# Unstable Cohen--Macaulay Algebras
Unstable Cohen--Macaulay Algebras
Unstable Cohen--Macaulay Algebras
Unstable Cohen--Macaulay Algebras
Unstable Cohen--Macaulay Algebras
Mara D. Neusel
Mara D. Neusel
Mara D. Neusel
Mara D. Neusel
Mara D. Neusel
UNIVERSITY OF NOTRE DAME
DEPARTMENT OF MATHEMATICS
370 CCMB
NOTRE DAME, IN 465565683
USA
EMAIL:NEUSEL.1@ND.EDU
MATHEMATICAL RESEARCH LETTERS
-- TO APPEAR --
AMS CODE: 55S10 Steenrod Algebra, 13XX Commutative Rings and Al
gebras, 55XX Algebraic Topology
KEYWORDS: Steenrod Algebra, Cohen--Macaulay Algebras, Unstable Al
gebras, P * Invariant Prime Ideal Spectrum, P * Inseparable Closure,
Polynomial Invariants of Finite Groups
Typeset by LST E X
SUMMARY :
SUMMARY :
SUMMARY :
SUMMARY :
SUMMARY : We characterize Cohen--Macaulay algebras in the category
K fg of unstable Noetherian algebras over the Steenrod algebra via the
depth of the P * invariant ideals. This allows us to transfer the Cohen--
Macaulay property to suitable subalgebras. We apply this to rings of
invariants of finite groups and to the P * inseparable closure.
For over 50 years the Steenrod algebra P * and algebras over it have
proven to be decisive tools in algebraic topology. Applications occur in
cohomology theory, invariant theory of finite groups, and most recently
in algebraic geometry. This makes it desirable to have a completely
algebraic (and not topological) approach to the subject.
In this paper we follow the path begun in [12], [13], [14] and [16] of
building a ``complete'' P * invariant commutative algebra. We find a P *
invariant version of the classical characterization of Cohen--Macaulay
algebras. This allows us to show that Cohen--Macaulayness is inherited
by subalgebras provided that the P * invariant prime ideal spectra are
in bijective correspondence and the extension is integral. This applies
in particular to an unstable algebra and its P * inseparable closure. It
also applies to rings of polynomial invariants of finite groups.
For an algebraic introduction to the Steenrod algebra and algebras (and
modules) over it we refer to the introduction of [13], Chapters 10 and 11
in [19], or Chapters 8 -- 10 in [17].
§1. Cohen--Macaulay Algebras
In this section we characterize unstable Cohen--Macaulay algebras.
We start with notation and a review of some terminology.
Let F be a Galois field of characteristic p with q elements. We denote
by H * a non--negatively graded, connected, commutative Noetherian F
algebra. Since all objects are graded, all homomorphisms are degree
preserving (unless otherwise stated) and all ideals are homogeneous.
Note also that the augmentation ideal is the unique homogeneous max
imal ideal in H * . Recall that a graded algebra is connected if and only
if it is equal to F in degree zero. Throughout the whole paper the Krull
dimension of H * is
dim(H * ) = n .
Let H * have the additional structure of an unstable algebra over the
Steenrod algebra P * . Namely, H * is a left P * module satisfying the Car
tan formulae
P i (hh ) = X
j+k=i
P j (h)P k (h ) " h, h V H * , " i ³ 0 ,
and the unstability condition
P i (h) = h q if i = deg(h)
0 if i > deg(h) " h V H * .
MARA D. NEUSEL
Of particular interest are unstable Noetherian algebras over P * that
arise as rings of invariants in the following way. Let
N : G GL(n, F)
be a faithful representation of a finite group G over the field F. The
group G acts via N on the ndimensional vector space V = F n . This
induces an action of G on the ring F[V ] = F[x 1 , . . . , x n ] of polynomial
functions in n variables via
g f (v) := f N(g) -1 (v) " f V F[V ], g V G, v V V .
The subring F[V ] G of polynomials invariant under this action is an un
stable Noetherian algebra over the Steenrod algebra. See [17] or [19] for
an introduction to invariant theory and the use of Steenrod technology
there. Since the ground field is finite, the general linear group GL(n, F)
is finite. Its ring of invariants (in the tautological representation) was
determined by L.E. Dickson and is therefore called the Dickson alge
bra
D * (n) = F[V ] GL(n, F) = F[d n,0 , . . . , d n,n-1 ].
It is a polynomial algebra generated by the Dickson classes
d n,0 , . . . , d n,n-1
(see Section 8.1 in [19], or the original [7]). The Dickson algebra plays
a central role in the invariant theory of finite groups, where it provides
us with a universal set of invariants present in every ring of invariants.
Even more crucial is its role in the entire category of unstable Noethe
rian algebras over P * . By the imbedding theorem, Theoren 8.1.5 in [13],
sufficiently high powers of the Dickson classes provide us with a univer
sal system of parameters for every unstable Noetherian algebra over P * .
We will use this fact frequently in the following.
Recall that the height of an ideal I is defined as
ht (I) = min{ht (p)I H p, p F H * prime }.
We denote by Proj (H * ) the spectrum of homogeneous prime ideals in
H * , and by Proj P * (H * ) the spectrum of P * invariant homogeneous prime
ideals, where an ideal is called P * invariant if it is stable under the
action of the Steenrod algebra P * .
In classical terms we can characterize Cohen--Macaulay algebras by
looking at the depth of the ideals in the algebra in question. A sequence
h 1 , . . . , h k V H * is called a regular sequence if
h 1 V H *
2
UNSTABLE COHEN--MACAULAY ALGEBRAS
and
h i V H *
(h 1 , . . . , h i-1 ) " i = 2 , . . . , k
are nonzero divisors, where
h i = pr i (h i )
is the image under the canonical projection
pr i : H * & H *
(h 1 , . . . , h i-1 ).
The depth of an ideal I is the length of a maximal regular sequence
with members in I. Recall that for every ideal I,
dp (I) £ ht (I).
We have equality for all ideals I if and only if the algebra is Cohen--
Macaulay (see Theorem 3.3.2 in [2]).
PROPOSITION 1.1
PROPOSITION 1.1
PROPOSITION 1.1
PROPOSITION 1.1
PROPOSITION 1.1: Let H * be an unstable Noetherian algebra over
the Steenrod algebra P * . The following statements are equivalent:
(1) H * is Cohen--Macaulay.
(2) dp (m) = ht (m), where m F H * is the augmentation ideal.
(3) dp (p) = ht (p) for every P * invariant prime ideal p F H * .
(4) dp (I) = ht (I) for every P * invariant ideal I F H * .
(5) Every P * invariant ideal I H H * of height k contains
d q t
n,0 , . . . , d q t
n,k-1
V I
for some suitably large t VN, and these elements form a regular
sequence of maximal length in I.
PROOF PROOF PROOF PROOF PROOF: Since the maximal ideal m F H * is P * invariant, the impli
cations
(4) Þ (3) Þ (2)
are obvious. The implication
(2) Þ (1)
follows from the corresponding classical result (cf. Theorem 3.3.2 in
[2]) that a ring is Cohen--Macaulay if the height of every maximal ideal
is equal to its depth. Since we are in the graded situation, we have
precisely one maximal ideal, namely the augmentation ideal.
The implication
(5) Þ (4)
3
MARA D. NEUSEL
is clear. It remains only to show the implication (1) Þ (5). To this end
assume that H * is Cohen--Macaulay. Since H * is Noetherian, it contains
a (q s )th power of the Dickson algebra for some s ³ 0 such that
D * (n) q s
= F[d q s
n,0 , . . . , d q s
n,n-1 ] H *
is an integral extension. This follows from the imbedding theorem (The
orem 8.1.5 in [13]). By Macaulay's theorem (Theorem 3.3.5 (vi) in [2])
the powers of the Dickson classes
d q s
n,0 , . . . , d q s
n,n-1
V H *
form a homogeneous system of parameters and hence a regular se
quence of maximal length in H * . By Theorem 8.3.4 in [13] every P *
invariant prime ideal p F H * of height k contains a P * invariant ideal
of height k generated by k elements. By the imbedding theorem (The
orem 8.1.5 in [13]) we can choose these k elements to be the fractals of
the top k Dickson classes
d q s
n,0 , . . . , d q s
n,k-1 .
Recall that a P * invariant ideal I F H * has a primary decomposition
I = q 1 Ç . . . Ç q l ,
where every primary ideal q i , i = 1 , . . . , l, as well as its radical
q i is
P * invariant (see Theorem 3.5 in [16]). By what we have seen above, we
know that
d q s
n,0 , . . . , d q s
n,k-1
V I .
Therefore
d q t
n,0 , . . . , d q t
n,k-1
V I F H *
for some t ³ s, and this is a regular sequence because d q s
n,0 , . . . , d q s
n,k-1 is
a regular sequence (see Exercise 12 of Section 3.1 in [2]). Y
REMARK REMARK REMARK REMARK REMARK: Note carefully that we can reformulate conditions (2) and
(3) as follows:
(2') The localization H *
m at the maximal ideal is a Cohen--Macaulay
ring.
(3') The localization H *
p at a P * invariant prime ideal is a Cohen--
Macaulay ring for every p V Proj P * (H * ).
These localizations inherit naturally an action of the Steenrod algebra,
but they are no longer unstable (cf. [20], [14]).
4
UNSTABLE COHEN--MACAULAY ALGEBRAS
§2. Integral Extensions
Let
v : H * K *
be an integral extension of unstable Noetherian algebras over the Steen
rod algebra P * . This map induces a map between the homogeneous
prime ideal spectra
v * : Proj (K * ) & Proj (H * ), p ) p ÇH * .
If we restrict this map to the subset of P * invariant homogeneous prime
ideals, we get
v * P * : Proj P *(K
* ) & Proj P *(H
* )
because the contraction of a P * invariant prime ideal is P * invariant
(see Lemma 2.1 in [12]). Note that
(v * ) -1 Proj P * (H * )
= Proj P *(K
* )
is always true by Theorem 2.3 in [12]. In particular this means that
v * P * is always surjective.
THEOREM 2.1
THEOREM 2.1
THEOREM 2.1
THEOREM 2.1
THEOREM 2.1: Keeping the above notation we assume that the in
duced map v * P * is bijective. If K * is Cohen--Macaulay then so is H * .
PROOF PROOF PROOF PROOF PROOF: By the imbedding theorem (Theorem 8.1.5 in [13]) we find
a fractal of the Dickson algebra inside H * such that
D * (n) q s
= F[d q s
n,0 , . . . , d q s
n,n-1 ] H *
is an integral extension. By the characterization of unstable Cohen--
Macaulay algebras given in Proposition 1.1 (5) it is enough to show that
for every P * invariant prime ideal q F H * of height ht (q) = k, we have
a regular sequence
d q s
n,0 , . . . , d q s
n,k-1
V q .
We proceed by induction on ht (q).
If ht (q) = 0 then
0 £ dp (q) £ ht (q) = 0 ,
and there is nothing to show.
However, we need to start our induction with ht (q) = 1. We have
D * (n) q s H * K *
È È È
q Ç D * (n) q s + q + q e ,
5
MARA D. NEUSEL
where (-) e denotes the extended ideal. The ideal q ÇD * (n) q s H D * (n) q s
is by construction P * invariant and prime of height 1. Therefore
q Ç D * (n) q s
= d q s
n,0
by Theorem 9.2.5 in [17]. In other words,
d q s
n,0
V q H q e .
Since K * is Cohen--Macaulay, the element d q s
n,0 is a nonzero divisor in
K * , so it is not a zero divisor in H * .
Assume that ht (q) = k > 1. First we note that there exists precisely one
prime ideal p F K * such that
q = p ÇH * .
Moreover, p is also P * invariant of height k, and
q e H q e = p
by Theorem 2.3 in [12]. By Proposition 8.3.3 in [13] there exists a P *
invariant prime ideal q F q of height k - 1. Hence, there is precisely
one prime ideal p F K * of height k - 1 such that
q = p ÇH * .
This ideal p is also P * invariant and we have
q e H p q e = p .
By induction, we know that q has depth k - 1, and we may choose the
fractals
d q s
n,0 , . . . , d q s
n,k-2
V q
of the top k - 1 Dickson classes as a regular sequence of maximal length
in q . Note that
ht (d q s
n,0 , . . . , d q s
n,k-2 ) = ht (q ) = k - 1.
Therefore also all ideals in the chain 1
d q s
n,0 , . . . , d q s
n,k-2 e
= d q s
n,0 , . . . , d q s
n,k-2 K *
H q e H p q e = p
have height k - 1. Since K * is Cohen--Macaulay, they also have depth
k - 1. In particular, the sequence
d q s
n,0 , . . . , d q s
n,k-2
V p
1 The notation (-) K * emphasizes that we are looking at an ideal in K * .
6
UNSTABLE COHEN--MACAULAY ALGEBRAS
is regular of maximal length for the prime ideal p . We want to extend
the regular sequence d q s
n,0 , . . . , d q s
n,k-2
V q by another element h k . For
that we assume to the contrary that every element in q \ (d q s
n,0 , . . . , d q s
n,k-2 )
is a zero divisor modulo (d q s
n,0 , . . . , d q s
n,k-2 ). Then q is contained in an as
sociated prime ideal of (d q s
n,0 , . . . , d q s
n,k-2 ). In other words, there exists
an element h V H * such that
q H (d q s
n,0 , . . . , d q s
n,k-2 ) : h .
Moreover,
q e H (d q s
n,0 , . . . , d q s
n,k-2 ) : h e
H (d q s
n,0 , . . . , d q s
n,k-2 ) e : h .
So, if h is a zero divisor in K *
(d q s
n,0 , . . . , d q s
n,k-2 ) e then
ht (d q s
n,0 , . . . , d q s
n,k-2 ) e : h = k - 1
because (d q s
n,0 , . . . , d q s
n,k-2 ) e = (d q s
n,0 , . . . , d q s
n,k-2 ) K * F K * is generated by a
regular sequence and is therefore height unmixed by Macaulay's un
mixedness theorem (see Definition 3.3.1 in [2]). However, the formula
k = ht (q e ) £ ht (d q s
n,0 , . . . , d q s
n,k-2 ) e : h = k - 1
shows that this is nonsense. On the other hand, if h is not a zero divisor
in K *
(d q s
n,0 , . . . , d q s
n,k-2 ) e , then
(d q s
n,0 , . . . , d q s
n,k-2 ) e : h = d q s
n,0 , . . . , d q s
n,k-2 K *
.
Again, we get a contradiction by comparing heights
k = ht (q e ) £ ht (d q s
n,0 , . . . , d q s
n,k-2 ) e : h = ht (d q s
n,0 , . . . , d q s
n,k-2 ) e
= k - 1.
Hence
d q s
n,0 , . . . , d q s
n,k-2
V q
is a regular sequence that can be extended to a regular sequence
d q s
n,0 , . . . , d q s
n,k-2 , h k V q
of length k in q. Since this is possible for every prime ideal q of height
k, this means that no prime ideal of height k is contained in a prime
ideal associated to
(d q s
n,0 , . . . , d q s
n,k-2 ) H * .
I.e., the ideal (d q s
n,0 , . . . , d q s
n,k-2 ) H * is height unmixed and every associated
prime ideal has height k - 1. Finally we need to show that we can choose
h k = d q s
n,k-1 .
7
MARA D. NEUSEL
To this end observe that the ideal
(d q s
n,0 , . . . , d q s
n,k-1 ) H H *
has height k because d q s
n,0 , . . . , d q s
n,k-1 is part of a homogeneous system
of parameters for H * by the imbedding theorem (Theorem 8.1.5 in [13]).
If d q s
n,k-1 were a zero divisor modulo (d q s
n,0 , . . . , d q s
n,k-2 ) in H * , then this
element would be contained in an associated prime ideal. So there is an
associated prime ideal of (d q s
n,0 , . . . , d q s
n,k-2 ) H * of height at least
ht (d q s
n,0 , . . . , d q s
n,k-1 ) = k - 1.
However, this contradicts that the ideal (d q s
n,0 , . . . , d q s
n,k-2 ) H * is height un
mixed. Hence we can choose h k = d q s
n,k-1 . This completes the induction
step. Y
REMARK REMARK REMARK REMARK REMARK: The converse of the preceding theorem is false. For ex
ample, choose a polynomial ring F[x] in one linear variable over a finite
field F. This is a unstable Cohen--Macaulay algebra. By Proposition
11.4.1 in [19] there are exactly two P * invariant prime ideals
(0) 5 (x) 5 F[x].
Let F[x, y] be the polynomial algebra in two linear variables, and con
sider its quotient by the ideal (xy, y q ). Since this ideal is P * invariant
we get an integral extension
v : F[x] F[x, y] (xy, y q )
in the category of unstable algebras. Note that the longest chain of P *
invariant prime ideals in the bigger algebra is
(y) 5 (y, x) 5 F[x, y]
(xy, y q ).
Hence v * P * is bijective. However, the polynomial algebra F[x] is
Cohen--Macaulay, but the bigger algebra is not. In fact, it has depth
zero since every element is a zero divisor.
Recall that the Steenrod algebra contains an infinite family of deriva
tions inductively defined by
P D 1 = P 1 and P D i = [P D i-1 , P q i-1
] " i ³ 2.
Set P D 0 (h) = deg(h)h. Then an unstable algebra H * is called P * inse
parably closed if there exists a pth root of h in H * whenever
P D i (h) = 0 " i ³ 0.
8
UNSTABLE COHEN--MACAULAY ALGEBRAS
Recall that a pth root of h is an element h V H * such that
h p = h .
The P * inseparable closure P *
H * of an unstable algebra H * is then
the smallest unstable algebra containing H * that is P * inseparably
closed. See Chapter 4 of [13] for an introduction to this notion and
its basic properties.
COROLLARY 2.2
COROLLARY 2.2
COROLLARY 2.2
COROLLARY 2.2
COROLLARY 2.2: Let H * be reduced and Noetherian. 2 If P *
H * is
Cohen--Macaulay, then so is H * .
PROOF PROOF PROOF PROOF PROOF: Let P *
H * be Cohen--Macaulay. By Proposition 4.2.1 (4) in
[13]
v : H * P *
H *
is an integral extension. By Theorem 4.3.1 in [13] it induces a bijection of
the respective prime ideal spectra. By Theorem 2.3 in [12] P * invariant
prime ideals correspond to P * invariant prime ideals, so we apply the
preceding result and conclude that H * is Cohen--Macaulay. Y
We illustrate these results with examples from invariant theory.
EXAMPLE 2.3
EXAMPLE 2.3
EXAMPLE 2.3
EXAMPLE 2.3
EXAMPLE 2.3 : Let N : G GL(n, F) be a faithful representation
of a finite group G containing the special linear group SL(n, F). Then
we have integral extensions
v : D * (n) = F[V ] GL(n, F) F[V ] G F[V ] SL(n, F) .
By Corollary 9.2.4 in [17] the P * invariant prime ideal spectrum of the
Dickson algebra D * (n) forms a single chain
(H) (0) F (d n,0 ) F (d n,0 , d n,1 ) F . . . F (d n,0 , . . . , d n,n-1 ).
The invariants of the special linear group are given by
F[E, d n,1 , . . . , d n.n-1 ] ,
where E q-1 = d n,0 is the Euler class of the SL(n, F)action on the dual
vector space V * (see Section 7.2 in [15] or Theorem 8.1.8 in [19]). The
P * invariant prime ideal spectrum of F[V ] SL(n, F) forms also just a single
chain
(0) F (E) F (E, d n,1 ) F . . . F (E, d n,1 , . . . , d n,n-1 )
2 The P * inseparable closure does not exist for algebras with nilpotent elements. The statement
would not make sense without this assumption. In contrast, P * H * does make sense for non--
Noetherian algebras. Indeed, P * H * is Noetherian if and only if H * is Noetherian by Theorem
6.3.1 in [13].
9
MARA D. NEUSEL
since these are precisely the prime ideals lying over those of the chain
(H) (cf. Example 1 of Section 2 in [12]). Therefore, v induces a bijection
v * P * : Proj P * F[V ] SL(n, F)
& Proj P * D * (n)
on the P * invariant prime ideal spectra. Hence also
v * P * : Proj P * F[V ] SL(n, F)
& Proj P * F[V ] G
is a bijection. Since the ring of invariants of the special linear group
is Cohen--Macaulay, the preceding result tells us that F[V ] G is also
Cohen--Macaulay. Now Syl p (G) = Syl p (GL(n , F)) has Cohen--Macaulay
invariants by a theorem of M.--J. Bertin [4], so G also has Cohen--
Macaulay invariants by Theorem 5.6.7 in [17]. As we explain next,
this is just a special case of a more general phenomenon.
Let
N : G GL(n, F)
be a faithful representation of a finite group G. Take a primitive (q - 1)st
root of unity g in F, where ïFï = q. Then the diagonal matrix
D = diag(g , . . . , g)
generates a cyclic group Z/(q - 1) of order q - 1 in GL(n, F). We consider
the invariants of the group G ° Z/(q - 1) F GL(n, F) generated by G and
D. We have integral extensions
F[V ] G ° Z/(q-1) F[V ] G F[V ].
Let p V Proj P *(F[V ] G ). Then the group G acts transitively on the set
of prime ideals p 1 , . . . , p k in F[V ] lying over p. These ideals are again
P * invariant [12]. By [18] the prime ideals p 1 , . . . , p k are generated by
linear forms, and moreover, every ideal generated by linear forms is
P * invariant and certainly prime. The affine variety associated to such
a prime ideal is therefore just a linear subspace U i H V of dimension
equal to n - ht (p) for i = 1 , . . . , k. This means that the orbit structure
of the action of G on the Grassmann varieties of V characterizes the
structure of Proj P * (F[V ] G ) and vice versa (cf. Chapter 9 in [17]). By
construction the orbit structure of the G ° Z/(q - 1)action on the Grass
mannians of V is the same as the one of G. Therefore we find that
Proj P * (F[V ] G ) @ Proj P * (F[V ] G ° Z/(q-1) ).
More generally, denote by G(G) the maximal subgroup of GL(n, F) con
taining G such that the orbit structure on the Grassmannians of V is
10
UNSTABLE COHEN--MACAULAY ALGEBRAS
the same as for G. 3 Then with the preceding argumentation we find
that
Proj P * (F[V ] G ) @ Proj P * (F[V ] G(G) ).
Hence we have proven the following proposition.
PROPOSITION 2.4
PROPOSITION 2.4
PROPOSITION 2.4
PROPOSITION 2.4
PROPOSITION 2.4: If F[V ] G is Cohen--Macaulay, then so is F[V ] H for
every group H with G H H H G(G).Y
The above Example 2.3 follows from this result with the observation
that G(SL(n, F)) = GL(n, F). Another pair of groups that fall into this
class are the symplectic group and the general symplectic group.
EXAMPLE 2.5
EXAMPLE 2.5
EXAMPLE 2.5
EXAMPLE 2.5
EXAMPLE 2.5 : Let Sp(2l, F) be the symplectic group with F a field of
odd characteristic. Choose a symplectic basis e 1 , . . . , e l , f 1 , . . . , f l such
that
f (e i , e j ) = 0 = f (f i , f j ), f (e i , f j ) = d i j
for a symplectic form f . Then the general symplectic group G(Sp(2l, F))
is generated by Sp(2l, F) and diagonal matrices
diag(g , . . . , g
!&&& l &&& , 1 , . . . , 1
!&& l && ) ,
where g V F × . Note that while the symplectic group is the group of f
isometries, the general symplectic group is the group of f similarities
(cf. Section 2.4 in [10] or the classic [8] Chapter II). 4 In any case, both
groups have the same orbit structure when acting on the Grassmanni
ans of V . Since the invariants of Sp(2l, F) form a complete intersection
([6], Section 8.2 in [3], or [11]) they are Cohen--Macaulay. Hence so are
the invariants of any group between Sp(2l, F) and G(Sp(2l, F)).
Of course, as in Example 2.3 the index of the symplectic group Sp(2l, F)
in G(Sp(2l, F)) is prime to the characteristic, so we could lift the Cohen--
Macaulayness of the invariants of the smaller group to the invariants of
the bigger group by observing that the relative transfer homomorphism
gives us a splitting of the natural inclusion (Theorem 5.6.7 in [17]). In
order to find an example such that we can't derive Cohen--Macaulayness
from older methods (cf. Section 5.5 in [17]) we have to find a group G H
GL(n, F) such that the index ïG(G) : Gï is divisible by the characteristic
and the dimension n is at least 4 (cf. Proposition 5.6.9 in [17]).
3 So G(G) is in many cases a subgroup of the group generated by G and all diagonal matrices of
GL(n,F). However, this rule of thumb can be very misleading. For example, take the field F = F 3
with 3 elements. If we restrict our attention to n = 2, then we find a quaternion group Q 8 in
SL(2, F 3 ), which acts transivitely on the hyperplanes. This is the only Grassmannian of interest
(see Example 1 of Section 2 in [12]). In oher words, G(Q 8 ) = GL(2, F 3 ).
4 L.E. Dickson calls these groups the special abelian linear group SA(2l, q) and the general abelian
linear group GA(2l, q).
11
MARA D. NEUSEL
EXAMPLE 2.6
EXAMPLE 2.6
EXAMPLE 2.6
EXAMPLE 2.6
EXAMPLE 2.6 : We consider groups G that act transitively on all
Grassmannians of V . This is the simplest possible orbit structure.
Namely, G H GL(n, F) with G(G) = GL(n, F). In Proposition 8.4 of [5]
we find a complete list of such groups consisting of three types:
(1) SL(n, F) H G. This is our Example 2.3.
(2) N : G = A 7 GL(4, F 2 ). The explicit embedding of the alternat
ing group A 7 into GL(4, F 2 ) can be found in Theorem 268 of [8]
or in Satz II 2.5 of [9]. The ring of invariants F[V ] A 7 is Cohen--
Macaulay by Theorem 3.13 in [1]. However, we can't derive any
thing new from this because the embedding N is just the natural
embedding of A 7 into A 8 @ GL(4, F 2 ).
(3) N : G = G(1, F 2 5 ) GL(5, F 2 ). The general semilinear group
G(1, F 2 5 ) admits a natural embedding into GL(5, F 2 ) (see Section
2.1 in [10]). Since G(1, F 2 5 ) has order 31 · 5, which is not divis
ible by 2, the ring of invariants F[V ] G(1,F 2 5 ) is Cohen--Macaulay
by Theorem 5.5.2 in [17]. By Proposition 8.4 in [5]
G G(1, F 2 5 )
= GL(5, F 2 ) ,
so all groups between them have Cohen--Macaulay invariants.
Note that the index ïGL(5, F 2 ) : G G(1, F 2 5 ) ï is divisible by
the characteristic. Finally, we can also describe the image of
G(1, F 2 5 ) in GL(5, F 2 ) as the normalizer subgroup of a Singer
cycle in GL(5, F 2 ) (cf. Satz II 7.3 in [9]).
Acknowledgment
I thank Dan Isaksen for proofreading and the referee for reference [1].
References
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[2] Stanislaw Balcerzyk and Tadeusz J’ ozefiak: Commutative Rings. Dimension,
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[7] Leonard E. Dickson: A Fundamental System of Invariants of the General Modular
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(1911), 75--98.
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UNSTABLE COHEN--MACAULAY ALGEBRAS
[8] Leonard E. Dickson: Linear Groups, Dover Publications Inc., New York 1958.
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[10] Peter Kleidman and Martin Liebeck: The Subgroup Structure of the Finite Clas
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[11] Mara D. Neusel: The Invariants of the Symplectic Groups, Appenzell 1998.
[12] Mara D. Neusel: Integral Extensions of Unstable Algebras over the Steenrod Al
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[13] Mara D. Neusel: Inverse Invariant Theory and Steenrod Operations, Memoirs of
the AMS No. 692 Vol. 146, AMS, Providence RI 2000.
[14] Mara D. Neusel: Localizations over the Steenrod Algebra. The Lost Chapter.
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[15] Mara D. Neusel: A Selection from the 1001 Examples. The Atlas of Polynomial
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[16] Mara D. Neusel and Larry Smith: The Lasker--Noether Theorem for P * Invariant
Ideals, Forum Mathematicum 10 (1998), 1--18.
[17] Mara D. Neusel and Larry Smith: Invariant Theory of Finite Groups, book
manuscript, to appear.
[18] Jean--Pierre Serre: Sur la dimension cohomologique des groupes profinis, Topol
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[19] Larry Smith: Polynomial Invariants of Finite Groups, 2nd corrected printing,
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[20] Clarence W. Wilkerson: Classifying Spaces, Steenrod Operations and Algebraic
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Mara D. Neusel
im mittleren Westen:
# Department of Mathematics
University of Notre Dame
Mittelweg 3 370 CCMB
D37133 Friedland Notre Dame, IN 465565683
Germany USA
mdn@sunrise.unimath.gwdg.de Neusel.1@nd.edu
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