Multilinear Algebra ---
Polynomial rings form a simple example of a graded algebra; such algebras occur
frequently and in Section 6.1 we define this concept. Another important example
is given by free algebras, which are discussed in Section 6.2, as well as the related
notions of tensor algebra and symmetric algebra on a K-module. A graded algebra
has an important invariant, its Hilbert series, essentially a power series whose coeffi-
cients indicate the dimensions of the components. In Section 6.3 we show that the
Hilbert series of a commutative Noetherian ring is a rational function and also prove
the Golod-Shafarevich theorem, giving a sufficient condition for a graded algebra to
be infinite-dimensional. The applications, to construct a finitely generated algebra
which is nil but not nilpotent, and a finitely generated infinite p-group, are sketched
in the exercises. Finally Section 6.4 deals with exterior algebras, providing a simple
derivation of determinants, and giving a brief geometrical application.
6.1 Graded Algebras
In a polynomial ring the degree of a polynomial is a useful concept which allows us to
analyse the ring in different ways. Thus we may write the polynomial ring as a direct
sum of its homogeneous components. There are many other rings which share this
property, and it is convenient to describe its general form before applying it.
Throughout, the coefficient ring K is an arbitrary commutative ring. The case
when there is no coefficient ring is included by taking K = Z, since every ring is a
Z-algebra. To avoid trivialities, we shall usually assume that K is non-trivial.
Definition. A graded K-algebra is a family (An) of K-modules indexed by Z, with a
bilinear mapping
(6.1.1)
denoted by (a, b) 1-+ ab, such that for any a E Am, bEAn> CE Ap ,
a(bc) = (ab)c (6.1.2)
and there is an element e E Ao such that ae = ea = a for all a E An (n E Z). An is
called the component of degree n; if a E An, we say that a is homogeneous of degree
n and write deg a = n.
165
,166 Multilinear Algebra
We shall assume that distinct An have only 0 in common, so that every non-zero
element has a unique degree, while 0 has all degrees. From the definition it is clear
that Ao is a K-algebra and each An is an Ao-bimodule.
Given a graded K-algebra (An), let us put A = EBnAn; we can define a multi-
plication on A, using (6.1.1) and the distributive laws. Thus if a = am + am + 1 + ...
+ aM and b = bn + bn + 1 + ... + bN , where ai, bi E Ai, then
It is clear that multiplication is associative, by (6.1.2), and distributive, with e as unit
element; thus A is a K-algebra. Sometimes A is called an internally graded algebra,
corresponding to the externally graded algebra (An). Of course we may be given
an algebra which is internally graded, i.e. A may have the form A = EBnAn> where
(An) is externally graded.
Example 1. Let A = K[XI, ... , Xd] be a polynomial ring in d indeterminates, and
denote by An the submodule of all homogeneous polynomials of total degree n;
then A is internally graded by the An. We observe that in this case An = 0 for
n < 0; this is expressed by saying that A is positively graded. In particular, for
d = 1 we have A = K[x]; here An = Kxn.
Example 2. The ring of 'Laurent polynomials' k[ t, t - I] is an example of a graded
ring with non-zero components for all n E Z. By contrast, the ring of all Laurent
series has no obvious grading (it has a filtration, see Further Exercise 4).
Example 3. Any ring R may be graded by putting Ao = R, An = 0 for n i= O. Such a
ring is said to be concentrated in degree O.
Let A be a graded ring. A graded A -module is a family of K- modules (Mn) such that
ArMs ~ Mr+s for all r, s E Z. We can again form an internally graded module by
writing M = EBMn; an element x of M is called homogeneous of degree n if x E Mn.
Every element of M can be uniquely expressed as a sum of finitely many homo-
geneous elements. We remark that for any graded ring A, a graded A-module M
which is finitely generated can always be generated by a finite set of homogeneous
elements; we need merely take all the components of the given set of generators.
The tensor product (over K) of two graded K-algebras is again a graded K-algebra
in an obvious sense: given A = (An), B = (Bn), we have A ® B = C, where
(6.1.3)
For example, the polynomial ring K[x, y] may be constructed as the tensor product
of K[x] and K[y]. We observe that when A and B are both positively graded, the sum
in (6.1.3) is necessarily finite, but this is not essential for the definition to make sense.
A graded algebra A is said to be anticommutative if
(6.1.4)
, 6.1 Graded Algebras 167
Thus to interchange a and b we change the sign if m and n are both odd and leave it
the same otherwise. For example, if An = 0 for all odd n, then (6.1.4) reduces to the
commutative law. Some writers omit the 'anti' and call (6.1.4) the commutative law
for graded algebras; although there is some justification for this practice, we shall not
follow it but refer to (6.1.4) as the anticommutative law. We remark that if A is anti-
commutative and K admits division by 2 (e.g. if K is a field of characteristic not 2),
then a2 = 0 for any a of odd degree; this is sometimes made part of the general
definition.
Let A = (An), B = (Bn) be two graded algebras. By a linear mapping of degree r
from A to B we understand a family of K-linear mappings fn : An --* Bn+r' A linear
mapping of degree 0 from A to B which is a ring homomorphism is called a homo-
morphism of graded algebras. Similarly we can define linear mappings of degree r
between graded A-modules.
The definition of sub algebra and ideal should be clear: if we regard A as internally
graded, a graded sub algebra B of A (also called a homogeneous subalgebra of A) is a
sub algebra B in the ordinary sense such that B = L (B n An); likewise a graded ideal
a of A is an ideal in the usual sense such that a = L (a n An). This condition can be
expressed by saying that for any finite family an E An (n E Z) we have L an E B (or
a) iff each an lies in B (or a). The reader should verify that the image and kernel of a
homomorphism of graded algebras are a graded sub algebra and ideal in this sense.
Conversely, given a graded algebra A and a graded ideal a in A, the quotient A/a
is again graded, and the natural homomorphism A --* Ala is a homomorphism of
graded algebras. Finally we observe that if X is a subset of A consisting of homo-
geneous elements, then the ideal a of A generated by X is graded, for each element
of a can be written as a sum of terms uxv, again in a, where x E X and u, v are
homogeneous.
In a positively graded ring A, the set A+ = L~ A is an ideal called the augmenta-
tion ideal; it is such that A = Ao EB A+. If Ao ~ K under the mapping a 1--* ae
(a E K), A is an augmented K-algebra in the sense of Section 5.1.
In the commutative case Noetherian graded rings have the following convenient
description.
Proposition 6.1.1. A positively graded commutative ring A is Noetherian if and only if
Ao is Noetherian and A is finitely generated as a ring over Ao.
Proof. If Ao is Noetherian and A is generated by XI, ... ,Xd over Ao, then it is a
homomorphic image of the polynomial ring AO[XI, ... ,xdl and hence is Noetherian
by the Hilbert basis theorem (see Section lOA below). Conversely, assume that A is
Noetherian; then so is Ao = AIA+ and A+ is finitely generated as an ideal. By an
earlier remark we may take a family of homogeneous generators XI, •.. ,Xd of A+,
of degrees rl, ... , rd say, and write C for the subring generated by the Xi over Ao.
To complete the proof we show that An S; C, by induction on n. Clearly Ao S; C;
if n > 0 and Ar S; C for r < n, take YEAn. We can write y as a linear combination
of the x's, say y = L aiXi, where ai is homogeneous of degree n - ri ::: O. Since
ri > 0, we can apply induction to conclude that ai E C and so An S; C. It follows
that A = C and so A is finitely generated as a ring over Ao. •
Polynomial rings form a simple example of a graded algebra; such algebras occur
frequently and in Section 6.1 we define this concept. Another important example
is given by free algebras, which are discussed in Section 6.2, as well as the related
notions of tensor algebra and symmetric algebra on a K-module. A graded algebra
has an important invariant, its Hilbert series, essentially a power series whose coeffi-
cients indicate the dimensions of the components. In Section 6.3 we show that the
Hilbert series of a commutative Noetherian ring is a rational function and also prove
the Golod-Shafarevich theorem, giving a sufficient condition for a graded algebra to
be infinite-dimensional. The applications, to construct a finitely generated algebra
which is nil but not nilpotent, and a finitely generated infinite p-group, are sketched
in the exercises. Finally Section 6.4 deals with exterior algebras, providing a simple
derivation of determinants, and giving a brief geometrical application.
6.1 Graded Algebras
In a polynomial ring the degree of a polynomial is a useful concept which allows us to
analyse the ring in different ways. Thus we may write the polynomial ring as a direct
sum of its homogeneous components. There are many other rings which share this
property, and it is convenient to describe its general form before applying it.
Throughout, the coefficient ring K is an arbitrary commutative ring. The case
when there is no coefficient ring is included by taking K = Z, since every ring is a
Z-algebra. To avoid trivialities, we shall usually assume that K is non-trivial.
Definition. A graded K-algebra is a family (An) of K-modules indexed by Z, with a
bilinear mapping
(6.1.1)
denoted by (a, b) 1-+ ab, such that for any a E Am, bEAn> CE Ap ,
a(bc) = (ab)c (6.1.2)
and there is an element e E Ao such that ae = ea = a for all a E An (n E Z). An is
called the component of degree n; if a E An, we say that a is homogeneous of degree
n and write deg a = n.
165
,166 Multilinear Algebra
We shall assume that distinct An have only 0 in common, so that every non-zero
element has a unique degree, while 0 has all degrees. From the definition it is clear
that Ao is a K-algebra and each An is an Ao-bimodule.
Given a graded K-algebra (An), let us put A = EBnAn; we can define a multi-
plication on A, using (6.1.1) and the distributive laws. Thus if a = am + am + 1 + ...
+ aM and b = bn + bn + 1 + ... + bN , where ai, bi E Ai, then
It is clear that multiplication is associative, by (6.1.2), and distributive, with e as unit
element; thus A is a K-algebra. Sometimes A is called an internally graded algebra,
corresponding to the externally graded algebra (An). Of course we may be given
an algebra which is internally graded, i.e. A may have the form A = EBnAn> where
(An) is externally graded.
Example 1. Let A = K[XI, ... , Xd] be a polynomial ring in d indeterminates, and
denote by An the submodule of all homogeneous polynomials of total degree n;
then A is internally graded by the An. We observe that in this case An = 0 for
n < 0; this is expressed by saying that A is positively graded. In particular, for
d = 1 we have A = K[x]; here An = Kxn.
Example 2. The ring of 'Laurent polynomials' k[ t, t - I] is an example of a graded
ring with non-zero components for all n E Z. By contrast, the ring of all Laurent
series has no obvious grading (it has a filtration, see Further Exercise 4).
Example 3. Any ring R may be graded by putting Ao = R, An = 0 for n i= O. Such a
ring is said to be concentrated in degree O.
Let A be a graded ring. A graded A -module is a family of K- modules (Mn) such that
ArMs ~ Mr+s for all r, s E Z. We can again form an internally graded module by
writing M = EBMn; an element x of M is called homogeneous of degree n if x E Mn.
Every element of M can be uniquely expressed as a sum of finitely many homo-
geneous elements. We remark that for any graded ring A, a graded A-module M
which is finitely generated can always be generated by a finite set of homogeneous
elements; we need merely take all the components of the given set of generators.
The tensor product (over K) of two graded K-algebras is again a graded K-algebra
in an obvious sense: given A = (An), B = (Bn), we have A ® B = C, where
(6.1.3)
For example, the polynomial ring K[x, y] may be constructed as the tensor product
of K[x] and K[y]. We observe that when A and B are both positively graded, the sum
in (6.1.3) is necessarily finite, but this is not essential for the definition to make sense.
A graded algebra A is said to be anticommutative if
(6.1.4)
, 6.1 Graded Algebras 167
Thus to interchange a and b we change the sign if m and n are both odd and leave it
the same otherwise. For example, if An = 0 for all odd n, then (6.1.4) reduces to the
commutative law. Some writers omit the 'anti' and call (6.1.4) the commutative law
for graded algebras; although there is some justification for this practice, we shall not
follow it but refer to (6.1.4) as the anticommutative law. We remark that if A is anti-
commutative and K admits division by 2 (e.g. if K is a field of characteristic not 2),
then a2 = 0 for any a of odd degree; this is sometimes made part of the general
definition.
Let A = (An), B = (Bn) be two graded algebras. By a linear mapping of degree r
from A to B we understand a family of K-linear mappings fn : An --* Bn+r' A linear
mapping of degree 0 from A to B which is a ring homomorphism is called a homo-
morphism of graded algebras. Similarly we can define linear mappings of degree r
between graded A-modules.
The definition of sub algebra and ideal should be clear: if we regard A as internally
graded, a graded sub algebra B of A (also called a homogeneous subalgebra of A) is a
sub algebra B in the ordinary sense such that B = L (B n An); likewise a graded ideal
a of A is an ideal in the usual sense such that a = L (a n An). This condition can be
expressed by saying that for any finite family an E An (n E Z) we have L an E B (or
a) iff each an lies in B (or a). The reader should verify that the image and kernel of a
homomorphism of graded algebras are a graded sub algebra and ideal in this sense.
Conversely, given a graded algebra A and a graded ideal a in A, the quotient A/a
is again graded, and the natural homomorphism A --* Ala is a homomorphism of
graded algebras. Finally we observe that if X is a subset of A consisting of homo-
geneous elements, then the ideal a of A generated by X is graded, for each element
of a can be written as a sum of terms uxv, again in a, where x E X and u, v are
homogeneous.
In a positively graded ring A, the set A+ = L~ A is an ideal called the augmenta-
tion ideal; it is such that A = Ao EB A+. If Ao ~ K under the mapping a 1--* ae
(a E K), A is an augmented K-algebra in the sense of Section 5.1.
In the commutative case Noetherian graded rings have the following convenient
description.
Proposition 6.1.1. A positively graded commutative ring A is Noetherian if and only if
Ao is Noetherian and A is finitely generated as a ring over Ao.
Proof. If Ao is Noetherian and A is generated by XI, ... ,Xd over Ao, then it is a
homomorphic image of the polynomial ring AO[XI, ... ,xdl and hence is Noetherian
by the Hilbert basis theorem (see Section lOA below). Conversely, assume that A is
Noetherian; then so is Ao = AIA+ and A+ is finitely generated as an ideal. By an
earlier remark we may take a family of homogeneous generators XI, •.. ,Xd of A+,
of degrees rl, ... , rd say, and write C for the subring generated by the Xi over Ao.
To complete the proof we show that An S; C, by induction on n. Clearly Ao S; C;
if n > 0 and Ar S; C for r < n, take YEAn. We can write y as a linear combination
of the x's, say y = L aiXi, where ai is homogeneous of degree n - ri ::: O. Since
ri > 0, we can apply induction to conclude that ai E C and so An S; C. It follows
that A = C and so A is finitely generated as a ring over Ao. •
