Filtered module

A module $ M $ endowed with an increasing or decreasing filtration, that is, an increasing or decreasing family of submodules $ ( M _ {n} ) _ {n \in \mathbf Z } $. A filtration is called exhaustive if $ M = \cup _ {n \in \mathbf Z } M _ {n} $, and separable if $ \cap _ {n \in \mathbf Z } M _ {n} = 0 $. If $ N $ is a submodule of a filtered module $ M $, then filtrations are defined on $ N $ and $ M/N $ in a natural way. If $ M = \sum _ {n \in \mathbf Z } M _ {(n)} $ is a graded module, then the submodules $ M _ {n} = \sum _ {i \geq n } M _ {(i)} $ define an exhaustive separable decreasing filtration on $ M $. Conversely, with every filtered module $ M $ endowed, for example, with a decreasing filtration, there is associated the graded module

$$ \mathop{\rm gr} M = \ \oplus _ {n \in \mathbf Z } \mathop{\rm gr} _ {n} M, $$

where $ \mathop{\rm gr} _ {n} M = M _ {n} /M _ {n + 1 } $. A filtration $ ( M _ {n} ) _ {n \in \mathbf Z } $ constitutes a fundamental system of neighbourhoods of zero. Its topology is separable if and only if the filtration is separable, and discrete if and only if $ M _ {n} = 0 $ for some $ n $.

References

[1]	N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)

Comments

In the theory of rings of differential operators, e.g. Weyl algebras (cf. Weyl algebra), and also in connection with enveloping algebras of Lie algebras (cf. Lie algebra), filtered modules play an important part. The notion most encountered there is that of a good filtration: An $ R $- module $ M $ over a filtered ring $ R $ has good filtration if $ M _ {n} = \sum _ {i=1}^ {m} R _ {n - d _ {i} } m _ {i} $ for some set of elements $ m _ {i} $ of $ M $ and $ d _ {1} \dots d _ {m} \in \mathbf Z $. A very nice class of good filtered modules consists of the holonomic modules, defined by a condition related to a bound on the growth of the $ M _ {n} $, $ n \in \mathbf Z $. The associated graded module of a filtered module with good filtration is a finitely-generated graded module. If the ring $ R $ has a discrete filtration and the associated graded ring is left-Noetherian, then a good filtration on $ M $ induces a good filtration on any submodule, and any filtration equivalent to a good one is again a good filtration on the module.

References

[a1]	J.-E. Björk, "Rings of differential operators" , North-Holland (1979)
[a2]	P. Schapira, "Microdifferential systems in the complex domain" , Springer (1985)