Cohen–Macaulay Modules and Holonomic Modules Over Filtered Rings

Abstract

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.

Key Words:

• Cohen–Macaulay module
• Filtered ring
• Gorenstein dimension
• Grade
• Holonomic module

2000 Mathematics Subject Classification:

• 13C14
• 13D05
• 16E10
• 16E30
• 16E65
• 16W70

ACKNOWLEDGMENT

Research of the second author is supported by Grant-in-Aid for Scientific Researches C(2) in Japan.

Notes

Communicated by S. Goto.