View source for Filtered module

<!--
f0401501.png
$#A+1 = 28 n = 0
$#C+1 = 28 : ~/encyclopedia/old_files/data/F040/F.0400150 Filtered module
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

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====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French)</TD></TR>
</table>

====Comments====
In the theory of rings of differential operators, e.g. Weyl algebras (cf. [[Weyl algebra|Weyl algebra]]), and also in connection with enveloping algebras of Lie algebras (cf. [[Lie algebra|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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.-E. Björk,   "Rings of differential operators" , North-Holland  (1979)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Schapira,   "Microdifferential systems in the complex domain" , Springer  (1985)</TD></TR></table>