Zariskian Filtrations

Cohen–Macaulay Modules and Holonomic Modules Over Filtered Rings

“…Then injective dimension of Λ is finite, say d, so that we can define a holonomic module. A filtered module M with a good filtration is holonomic, if gradeM = d. We generalize some results in [14], Chapter III, §4 and give a characterization of a holonomic module M by a property of Min(grM). An example of a filtered (non-regular) Gorenstein ring is given in 3.8.…”

“…As for filtered rings and module, the reader is referred to [14] or [20]. We only state here some definitions and facts.…”

“…Then the following three conditions are equivalent ( [14], Chapter I, 5.2 and [20], Chapter D, IV.3) (a) M has a good filtration.…”

“…[4], [17] etc.). The use of an invariant 'grade' is a core of the theory for Auslander regular or Gorenstein filtered rings ( [4], [5], [6], [7], [14]). In particular, its invariance under forming associated graded modules is essential.…”

“…Assume further that grΛ is a * local ring with the condition (P) (see Appendix), then 'Auslander-Bridger formula' holds for a filtered module M such that grM has finite Gdimension and G-dimM = G-dim grM: G-dimM + * depth grM = * depth grΛ (Proposition 2.6). To handle grade in the literatures, a kind of 'finitary' condition over a ring such as 'regularity' or 'Gorensteiness' is setted ( [7], §5 and [14], Chapter III, §2, 2.5). We find out that only the finiteness of G-dimension of grM implies gradeM = grade grM for a filtered module with a good filtration (Theorem 2.8).…”