)
  • [T] B. Teissier, Cycles évanescents, sections planes, et conditions de Whitney, Singularitiés à Cargèse, 1972, Astérisque 7-8 (1973), 285-362. MR 0374482 (51:10682)
  • [V] G. Valla, Certain graded algebras are always Cohen-Macaulay, J. Algebra 42 (1976), 537-548. MR 0422249 (54:10240)
  • [Z] O. Zariski, Polynomial ideals defined by infinitely near base points, Amer. J. Math. 60 (1938), 151-204. MR 1507308
  • [ZS] O. Zariski and P. Samuel, Commutative algebra, Vol. II, Van Nostrand, Princeton, N. J., 1960. MR 0120249 (22:11006)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 13H15, 13H10

Retrieve articles in all journals with MSC: 13H15, 13H10

Additional Information

Keywords: Bhattacharya polynomial, contracted ideal, integral closure of an ideal, joint reduction, mixed multiplicities, Rees algebra, regular local ring
Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society