Comparison of multigraded and ungraded Cousin complexes

This paper generalizes, in two senses, work of Petzl and Sharp, who showed that, for a Z-graded module M over a Z-graded commutative Noetherian ring R , the graded Cousin complex for M introduced by Goto and Watanabe can be re garded as a subcomplex of the ordinary Cousin complex studied by Sharp, and that the resulting quotient complex is always exact. The generalizations c onsidered in this paper are, firstly, to multigraded situations and, second ly, to Cousin complexes with respect to more general filtrations than the b asic ones considered by Petzl and Sharp. New arguments are presented to pro vide a sufficient condition for the exactness of the quotient complex in th is generality, as the arguments of Petal and Sharp will not work for this s ituation without additional input.