Cohen–Macaulay Modules and Holonomic Modules Over Filtered Rings

We study Gorenstein dimension and grade of a module M over a filtered ring whose associated graded ring is a commutative Noetherian ring. An equality or an inequality between these invariants of a filtered module and its associated graded module is the most valuable property for an investigation of filtered rings. We prove an inequality G?dim M ≤ G?dim gr M and an equality grade M = grade gr M, whenever Gorenstein dimension of gr M is finite (Theorems 2.3 and 2.8). We would say that the use of G-dimension adds a new viewpoint for studying filtered rings and modules. We apply these results to a filtered ring with a Cohen–Macaulay or Gorenstein associated graded ring and study a Cohen–Macaulay, perfect, or holonomic module.     

On the Existence of Certain Modules of Finite Gorenstein Homological Dimensions

Let (R, 𝔪) be a commutative Noetherian local ring. It is known that R is Cohen–Macaulay if there exists either a nonzero finitely generated R-module of finite injective dimension or a nonzero Cohen–Macaulay R-module of finite projective dimension. In this article, we investigate the Gorenstein analogues of these facts.     

GRADE AND GORENSTEIN DIMENSION

Let R be a commutative Noetherian ring and let M be a finite (that is, finitely generated) R-module. The notion grade of M, grade M, has been introduced by Rees as the least integer t ≥ 0 such that Ext ^t _R (M,R) ≠ 0, see [11] Rees, D. 1957. The Grade of an Ideal or Module. Proc. Camb. Phil. Soc., 53: 28–42. [Crossref] , [Google Scholar]. The Gorenstein dimension of M, G-dim M, has been introduced by Auslander as the largest integer t ≥ 0 such that Ext ^t _R (M, R) ≠ 0, see [3] Auslander, M. 1967. Anneaux De Gorenstein Et Torsion En Algebre Commutative Edited by: Mangeney, M., Peskine, C. and Szpiro, L. Paris: Ecole Normale Superieure de Jeunes Filles.  [Google Scholar]. In this paper the R-module M is called G-perfect if grade M = G-dim M. It is a generalization of perfect module. We prove several results for the new concept similar to the classical results.