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arXiv · 0711.0057

Cohen-Macaulay modules and holonomic modules over filtered rings

Hiroki Miyahara·Kenji Nishida
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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$M\leq{G-dim gr}M$ and an equality ${\rm grade}M={\rm grade gr}M$, whenever Gorenstein dimension of ${\rm gr}M$ 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.

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Hiroki Miyahara, Kenji Nishida. 2007-11-01. Cohen-Macaulay modules and holonomic modules over filtered rings. https://arxiv.org/abs/0711.0057

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