On the depth of the associated graded ring

Guerrieri, Anna

doi:10.1090/S0002-9939-1995-1211580-9

On the depth of the associated graded ring
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Proc. Amer. Math. Soc. 123 (1995), 11-20 Request permission

Abstract:

Let (R, m) be a Cohen-Macaulay local ring of positive dimension d, let I be an $m -$ primary ideal of R. In this paper we individuate some conditions on I that allow us to determine a lower bound for depth ${\text {gr}_I}(R)$. It is proved that if $J \subseteq I$ is a minimal reduction of I such that $\lambda ({I^2} \cap J/IJ) = 2$ and ${I^n} \cap J = {I^{n - 1}}J$ for all $n \geq 3$, then depth ${\text {gr}_I}(R) \geq d - 2$; let us remark that $\lambda$ denotes the length function.

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