Comparison of graded and ungraded Cousin complexes

Abstract

Let R = ©,, 6Z R, be a Z-graded commutative Noetherian ring and let M be a Z-graded R-module. S. Goto and K.. Watanabe introduced 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 author. 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).

Comparison of Multigraded and Ungraded Cousin Complexes

“…It is natural to ask whether the analogues of the results of [6] hold in this G-graded situation. One of the aims of this paper is to show that they do; however, while the fact that *…”

“…• can be proved by straightforward modification of arguments in [6], we needed fresh ideas to prove, in this G-graded case, that the resulting quotient complex is always exact, as the argument in [6] does not deal with the case where rank G > 1.…”

“…The other way in which this paper generalizes work from [6] is that here we consider the Cousin complex C(F, M)…”

“…• is exact, straightforward generalization of the work in [6] is insufficient, on its own, to achieve our goals.…”

“…In [6], Petzl and Sharp compared the graded Cousin complex * C(M ) • (introduced by Goto and Watanabe in [4]) for a graded module M over a Z-graded commutative Noetherian ring (Z denotes the set of integers) with the ordinary Cousin complex C(M ) • (studied by Sharp in [7]) obtained by ignoring the gradings. The main results of [6] are that * C(M ) • can be viewed as a subcomplex of C(M ) • , and that the resulting quotient complex is always exact.…”