G-Gorenstein Modules

Let R be a commutative Noetherian ring. In this paper, we study those finitely generated R-modules whose Cousin complexes provide Gorenstein injective resolutions. We call such a module a G-Gorenstein module. Characterizations of G-Gorenstein modules are given and a class of such modules is determined. It is shown that the class of G-Gorenstein modules strictly contains the class of Gorenstein modules. Also, we provide a Gorenstein injective resolution for a balanced big Cohen-Macaulay R-module. Finally, using the notion of a G-Gorenstein module, we obtain characterizations of Gorenstein and regular local rings.

Gorenstein Injective Resolutions, Cousin Complexes and Top Local Cohomology Modules

Let R be a Gorenstein local ring. We show that for a balanced big Cohen–Macaulay module M over R, the Cousin complex [Formula: see text] provides a Gorenstein injective resolution of M. Also, over a d-dimensional Gorenstein local ring R with maximal ideal 𝔪, we show that [Formula: see text], the dth local cohomology module of M with respect to 𝔪, is Gorenstein injective if (a) M is a balanced big Cohen–Macaulay R-module, or (b) M ∈ G(R), where G(R) is the Auslander's G-class of R.

COMPARISON OF MULTIGRADED AND UNGRADED COUSIN COMPLEXES

This paper generalizes, in two senses, work of Petzl and Sharp, who showed that, for a $\mathbb{Z}$-graded module $M$ over a $\mathbb{Z}$-graded commutative Noetherian ring $R$, the graded Cousin complex for $M$ introduced by Goto and Watanabe can be regarded as a subcomplex of the ordinary Cousin complex studied by Sharp, and that the resulting quotient complex is always exact. The generalizations considered in this paper are, firstly, to multigraded situations and, secondly, to Cousin complexes with respect to more general filtrations than the basic ones considered by Petzl and Sharp. New arguments are presented to provide a sufficient condition for the exactness of the quotient complex in this generality, as the arguments of Petzl and Sharp will not work for this situation without additional input.AMS 2000 Mathematics subject classification: Primary 13A02; 13E05; 13D25; 13D45

Comparison of graded and ungraded Cousin complexes

Let R = ⊕n∈zRn 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).