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 assosiated 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$M\le G-dimgrM$ and an equality $\mathrm{grade}M=\mathrm{gradegr}M$, whenever Gorenstein dimension of $\mathrm{gr}M$ 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.

21 pages, to appear in Communications in Algebra