# Local Cohomology of Bigraded Rees Algebras and Normal Hilbert Coefficients

Source: [Local Cohomology of Bigraded Rees Algebras and Normal Hilbert Coefficients](<https://www.sota2.com/research/paper/68fa505c1e209131ce96a8d9>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T10:47:44.994Z

Authors: Shreedevi K. Masuti, J. K. Verma

Published: 2013-07-12T12:20:36.000Z

## Abstract

Let $(R,\\m)$ be an analytically unramified Cohen-Macaulay local ring of dimension 2 with infinite residue field and $\\ov{I}$ be the integral closure of an ideal $I$ in $R$. Necessary and sufficient conditions are given for $\\ov{I^{r+1}J^{s+1}}=a\\ov{I^rJ^{s+1}}+b\\ov{I^{r+1}J^s}$ to hold ${for all}r \\geq r\_0{and}s \\geq s\_0$ in terms of vanishing of $\[H^2\_{(at\_1,bt\_2)}(\\ov{\\mathcal{R}^\\prime}(I,J))\]\_{(r\_0,s\_0)}$, where $a \\in I,b \\in J$ is a good joint reduction of the filtration $\\{\\ov{I^rJ^s}\\}.$ This is used to derive a theorem due to Rees on normal joint reduction number zero. The vanishing of $\\ov{e}\_2(IJ)$ is shown to be equivalent to Cohen-Macaulayness of $\\ov{\\mathcal{R}}(I,J)$.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/1307.3431>)

[PDF](<https://arxiv.org/pdf/1307.3431.pdf>)