This is going to be a blog post on Filtrations and Gradings. We’re going to closely follow the development in Local Algebra by Serre.
A filtered ring is a ring with the set of ideals such that
,
, and
. An example would be
, where
is the ideal generated by
in
.
Similarly, a filtered module over a filtered ring
is defined as a module with a set of submodules
such that
,
, and
. Why not just have
? This is because multiplication between elements of a module may not be defined. An example would be the module generated by by the element
over
, where
.
Filtered modules form an additive category with morphisms
such that
. A trivial example is
, defined using the grading above, and the map being defined as
.
If is a submodule, then the induced filtration is defined as
. Is every
a submodule of
? Yes, because every
is by definition a submodule of
, and the intersection of two submodules (
and
in particular) is always a submodule. Simialrly, the quotient filtration
is also defined. As the quotient of two modules, the meaning of
is clear. However, what about the filtration of
? Turns out the filtration of
is defined the following way:
. We need to have
as the object under consideration because it is not necessary that
.
An important example of filtration is the -adic filtration. Let
be an ideal of
, and let the filtration of
be defined as
. Similarly, for a module
over
, the
-adic filtration of
is defined by
.
Now we shall discuss the topology defined by filtration. If is a filtered module over the filtered ring, then
form a basis for neighbourhoods around
. This obviously is a nested set of neighbourhoods, and surely enough the intersection of a finite number of neighbourhoods is also a neighbourhood, and so is the union of any set of neighbourhoods. Hence, the usual topological requirements for a basis is satisfied. But why
?
Proposition: Let be a submodule of a filtered module
. Then the closure of
of
is defined as
. How does this work? If one were to hand wave a bit, we are essentially finding the intersection of all neighbourhoods of
. Remember that each
is a neighbourhood of
. We’re translating each such neighbourhood by
, which is another way of saying we’re now considering all neighbourhoods of
. And then we find the intersection of all such neighbourhoods to find the smallest closed set containing
. There is an analogous concept in metric spaces- the intersection of all open sets containing
, for instance, is the closed set
. The analogy is not perfect, as the intersection of all neighbourhoods of
is
itself, which is not a closed set. But hey. We at least have something to go by.
Corollary: is Hausdorff if and only if
.