2007/11/01 by Hiroki Miyahara, Miyahara, Hiroki, Kenji Nishida +1
Mathematics · #13C14 #13D05 #16E10 #16E30 #16E65 #16W70 #Commutative Algebra (math.AC) #FOS: Mathematics #Rings and Algebras (math.RA) #math.AC #math.RA #msc:13C14 #msc:13D05 #msc:16E10 #msc:16E30 #msc:16E65 #msc:16W70
paper · pdf · doi:10.48550/arxiv.0711.0057
21 pages, to appear in Communications in Algebra
arxiv created 2007/11/01 · arxiv updated 2009/12/01
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-dimM≤G-dim grM and an equality \rm gradeM=\rm grade grM, whenever Gorenstein dimension of \rm grM 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.