Reduction numbers and multiplicities of multigraded structures. (English) Zbl 0931.13002
Let \(A\) be a local ring and let \(I_1,\dots,I_r\) be ideals in \(A\). The multi-Rees algebra with respect to \(I_1,\dots ,I_r\) is the ring \(R_A(I_1,\dots,I_r) = A[I_1t_1,\dots,I_rt_r]\) where \(t_1,\dots,t_r\) are indeterminates, and the multiform rings are the rings \(\text{gr}_A(I_1,\dots,I_r;I_i)= R_A(I_1,\dots,I_r)/I_iR_A(I_1,\dots,I_r)\), \(i=1,\dots,r\). We quote from the authors’ introduction:
The aim of this paper is twofold. First we want to relate the Cohen-Macaulay property of \(R_A(I_1,\dots,I_r)\) to the Cohen-Macaulay property of the ordinary Rees rings \(R_A(I_j)\) \((j = 1,\dots,r)\). Then we determine the multiplicities of multi-Rees and form rings. In particular, we ask when minimal multiplicity occurs.
The aim of this paper is twofold. First we want to relate the Cohen-Macaulay property of \(R_A(I_1,\dots,I_r)\) to the Cohen-Macaulay property of the ordinary Rees rings \(R_A(I_j)\) \((j = 1,\dots,r)\). Then we determine the multiplicities of multi-Rees and form rings. In particular, we ask when minimal multiplicity occurs.
Reviewer: U.Vetter (Oldenburg)
MSC:
| 13A30 | Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics |
| 13C14 | Cohen-Macaulay modules |
| 13H15 | Multiplicity theory and related topics |
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