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
GUERRIERI, ANNA
1995-01-01
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.File in questo prodotto:
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