Rees algebras and mixed multiplicities

Verma, J. K.

doi:10.1090/S0002-9939-1988-0929432-4

Rees algebras and mixed multiplicities
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by J. K. Verma

Proc. Amer. Math. Soc. 104 (1988), 1036-1044

DOI: https://doi.org/10.1090/S0002-9939-1988-0929432-4

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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 \[ e((I,Jt)T = {e_0}(I|J) + {e_1}(I|J) + \cdots + {e_{d - 1}}(I|J),\] where ${e_i}(I|J),i = 0,1, \ldots ,d - 1$ 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

P. B. Bhattacharya, The Hilbert function of two ideals, Proc. Cambridge Philos. Soc. 53 (1957), 568–575. MR 89835
Craig Huneke, Complete ideals in two-dimensional regular local rings, Commutative algebra (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 15, Springer, New York, 1989, pp. 325–338. MR 1015525, DOI 10.1007/978-1-4612-3660-3_{1}6
Craig Huneke and Judith D. Sally, Birational extensions in dimension two and integrally closed ideals, J. Algebra 115 (1988), no. 2, 481–500. MR 943272, DOI 10.1016/0021-8693(88)90274-8
Joseph Lipman, On complete ideals in regular local rings, Algebraic geometry and commutative algebra, Vol. I, Kinokuniya, Tokyo, 1988, pp. 203–231. MR 977761
D. G. Northcott and D. Rees, Reductions of ideals in local rings, Proc. Cambridge Philos. Soc. 50 (1954), 145–158. MR 59889, DOI 10.1017/s0305004100029194
D. Rees, ${\mathfrak {a}}$-transforms of local rings and a theorem on multiplicities of ideals, Proc. Cambridge Philos. Soc. 57 (1961), 8–17. MR 118750, DOI 10.1017/s0305004100034800
D. Rees, Hilbert functions and pseudorational local rings of dimension two, J. London Math. Soc. (2) 24 (1981), no. 3, 467–479. MR 635878, DOI 10.1112/jlms/s2-24.3.467
D. Rees, The general extension of a local ring and mixed multiplicities, Algebra, algebraic topology and their interactions (Stockholm, 1983) Lecture Notes in Math., vol. 1183, Springer, Berlin, 1986, pp. 339–360. MR 846458, DOI 10.1007/BFb0075469
D. Rees, Generalizations of reductions and mixed multiplicities, J. London Math. Soc. (2) 29 (1984), no. 3, 397–414. MR 754926, DOI 10.1112/jlms/s2-29.3.397
Hans Rademacher, Topics in analytic number theory, Die Grundlehren der mathematischen Wissenschaften, Band 169, Springer-Verlag, New York-Heidelberg, 1973. Edited by E. Grosswald, J. Lehner and M. Newman. MR 0364103
John Riordan, Combinatorial identities, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0231725
Bernard Teissier, Cycles évanescents, sections planes et conditions de Whitney, Singularités à Cargèse (Rencontre Singularités Géom. Anal., Inst. Études Sci., Cargèse, 1972) Astérisque, Nos. 7 et 8, Soc. Math. France, Paris, 1973, pp. 285–362 (French). MR 0374482
G. Valla, Certain graded algebras are always Cohen-Macaulay, J. Algebra 42 (1976), no. 2, 537–548. MR 422249, DOI 10.1016/0021-8693(76)90112-5
Oscar Zariski, Polynomial Ideals Defined by Infinitely Near Base Points, Amer. J. Math. 60 (1938), no. 1, 151–204. MR 1507308, DOI 10.2307/2371550
Oscar Zariski and Pierre Samuel, Commutative algebra. Vol. II, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0120249