A bound on the reduction number of a primary ideal

Rossi, M.

doi:10.1090/S0002-9939-99-05393-9

A bound on the reduction number of a primary ideal
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Proc. Amer. Math. Soc. 128 (2000), 1325-1332 Request permission

Abstract:

Let $( A,\mathcal {M})$ be a local ring of positive dimension $d$ and let $I$ be an $\mathcal {M}$-primary ideal. We denote the reduction number of $I$ by $r(I)$, which is the smallest integer $r$ such that $I^{r+1}=JI^r$ for some reduction $J$ of $I.$ In this paper we give an upper bound on $r(I)$ in terms of numerical invariants which are related with the Hilbert coefficients of $I$ when $A$ is Cohen-Macaulay. If $d=1$, it is known that $r(I) \le e(I) -1$ where $e(I)$ denotes the multiplicity of $I.$ If $d \le 2,$ in Corollary 1.5 we prove $r(I) \le e_1(I) - e(I) + \lambda (A/I) + 1$ where $e_1(I)$ is the first Hilbert coefficient of $I.$ From this bound several results follow. Theorem 1.3 gives an upper bound on $r(I)$ in a more general setting.

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