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.

### On the depth of the associated graded ring

#### 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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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11697/21082
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