2Noetherian algebras
III Algebras
2.1 Noetherian algebras
In the introduction, we met the definition of Noetherian algebras.
Definition
(Noetherian algebra)
.
An algebra is left Noetherian if it satisfies
the ascending chain condition (ACC ) on left ideals, i.e. if
I
1
≤ I
2
≤ I
3
≤ · · ·
is an ascending chain of left ideals, then there is some
N
such that
I
N+m
=
I
N
for all m ≥ 0.
Similarly, we say an algebra is Noetherian if it is both left and right Noethe-
rian.
We’ve also met a few examples. Here we are going to meet lots more. In
fact, most of this first section is about establishing tools to show that certain
algebras are Noetherian.
One source of Noetherian algebras is via constructing polynomial and power
series rings. Recall that in IB Groups, Rings and Modules, we proved the Hilbert
basis theorem:
Theorem
(Hilbert basis theorem)
.
If
A
is Noetherian, then
A
[
X
] is Noetherian.
Note that our proof did not depend on
A
being commutative. The same
proof works for non-commutative rings. In particular, this tells us
k
[
X
1
, · · · , X
n
]
is Noetherian.
It is also true that power series rings of Noetherian algebras are also Noethe-
rian. The proof is very similar, but for completeness, we will spell it out
completely.
Theorem. Let A be left Noetherian. Then A[[X]] is Noetherian.
Proof.
Let
I
be a left ideal of
A
[[
X
]]. We’ll show that if
A
is left Noetherian,
then I is finitely generated. Let
J
r
= {a : there exists an element of I of the form aX
r
+ higher degree terms}.
We note that J
r
is a left ideal of A, and also note that
J
0
≤ J
1
≤ J
2
≤ J
3
≤ · · · ,
as we can always multiply by
X
. Since
A
is left Noetherian, this chain terminates
at
J
N
for some
N
. Also,
J
0
, J
1
, J
2
, · · · , J
N
are all finitely generated left ideals.
We suppose
a
i1
, · · · , a
is
i
generates
J
i
for
i
= 1
, · · · , N
. These correspond to
elements
f
ij
(X) = a
ij
X
j
+ higher odder terms ∈ I.
We show that this finite collection generates
I
as a left ideal. Take
f
(
X
)
∈ I
,
and suppose it looks like
b
n
X
n
+ higher terms,
with b
n
6= 0.
Suppose n < N . Then b
n
∈ J
n
, and so we can write
b
n
=
X
c
nj
a
nj
.
So
f(X) −
X
c
nj
f
nj
(X) ∈ I
has zero coefficient for X
n
, and all other terms are of higher degree.
Repeating the process, we may thus wlog
n ≥ N
. We get
f
(
X
) of the form
d
N
X
N
+ higher degree terms. The same process gives
f(X) −
X
c
Nj
f
Nj
(X)
with terms of degree
N
+ 1 or higher. We can repeat this yet again, using the
fact J
N
= J
N+1
, so we obtain
f(X) −
X
c
Nj
f
Nj
(x) −
X
d
N+1,j
Xf
Nj
(X) + · · · .
So we find
f(X) =
X
e
j
(X)f
Nj
(X)
for some
e
j
(
X
). So
f
is in the left ideal generated by our list, and hence so is
f.
Example.
It is straightforward to see that quotients of Noetherian algebras are
Noetherian. Thus, algebra images of the algebras
A
[
x
] and
A
[[
x
]] would also be
Noetherian.
For example, finitely-generated commutative
k
-algebras are always Noethe-
rian. Indeed, if we have a generating set
x
i
of
A
as a
k
-algebra, then there is an
algebra homomorphism
k[X
1
, · · · , X
n
] A
X
i
x
i
We also saw previously that
Example. Any Artinian algebra is Noetherian.
The next two examples we are going to see are less obviously Noetherian,
and proving that they are Noetherian takes some work.
Definition
(
n
th Weyl algebra)
.
The
n
th Weyl algebra
A
n
(
k
) is the algebra
generated by X
1
, · · · , X
n
, Y
1
, · · · , Y
n
with relations
Y
i
X
i
− X
i
Y
i
= 1,
for all i, and everything else commutes.
This algebra acts on the polynomial algebra
k
[
X
1
, · · · , X
n
] with
X
i
acting
by left multiplication and
Y
i
=
∂
∂X
i
. Thus
k
[
X
1
, · · · , X
n
] is a left
A
n
(
k
) module.
This is the prototype for thinking about differential algebras, and
D
-modules in
general (which we will not talk about).
The other example we have is the universal enveloping algebra of a Lie
algebra.
Definition
(Universal enveloping algebra)
.
Let
g
be a Lie algebra over
k
, and
take a
k
-vector space basis
x
1
, · · · , x
n
. We form an associative algebra with
generators x
1
, · · · , x
n
with relations
x
i
x
j
− x
j
x
i
= [x
i
, x
j
],
and this is the universal enveloping algebra U(g).
Example.
If
g
is abelian, i.e. [
x
i
, x
j
] = 0 in
g
, then the enveloping algebra is
the polynomial algebra in x
1
, · · · , x
n
.
Example. If g = sl
2
(k), then we have a basis
0 1
0 0
,
0 0
1 0
,
1 0
0 −1
.
They satisfy
[e, f ] = h, [h, e] = 2e, [h, f ] = −2f,
To prove that
A
n
(
k
) and
U
(
g
) are Noetherian, we need some machinery, that
involves some “deformation theory”. The main strategy is to make use of a
natural filtration of the algebra.
Definition
(Filtered algebra)
.
A (
Z
-)filtered algebra
A
is a collection of
k
-vector
spaces
· · · ≤ A
−1
≤ A
0
≤ A
1
≤ A
2
≤ · · ·
such that A
i
· A
j
⊆ A
i+j
for all i, j ∈ Z, and 1 ∈ A
0
.
For example a polynomial ring is naturally filtered by the degree of the
polynomial.
The definition above was rather general, and often, we prefer to talk about
more well-behaved filtrations.
Definition (Exhaustive filtration). A filtration is exhaustive if
S
A
i
= A.
Definition (Separated filtration). A filtration is separated if
T
A
i
= {0}.
Unless otherwise specified, our filtrations are exhaustive and separated.
For the moment, we will mostly be interested in positive filtrations.
Definition (Positive filtration). A filtration is positive if A
i
= 0 for i < 0.
Our canonical source of filtrations is the following construction:
Example. If A is an algebra generated by x
1
, · · · , x
n
, say, we can set
– A
0
is the k-span of 1
– A
1
is the k-span of 1, x
1
, · · · , x
n
– A
1
is the k-span of 1, x
1
, · · · , x
n
, x
i
x
j
for i, j ∈ {1, · · · , n}.
In general,
A
r
is elements that are of the form of a (non-commutative) polynomial
expression of degree ≤ r.
Of course, the filtration depends on the choice of the generating set.
Often, to understand a filtered algebra, we consider a nicer object, known as
the associated graded algebra.
Definition
(Associated graded algebra)
.
Given a filtration of
A
, the associated
graded algebra is the vector space direct sum
gr A =
M
A
i
A
i−1
.
This is given the structure of an algebra by defining multiplication by
(a + A
i−1
)(b + A
j−1
) = ab + A
i+j−1
∈
A
i+j
A
i+j−1
.
In our example of a finitely-generated algebra, the graded algebra is generated
by x
1
+ A
0
, · · · , x
n
+ A
0
∈ A
1
/A
0
.
The associated graded algebra has the natural structure of a graded algebra:
Definition
(Graded algebra)
.
A (
Z
-)graded algebra is an algebra
B
that is of
the form
B =
M
i∈Z
B
i
,
where
B
i
are
k
-subspaces, and
B
i
B
j
⊆ B
i+j
. The
B
i
’s are called the homogeneous
components.
A graded ideal is an ideal of the form
M
J
i
,
where J
i
is a subspace of B
i
, and similarly for left and right ideals.
There is an intermediate object between a filtered algebra and its associated
graded algebra, known as the Rees algebra.
Definition
(Rees algebra)
.
Let
A
be a filtered algebra with filtration
{A
i
}
. Then
the Rees algebra
Rees
(
A
) is the subalgebra
L
A
i
T
i
of the Laurent polynomial
algebra A[T, T
−1
] (where T commutes with A).
Since 1 ∈ A
0
⊆ A
1
, we know T ∈ Rees(A). The key observation is that
– Rees(A)/(T )
∼
=
gr A.
– Rees(A)/(1 − T )
∼
=
A.
Since
A
n
(
k
) and
U
(
g
) are finitely-generated algebras, they come with a
natural filtering induced by the generating set. It turns out, in both cases, the
associated graded algebras are pretty simple.
Example.
Let
A
=
A
n
(
k
), with generating set
X
1
, · · · , X
n
and
Y
1
, · · · , Y
n
. We
take the filtration as for a finitely-generated algebra. Now observe that if
a
i
∈ A
i
,
and a
j
∈ A
j
, then
a
i
a
j
− a
j
a
i
∈ A
i+j−2
.
So we see that gr A is commutative, and in fact
gr A
n
(k)
∼
=
k[
¯
X
1
, · · · ,
¯
X
n
,
¯
Y
1
, · · · ,
¯
Y
n
],
where
¯
X
i
,
¯
Y
i
are the images of
X
i
and
Y
i
in
A
1
/A
0
respectively. This is not
hard to prove, but is rather messy. It requires a careful induction.
Example.
Let
g
be a Lie algebra, and consider
A
=
U
(
g
). This has generating
set
x
1
, · · · , x
n
, which is a vector space basis for
g
. Again using the filtration for
finitely-generated algebras, we get that if a
i
∈ A
i
and a
j
∈ A
j
, then
a
i
a
j
− a
j
a
i
∈ A
i+j−1
.
So again gr A is commutative. In fact, we have
gr A
∼
=
k[¯x
1
, · · · , ¯x
n
].
The fact that this is a polynomial algebra amounts to the same as the Poincar´e-
Birkhoff-Witt theorem. This gives a k-vector space basis for U(g).
In both cases, we find that
gr A
are finitely-generated and commutative, and
therefore Noetherian. We want to use this fact to deduce something about
A
itself.
Lemma.
Let
A
be a positively filtered algebra. If
gr A
is Noetherian, then
A
is
left Noetherian.
By duality, we know that A is also right Noetherian.
Proof. Given a left ideal I of A, we can form
gr I =
M
I ∩ A
i
I ∩ A
i−1
,
where I is filtered by {I ∩ A
i
}. By the isomorphism theorem, we know
I ∩ A
i
I ∩ A
i−1
∼
=
I ∩ A
i
+ A
i−1
A
i−1
⊆
A
i
A
i−1
.
Then gr I is a left graded ideal of gr A.
Now suppose we have a strictly ascending chain
I
1
< I
2
< · · ·
of left ideals. Since we have a positive filtration, for some
A
i
, we have
I
1
∩ A
i
(
I
2
∩ A
i
and I
1
∩ A
i−1
= I
2
∩ A
i−1
. Thus
gr I
1
( gr I
2
( gr I
3
( · · · .
This is a contradiction since
gr A
is Noetherian. So
A
must be Noetherian.
Where we need positivity is the existence of that transition from equality to
non-equality. If we have a
Z
-filtered algebra instead, then we need to impose
some completeness assumption, but we will not go into that.
Corollary. A
n
(k) and U(g) are left/right Noetherian.
Proof. gr A
n
(
k
) and
gr U
(
g
) are commutative and finitely generated algebras.
Note that there is an alternative filtration for
A
n
(
k
) yielding a commutative
associated graded algebra, by setting A
0
= k[X
1
, · · · , X
n
] and
A
1
= k[X
1
, · · · , X
n
] +
n
X
j=1
k[X
1
, · · · , X
n
]Y
j
,
i.e. linear terms in the
Y
, and then keep on going. Essentially, we are filtering on
the degrees of the
Y
i
only. This also gives a polynomial algebra as an associative
graded algebra. The main difference is that when we take the commutator,
we don’t go down by two degrees, but one only. Later, we will see this is
advantageous when we want to get a Poisson bracket on the associated graded
algebra.
We can look at further examples of Noetherian algebras.
Example.
The quantum plane
k
q
[
X, Y
] has generators
X
and
Y
, with relation
XY = qY X
for some
q ∈ k
×
. This thing behaves differently depending on whether
q
is a
root of unity or not.
This quantum plane first appeared in mathematical physics.
Example.
The quantum torus
k
q
[
X, X
−1
, Y, Y
−1
] has generators
X
,
X
−1
,
Y
,
Y
−1
with relations
XX
−1
= Y Y
−1
= 1, XY = qY X.
The word “quantum” in this context is usually thrown around a lot, and
doesn’t really mean much apart from non-commutativity, and there is very little
connection with actual physics.
These algebras are both left and right Noetherian. We cannot prove these
by filtering, as we just did. We will need a version of Hilbert’s basis theorem
which allows twisting of the coefficients. This is left as an exercise on the second
example sheet.
In the examples of
A
n
(
k
) and
U
(
g
), the associated graded algebras are
commutative. However, it turns out we can still capture the non-commutativity
of the original algebra by some extra structure on the associated graded algebra.
So suppose
A
is a (positively) filtered algebra whose associated graded algebra
gr A
is commutative. Recall that the filtration has a corresponding Rees algebra,
and we saw that Rees A/(T )
∼
=
gr A. Since gr A is commutative, this means
[Rees A, Rees A] ⊆ (T ).
This induces a map
Rees
(
A
)
× Rees
(
A
)
→
(
T
)
/
(
T
2
), sending (
r, s
)
7→ T
2
+ [
r, s
].
Quotienting out by (T ) on the left, this gives a map
gr A × gr A →
(T )
(T
2
)
.
We can in fact identify the right hand side with gr A as well. Indeed, the map
gr A
∼
=
Rees(A)
(T )
(T )
(T
2
)
mult. by T
,
is an isomorphism of gr A
∼
=
Rees A/(T )-modules. We then have a bracket
{ · , · } : gr A × gr A gr A
(¯r, ¯s) {r, s}
.
Note that in our original filtration of the Weyl algebra
A
n
(
k
), since the commu-
tator brings us down by two degrees, this bracket vanishes identically, but using
the alternative filtration does not give a non-zero { · , · }.
This { · , · } is an example of a Poisson bracket.
Definition
(Poisson algebra)
.
An associative algebra
B
is a Poisson algebra if
there is a k-bilinear bracket { · , · } : B × B → B such that
– B is a Lie algebra under { · , · }, i.e.
{r, s} = −{s, r}
and
{{r, s}, t} + {{s, t}, r} + {{t, r}, s} = 0.
– We have the Leibnitz rule
{r, st} = s{r, t} + {r, s}t.
The second condition says {r, · } : B → B is a derivation.