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 PDF

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

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.

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