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The reduction number of an ideal and the local cohomology of the associated graded ring


Author: Thomas Marley
Journal: Proc. Amer. Math. Soc. 117 (1993), 335-341
MSC: Primary 13D45; Secondary 13D40, 13H15
DOI: https://doi.org/10.1090/S0002-9939-1993-1112496-7
MathSciNet review: 1112496
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Abstract: Let $ (R,m)$ be a local ring and $ I$ an $ m$-primary ideal. A result of Trung shows that if the local cohomology of $ g{r_I}(R)$ satisfies certain conditions, then the reduction number of $ I$ is independent of the minimal reduction chosen. These conditions consist of $ t = \dim R - \operatorname{grade} \;g{r_I}{(R)^ + }$ inequalities. We show that if $ R$ is Cohen-Macaulay, then one of these inequalities is always satisied, while another can often be easily checked. Applications are then given in two-dimensional Cohen-Macaulay rings. For instance, we show that if the Hilbert function of $ I$ equals the Hilbert polynomial of $ I$ for all integers greater than $ 1$, then the reduction number is independent of the choice of minimal reduction.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1993-1112496-7
Keywords: Local cohomology, reduction, Hilbert function
Article copyright: © Copyright 1993 American Mathematical Society

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