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

Hyry, Eero

doi:10.1090/S0002-9947-99-02143-1

The diagonal subring and the Cohen-Macaulay property of a multigraded ring
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Trans. Amer. Math. Soc. 351 (1999), 2213-2232 Request permission

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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