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

Abstract

Let T T be a multigraded ring defined over a local ring ( A , m ) (A,\mathfrak {m}) . This paper deals with the question how the Cohen-Macaulay property of T T is related to that of its diagonal subring T Δ T^\Delta . In the bigraded case we are able to give necessary and sufficient conditions for the Cohen-Macaulayness of T T . If I 1 , … , I r ⊂ A 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 , … , I r ) R_A(I_1,\dotsc ,I_r) with the Cohen-Macaulay property of the usual Rees algebra R A ( I 1 ⋯ I r ) R_A(I_1\cdots I_r) . We also obtain a bound for the joint reduction numbers of two m \mathfrak {m} -primary ideals in the case the corresponding multi-Rees algebra is Cohen-Macaulay.