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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The diagonal subring
and the Cohen-Macaulay property
of a multigraded ring


Author: Eero Hyry
Journal: Trans. Amer. Math. Soc. 351 (1999), 2213-2232
MSC (1991): Primary 13A30; Secondary 14B15, 14M05
Published electronically: February 23, 1999
MathSciNet review: 1467469
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Abstract: Let $T$ be a multigraded ring defined over a local ring $(A,\mathfrak{m})$. This paper deals with the question how the Cohen-Macaulay property of $T$ is related to that of its diagonal subring $T^\Delta$. In the bigraded case we are able to give necessary and sufficient conditions for the Cohen-Macaulayness of $T$. If $I_1,\dotsc,I_r\subset A$ are ideals of positive height, we can then compare the Cohen-Macaulay property of the multi-Rees algebra $R_A(I_1,\dotsc,I_r)$ with the Cohen-Macaulay property of the usual Rees algebra $R_A(I_1\cdots I_r)$. We also obtain a bound for the joint reduction numbers of two $\mathfrak{m}$-primary ideals in the case the corresponding multi-Rees algebra is Cohen-Macaulay.


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Additional Information

Eero Hyry
Affiliation: National Defence College, Santahamina, FIN-00860 Helsinki, Finland
Email: eero.hyry@helsinki.fi

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02143-1
PII: S 0002-9947(99)02143-1
Received by editor(s): June 1, 1996
Published electronically: February 23, 1999
Article copyright: © Copyright 1999 American Mathematical Society