Note on multiplicity

Katz, Daniel

doi:10.1090/S0002-9939-1988-0929434-8

Note on multiplicity
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by Daniel Katz

Proc. Amer. Math. Soc. 104 (1988), 1021-1026

DOI: https://doi.org/10.1090/S0002-9939-1988-0929434-8

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Abstract:

Let $(R,M)$ be a local ring with infinite residue field and $I = ({x_1}, \ldots ,{x_d})R$ an ideal generated by a system of parameters. It is shown that the multiplicity of $I$ equals the multiplicity of $IT$ where \[ T = \tilde R{[{x_1}/{x_d}, \ldots ,{x_{d - 1}}/{x_d}]_{M\tilde R[{x_1}/{x_d}, \ldots ,{x_{d - 1}}/{x_d}]}}\] and $\tilde R = R/(0:x_d^N),N$ large.

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