COMPARISON OF GRADED AND UNGRADED COUSIN COMPLEXES

Let R = +R-n is an element of Z(n) be a Z-graded commutative Noetheria n ring and let M be a Z-graded R-module. S. Goto and K. Watanabe intro duced the graded Cousin complex C(M)(.) for M, a complex of graded R- modules. Also one can ignore the grading on M and construct the Cousin complex C(M)(.) For M, discussed in earlier papers by the second auth or. The main results in this paper are that C(M)(.) can be considered as a subcomplex of C(M)(.) and that the resulting quotient complex is always exact. This sheds new light on the known facts that, when M is non-zero and finitely generated, C(M)(.) is exact if and only if C(M )(.) is (and this is the case precisely when M is Cohen-Macaulay).