Bourbaki has a whole chapter in Commutative Algebra devoted to “graduations, filtrations, and topologies,” which indicates the importance of these concepts. That’s the theme for the next few posts I’ll do here, although I will (of course) be more concise.
In general, all rings will be commutative.
Gradings
The idea of a graded ring is necessary to define projective space.
Definition 1 A graded ring is ring
together with a decomposition
such that
. The set
is said to consist of homogeneous elements of degree
.
Note that is a subring (containing
) and
an
-algebra. Also, in many cases we actually have
, so the negative elements don’t matter, but to talk about localization, we want the greater generality.
The example to keep in mind here is the polynomial ring associated to any commutative ring
. Here the homogeneous elements of degree
are the monomials of degree
, or multiples of
.
More generally, consider the polynomial ring and let the homogeneous elements of degree
be the polynomials which are sums of monomials of degree
. In this way, homogeneous elements correspond to homogeneous polynomials. This is the example that leads to projective space.
Naturally, we can form a category of graded rings, but we need the appropriate morphisms:
Definition 2 Let
be graded rings. Then a homomorphism of graded rings is a ring-homomorphism
such that
for all
, in other words
preserves homogeneous elements and degrees.
To keep up with the theme from my previous posts, let’s do the standard test for when a graded ring is Noetherian:
Theorem 3 If is a graded ring with
for
, then
is Noetherian if and only if
is Noetherian and
is a finitely generaed
-algebra.
One direction is the Hilbert basis theorem. Conversely, suppose is Noetherian. First I claim
is Noetherian. Indeed, otherwise given ideals
, we consider
note that strict inclusion holds, by considering the components of degree (which are just the ideals
). This is a contradiction, so
is Noetherian.
Now we need to check is a finitely generated
-algebra. For this, consider the
-ideal
and choose generators . By splitting them into components, assume each
is homogeneous of degree
.
I claim that . To prove this, we will check inductively that
which will imply the claim. This is clearly true for . Assume it true for
, and let
. We can write
and by taking the -th homogeneous component, we may assume each
is actually homogeneous of degree
. Then by (complete) induction each
, so the same is true for
.
Now we can look at graded modules:
Definition 4 If
is a graded ring, a graded
-module is an
-module
together with a decomposition
such that
.
So in particular is a graded
-module. In the same vein, there is a category of graded
-modules with homomorphisms preserving the grading. This is all essentially a repetition of what was already said.
Filtrations
Filtrations are a more general concept. Basically, you don’t have a notion of “degree ,” but you instead have a notion of “degree
.”
Definition 5
A filtered ring is a ringtogether with subgroups
with
,
, and
![]()
For simplicity I am only looking at filtrations for nonnegative integers. It is as usual possible to define filtered modules (we want subgroups
with
and
) and homomorphisms preserving filtrations.
As an example, if is graded, we can let
This is a filtration. A more interesting example comes from the theory of Lie algebras. If is a Lie algebra over a field with
a basis, then the Poincaré-Birkhoff-Witt theorem (which I hope to discuss, eventually) states that products of the form
are a basis of the enveloping algebra . So we can filter this ring by letting
Nevertheless, this is a noncommutative ring in general, so we have slightly violated our conventions.
Conversely, we can get from a filtered ring a graded ring as follows:
Definition 6
Ifis a filtered ring, we define the associated graded ring
by
where
. To define the product of
with
, lift to
, and take the image
of
.If
is a filtered
-module, define the associated graded module
similarly.
It is easy to check the above definition is legitimate.
Similarly, we can define a descending filtration on a ring (resp. module) by reversing the inclusions: thus (resp.
). There is a similar definition for the associated graded ring
(resp. ). An important example of this is obtained as follows. If
is an ideal, then the filtration
is called the
-adic filtration.
As before, there is a category of filtered rings, and given a filtered ring , a category of filtered
-modules. [Edit: For some reason I forgot to add the next comment in the post at first. AM] Then
is a functor from filtered rings to graded rings, or filtered
-modules to graded
-modules.
As an aside, Anirudha Balasubramanian suggested during a talk (to a non-mathematical audience) a clever analogy: a descending filtration (in his example a lower central series) is like a Matryoshka doll, or a rock with many layers.
So, next will be some discussion of topologies, the Artin-Rees lemma and its applications, and completions. Today we finished the basics.
August 18, 2009 at 12:01 pm
In definition 5, it’s said that A_i are subsets. Then what does A_i/A_{i-1} mean in definition 6?
August 18, 2009 at 12:29 pm
The condition that
implies that
is a subring, and the condition that
implies that each
is a
-module. The quotient in definition 6 is the module quotient.
August 18, 2009 at 1:24 pm
Qiaochu was right. Thanks for the correction!
August 19, 2009 at 10:04 pm
[…] filtered rings, filtrations, I-adic filtration trackback Today I’ll continue the series on graded rings and filtrations by discussing the resulting topologies and the Artin-Rees […]