Invariant Theory and the Koszul Complex
Invariant Theory and the Koszul Complex
Invariant Theory and the Koszul Complex
Invariant Theory and the Koszul Complex
Invariant Theory and the Koszul Complex
Representations of Z/p in Characteristic p
Representations of Z/p in Characteristic p
Representations of Z/p in Characteristic p
Representations of Z/p in Characteristic p
Representations of Z/p in Characteristic p
Applications Applications Applications Applications Applications
Larry Smith
Larry Smith
Larry Smith
Larry Smith
Larry Smith
YALE UNIVERSITY
NEW HAVEN, CT, USA
AND
#
G ˜
OTTINGEN, GERMANY
SUMMARY :
SUMMARY :
SUMMARY :
SUMMARY :
SUMMARY : We study the ring of invariants F[V ] Z/p , and its derived functors H i (Z/p ; F[V ]),
of the cyclic group Z/p of prime order p over a field F of characteristic p. We verify a formula
of Ellingsrud and Skjelbred [13] for the homological codimension, show the quotient algebra
F[V ] Z/p /Im(Tr Z/p ) is CohenMacaulay, and that the ideal generated by the elements in the
image of the transfer homomorphism, Im(Tr Z/p ) F F[V ] Z/p , is primary of height n - 1 when V
is an ndimensional irreducible representation of Z/p. Using our cohomological computations
and a previous result [34] about permutation representations we are able to obtain an upper
bound for the degree of homogeneous forms in a minimal algebra generating set for F[V ] Z/p .
The final work on this manuscript was done at Yale University, and I would like to thank the Yale
Mathematics department for providing me with an office, library facilities, and atmosphere sup
portive to my research efforts.
MATHEMATICS SUBJECT CLASSIFICATION : 13A50 Invariant Theory
Typeset by LST E X
This note is a continuation of [33]. We will first be concerned with the rings of invariants
of the irreducible representations of the cyclic group Z/p of prime order p over a field F of
characteristic p. Specifically, for 2 £ n £ p the matrix (a single Jordan block)
J n =
2 6 6 6 6 4
1 1
0 1 1
0 . . . . . . 0
1 1
1
3 7 7 7 7 5
V GL(n , F) 2 £ n £ p ,
where F is a field of characteristic p is of order p and implements a faithful representation
N n : Z/p GL(n , F). Being a single Jordan block it is irreducible, and N 2 , . . . , N p is a complete
list of irreducible representations of Z/p over F. Since J n is defined over the Galois field F p
with p elements, and F p is a splitting field for Z/p, we may, and will, suppose that F = F p . Any
finite dimensional representation of Z/p over F is a finite direct sum 1 of copies of these.
We set V n = F n and denote by F[V n ] the graded algebra of homogeneous polynomial functions
on V n (see e.g., [30] Section 1.1).Via J n , or N n , the group Z/p acts on F[V n ] and there is the
ring of invariants F[V n ] Z/p , which has been the subject of numerous 2 investigations [4], [5],
[19], [13], [33], and [9] amongst others. In this note we are concerned with the depth, or
homological codimension hom--codim(F[V n ] Z/p ) and the transfer homomorphism ([30]
Section 4.2)
Tr Z/p : F[V n ] & F[V n ] Z/p ,
which in this case is defined by Tr Z/p = J 0
n + J 1
n + · · · + J p-1
n . The transfer homomorphism along
with . := 1 - J n appears in the standard cocomplex
0 & F[V n ] .
&&&&&&& F[V n ] Tr Z/p
&&&&&&& F[V n ] .
&&&&&&& F[V n ] & · · ·
used to compute H * (Z/p, F[V n ]), which in turn intervenes via the methods of [33] or [13] for
studying hom--codim(F[V n ] Z/p ). From this cocomplex one sees ([10] or [14])
H 0 (Z/p ; F[V n ]) = F[V n ] Z/p
H 1 (Z/p ; F[V n ]) = ker(Tr Z/p )/Im(. )
H 2 (Z/p ; F[V n ]) = F[V n ] Z/p /I m(Tr Z/p )
and H i+2 (Z/p ; F[V n ]) = H i (Z/p ; F[V n ]) for all i > 0. To make effective use of the tools of [33]
requires a knowledge of the cohomology H i (Z/p ; F[V n ]) for i > 0 as a module over a suitable
subalgebra, such as D * (n), the Dickson algebra, of F[V n ] Z/p .
Let us abbreviate H i (Z/p ; F[V n ]) to H ev (n) for i > 0 and i even, and H od (n) for i odd. Although
not profusely published, nor well known, it is not hard to see that H od (n) and H ev (n) have
the same Poincar’ e series as graded vector spaces. This is a special case of Herbrand's Lemma
(see e.g., [21] Theorem 5.2 or [28] VIII Proposition 8) and follows from the two exact sequences
0 & F[V n ] Z/p & F[V n ] .
&&&&&&& F[V n ] & coker(.) & 0
0 & H od (n) & coker(.) Tr Z/p
&&&&&&& F[V n ] Z/p & H ev (n) & 0
by taking Euler characteristics. Somewhat more suprising is the identity for Poincar’ e series
of graded vector spaces
n
X s=0
(-1) s P Tor s
D * (n) F , H od (n)
, t =
n
X s=0
(-1) s P Tor s
D * (n) F , H ev (n)
, t .
1 But do not jump the conclusion that anything analagous holds for the rings of invariants! See e.g. [34].
2 Indeed, the study of these rings appears to be addictive.
LARRY SMITH
This follows from Smoke's [35] formula (section 3) for the multiplicity symbols: this amounts
to saying that H od (n) and H ev (n) are equal in the Grothendieck group of finitely generated
modules over D * (n). In this note we will show that in fact H od (n) and H ev (n) also have the
same homological dimension as modules over D * (n). This has led the author to pose the fol
lowing:
PROBLEM PROBLEM PROBLEM PROBLEM PROBLEM : Let A be a graded, connected, commutative algebra over the field F of char
acteristic p and Z/p Aut * (A) a representation of Z/p by graded automorphisms of A. Are
H 1 (Z/p ; A) and H 2 (Z/p ; A) isomorphic as modules over A Z/p ?, or at least, over a suitable sub
algebra S H A Z/p ?
In [33] Proposition 4.3 we showed this to be the case when Z/p acts on A = F[V ] via a permu
tation representation on a basis for V .
As a final comment on this problem we note that Tr Z/p = . p-1 V F(Z/p), where F(Z/p) denotes
the group ring of Z/p over F. Since . p = 0 one has Im(. p-i ) H ker(. i ) for i = 1 , . . . , p - 1, and
one might wonder if the modules ker(. i )/Im(. p-i ) are all isomorphic for i = 1 , . . . , p - 1.
Section 1 contains the cohomological computations that are applied in later sections, first
for irreducible representations, and then for general finite dimensional representations, to
verify a formula for hom--codim(F[V ] Z/p ) from [13] and to prove that Im(Tr Z/p ) F F[V n ] Z/p is
a primary ideal of height n - 1 when V n is irreducible. In Section 4 we establish for a general
finite dimensional Z/prepresentation an upper bound for the degree of an algebra generator
of F[V ] Z/p in a minimal generating set. The proof of the depth formula uses [33] and the
transfer computation uses [18]. The degree bound depends on results in [34] for permutation
representations, some computations from [29], and some elmentary properties of Dade bases
[25] shared by Jordan bases, as explained here.
§1. Some Cohomology Computations
We adhere to the notations introduced above, so in particular V n is the irreducible ndimensional
representation of Z/p over the Galois field F = F p with pelements; whence 2 £ n £ p. Let
x 1 , . . . , x n be the standard basis for V n , i.e., the basis in which the representation of Z/p on
V n is implemented by the matrix J n , and denote by z 1 , . . . , z n V V *
n the dual basis for the dual
vector space V *
n of V n . For two consecutive values of n the cohomology modules H ev/od (n) are
related, since there is the short exact sequence
0 & F[z 1 , . . . , z n ] ·z n
& F[z 1 , . . . , z n ] & F[z 1 , . . . , z n-1 ] & 0
where ·z n denotes multiplication by the fixed vector z n : this yields an exact hexagon
H ev (n) & H ev (n)
i u
H od (n - 1) Y H ev (n - 1)
m q
H od (n) !& H od (n)
DIAGRAM DIAGRAM DIAGRAM DIAGRAM DIAGRAM 1.1: Exact Hexagon
The cohomology H ev/od (n) are modules over F[V n ] Z/p . The following lemma will help us to
prove several results about H ev/od (n) when 2 £ n £ p by downward induction on n starting
with the case n = p and results of [33].
LEMMA 1.1
LEMMA 1.1
LEMMA 1.1
LEMMA 1.1
LEMMA 1.1: The map
·z n : H ev/od (n) & H ev/od (n)
is the zero map.
2
INVARIANT THEORY AND THE KOSZUL COMPLEX
PROOF PROOF PROOF PROOF PROOF: We are going to need a couple of formulae. We begin with the list
.(z 1 ) = -z 2
.(z 2 ) = -z 3
*
* . . . = . . .
.(z n-1 ) = -z n
.(z n ) = 0
from which we obtain
Tr Z/p (z 1 ) = . p-1 (z 1 ) = (-1) p-1 z n = z n
and therefore z n V Im(Tr Z/p ). From [13] we recall the formula (Lemma 1.1 (ii))
(**) f · Tr Z/p (h) - Tr Z/p (f ) · h V Im(. )
where f , h V F[V n ] are arbitrary.
We divide the proof of the lemma into two cases:
Case H ev : we then have
z n V Im(Tr Z/p ),
H ev (n) = F[V ] Z/p /Im(Tr Z/p )
from which it is immediate that
·z n : H ev (n) K
is the zero map.
Case H od : we have
H od (n) = ker(Tr Z/p )/Im(.)
so if h V ker(Tr Z/p ) we obtain from the formual (**) and *
*
-z n · h = -Tr Z/p (z 1 ) · h V Im(.)
so
·z n : H od (n) K
is also the zero map.
From Lemma 1.1 it follows that the hexagon Ysplits into short exact sequences, namely:
LEMMA 1.2
LEMMA 1.2
LEMMA 1.2
LEMMA 1.2
LEMMA 1.2: For 2 £ n £ p there are exact sequences
0 & H(n) i *
& H(n - 1) d
& H(n) & 0
where i * has degree 0 amd d degree 1.
PROPOSITION 1.3
PROPOSITION 1.3
PROPOSITION 1.3
PROPOSITION 1.3
PROPOSITION 1.3: For 2 £ n £ p the element d n,n-1 V D * (n) is a regular element on H(n).
PROOF PROOF PROOF PROOF PROOF: For n = p this follows from [34] Theorem 2, or [33] the discussion preceeding
Proposition 4.1. We proceed by downward induction from p and assume the result estab
lished for n. Consider the exact sequence of Lemma 1.2
0 & H(n) i *
& H(n - 1) d
& H(n) & 0.
Since d n,n-1 VD * (n) is regular on H(n), H(n) is a free module over F[d n,n-1 ]. As modules over
F[d n,n-1 ] the preceeding sequence must then split, and H(n - 1) is also a free module over
F[d n,n-1 ]. This of course is equivalent to d n,n-1 being regular on H(n - 1).
PROPOSITION 1.4
PROPOSITION 1.4
PROPOSITION 1.4
PROPOSITION 1.4
PROPOSITION 1.4: For 2 £ n £ p the elements d n,n-2 , . . . , d n,0 V D * (n) act nilpotently on
H(n).
3
LARRY SMITH
PROOF PROOF PROOF PROOF PROOF: For n = p the elements d n,n-2 , . . . , d n,0 act trivially on H(p) by [34] Theorem 2,
and as in Proposition 1.3 we again proceed by downward induction. From the exact sequence
0 & H(n) i *
& H(n - 1) d
& H(n) & 0
of Lemma 1.2 we see that regarded as a D * (n)module the elements d n,n-2 , . . . , d n,0 V D * (n)
act nilpotently on H(n - 1) also.
Next, note that the D * (n)module structure on H(n - 1) arises via the change of rings map
i * : D * (n) & D * (n - 1) induced by the inclusion V n-1 V n . The map i * satisfies ([30] Section
8.1)
i * (d n,i ) = 0 for i = 0
d p
n-1,i-1 for i = 1 , . . . , n - 1.
Therefore the nilpotence of d n,n-2 , . . . , d n,0 V D * (n) entails that of d n-1,n-3 , . . . , d n-1,0 V
D * (n - 1).
Combining Propositions 1.3 and 1.2 leads to:
THEOREM 1.5
THEOREM 1.5
THEOREM 1.5
THEOREM 1.5
THEOREM 1.5: For 2 £ n £ p hom--dim D * (n) H(n) = n - 1.
PROOF PROOF PROOF PROOF PROOF: By Proposition 1.4 the elements
d n,n-2 , . . . , d n,0 V D * (n)
act nilpotently on D * (n). Therefore for large V N the elements
d n,n-2 , . . . , d n,0
V D * (n)
act trivially. Consider the subalgebra S := F[d n,n-1 , d n,n-2 , . . . , d n,0 ] H D * (n). Then D * (n) is a
free Smodule of finite rank, so
hom--dim D * (n) H(n)
= hom--dim S H(n)
.
Since, in addition H(n) is a free F[d n,n-1 ]module by Proposition 1.3, a simple Koszul complex
computation shows
Tor S F , H(n) @ F Ä S H(n) Ä E[s -1 (d n,n-2 ) , . . . , s -1 (d n,0 )]
so
Tor -(n-1)
S F , H(n)
W s -1 (d n,n-2 ) · · · s -1 d n,0
Q= 0
Tor -n
S F , H(n)
= 0
from which the result follows.
COROLLARY 1.6
COROLLARY 1.6
COROLLARY 1.6
COROLLARY 1.6
COROLLARY 1.6: For 2 £ n £ p, hom--codim D * (n) H(n) = 1.
PROOF PROOF PROOF PROOF PROOF: This follows from Theorem 1.5 and the AuslanderBuchsbaum formula [8] Theo
rem 1.3.3.
§2. Invariants of Irreducible Z/prepresentations
Using the results of the previous section and the spectral sequence of [33] leads to a proof of
one of the results of [13].
PROPOSITION 2.1
PROPOSITION 2.1
PROPOSITION 2.1
PROPOSITION 2.1
PROPOSITION 2.1(Ellingsrud and Skjelbred): Let p ³ 5 be an odd prime, F = F p , and N n :
Z/p GL(n , F) the representation implemented by the Jordan block J n , 3 £ n £ p. Then
hom--codim(F[V ] Z/p ) = 3.
4
INVARIANT THEORY AND THE KOSZUL COMPLEX
PROOF PROOF PROOF PROOF PROOF: Consider the spectral sequence {E r , d r } of [33] with
E r Þ H * Z/p , F[V ] GL(n ,F)
E s,t
2 = Tor s
D * (n) F, H t (Z/p ; F[V ])
.
From Theorem 1.5 the E 2 term of this spectral sequence has the form
t
s
0
-n + 3
-n + 2
-n + 1
0
1
2
3
. . . · · ·
. . . · · ·
. . . · · ·
. . . · · ·
0
E -n+1,1
2
E -n+3,0
2
d 2
where the term E -(n-1),1
2
Q= 0. The total degree of this term is 1 + (-(n - 1)) = 2 - n which
is negative. Hence E -(n-1),1 O = 0. The only nonzero differential either starting or ending at
E -(n-1),1
2 is the indicated one, namely
d 2 : E -(n-1),1
2
& E -(n-3),0
2 ,
which must therefore be an isomorphism. Hence
0 Q= E -(n-3),0
2 = Tor -(n-3)
D * (n) (F , F[V ] Z/p )
and therefore
hom--dim D * (n) (F[V ] Z/p ) ³ n - 3.
On the otherhand, the picture also shows that E -(n-2),0
2 must be zero since it too has negative
total degree and no nonzero differential can either arrive or terminate at it. Therefore
0 = E -(n-2),0
2 = Tor -(n-2)
D * (n) (F , F[V ] Z/p )
so
hom--dim D * (n) (F[V ] Z/p ) £ n - 3.
Combining these two inequalities gives
hom--dim D * (n) (F[V ] Z/p ) = n - 3.
and the result follows from the AuslanderBuchsbaum equality, loc.cit., .
REMARK REMARK REMARK REMARK REMARK: The analagous result for p =3 follows from the fact that rings invariants in three
or less variables are always CohenMacaulay [31].
We can also apply our cohomological computations from Section 1 to study the ideal Im(Tr Z/p ) F
F[V ] Z/p when V is an irreducible Z/prepresentation over F. Specifically (see also [18] Propo
sition 2.9)
PROPOSITION 2.2
PROPOSITION 2.2
PROPOSITION 2.2
PROPOSITION 2.2
PROPOSITION 2.2: Let p be an odd prime and V an irreducible representation of Z/p over
the field F of characteristic p. Then Im(Tr Z/p ) F F[V ] Z/p is a primary ideal of height dim F (V ) -
1.
5
LARRY SMITH
PROOF PROOF PROOF PROOF PROOF: There is no loss in generality in supposing that V is the representation over F =F p
implemented by the Jordan block J n V GL(n , F) where 2 £ n £ p. By [18] the transfer variety,
X Z/p , i.e., the variety defined by the extended ideal Im(Tr Z/p ) e F F[V ], is the fixed point set
V Z/p . If x 1 , . . . , x n V V is a Jordan basis with dual basis z 1 , . . . , z n V V * this means
X Z/p = V Z/p = Span F {x 1 } = n
Ç
i=2
ker(z i ) ,
so by Hilbert's Nullstellensatz
q Im(Tr Z/p )
e = (z 2 , . . . , z n ) F F[V ].
This is a prime ideal in F[V ] and hence so is
(z 2 , . . . , z n ) Ç F[V ] Z/p = q Im(Tr Z/p ) e = q Im(Tr Z/p ) F F[V ] Z/p .
Therefore Im Tr Z/p ) FF[V ] Z/p has a unique minimal associated prime ideal, say p. The ideal
p is invariant under the action of the Steenrod algebra P * (see e.g., [30] Chapter 11 Section
5). This ideal has height n - 1, hence p Ç D * (n) is also a prime ideal of height n - 1 and
P * invariant. By [30] Theorem 11.4.6 the only height n - 1, P * invariant, ideal in D * (n) is
(d n,0 , . . . , d n,n-2 ), and the only height n, P * invariant, ideal is the maximal ideal. If Im(Tr Z/p )
has an embedded prime ideal in F[V ] Z/p , say ”
p, then ”
p has height n and is P * invariant [23]
and [22]. Then ”
p Ç D * (n) must be the maximal ideal of D * (n) so by lying over [30] Lemma
5.4.1 ”
p is the maximal ideal of F[V ] Z/p .
Consider the quotient algebra F[V ] Z/p /Im(Tr Z/p ) = H ev (n). The associated primes of (0) in H ev
are the quotients of the associated primes of Im(Tr Z/p ) in F[V ] Z/p by Im(Tr Z/p ). The preceeding
argument therefore shows: if Im(Tr Z/p ) has an embedded prime in F[V ] Z/p , then (0) F H ev (n)
has the maximal ideal as an associated prime ideal. By [3] Theorem 2.3.22 this means that
every element in H ev (n) of positive degree is a zero divisor, contrary to Proposition 1.3. There
fore Im(Tr Z/p ) F F[V ] Z/p cannot have any embedded prime ideals, and since it has a unique
minimal associated prime ideal it must be primary.
REMARK REMARK REMARK REMARK REMARK: Representations of Z/2 in characteristic 2 are permutation representations, so
Im(Tr Z/2 ) is quite explicitly described in [34].
§3. Invariants of General Z/prepresentations
In this section we apply the results of the preceeding sections to general finite dimensional
representations N : Z/p GL(n , F) of Z/p over a field of characteristic p. Again, since F p is a
splitting field for Z/p, we may assume that F = F p . It will be convenient to fix some notations
for this section.
NOTATION NOTATION NOTATION NOTATION NOTATION: N : Z/p GL(n , F) will denote a fixed faithful representation of Z/p over the
Galois field F =F p , p an odd prime. V =F n and V = V n 1
Å · · · ÅV n k
will be a fixed decomposition
of V into irreducible representations of dimensions n 1 , . . . , n k . Each fixed point set V *
n i
is 1
dimensional and i V (V *
n i
) Z/p is a fixed nonzero linear form for i = 1 , . . . , k. N.b., 1 £ n i £ p,
i = 1 , . . . , k and Z/p acts on V n i
in an appropriate choice of basis via the Jordan block J n i
V
GL(n i , F).
We first need to adapt Lemma 1.1 to this more general context.
LEMMA 3.1
LEMMA 3.1
LEMMA 3.1
LEMMA 3.1
LEMMA 3.1: With the preceeding notations the maps
· i : H j (Z/p ; F[V ]) & H j (Z/p ; F[V ]) j > 0
are trivial for i = 1 , . . . , k.
6
INVARIANT THEORY AND THE KOSZUL COMPLEX
PROOF PROOF PROOF PROOF PROOF: Let z 1 (i) , . . . , z n i
(i) = i be the dual Jordan basis for V *
n i
. The formula *
* from
the proof of Lemma 1.1 shows that
i V Im(Tr Z/p ) F F[V ] Z/p
and the formula (**) shows
i · h V Im(.) " h V ker(Tr Z/p ).
The result then follows as in the proof of Lemma 1.1.
PROPOSITION 3.2
PROPOSITION 3.2
PROPOSITION 3.2
PROPOSITION 3.2
PROPOSITION 3.2: With the preceeding notations any two distinct elements of the collec
tion 1 , . . . , k V F[V ] Z/p are a regular sequence but no three are.
PROOF PROOF PROOF PROOF PROOF: We may as well suppose that k is at least two. Consider the Koszul complex
L = F[V ] Ä E[s -1 1 , . . . , s -1 k ]
. (f Ä 1) = 0 , .(1 Ä s -1 i ) = i Ä 1 i = 1 , . . . , k .
As in [33] Section 2 we have the cohomology H * Z/p ; (L , . )
of Z/p with coefficients in the
complex (L , . ), from which we obtain (loc.cit.) a spectral sequence
E r Þ H * (Z/p ; F[V n 1 -1 Å · · · Å V n k -1 ])
E s,t
2 = Tor s
F[ 1 ,..., k ]
F , H t (Z/p ; F[V ])
.
Here we have identified
F[V ]
( 1 , . . . , k )
via the obvious isomorphism with
F[V n 1 -1 Å · · · Å V n k -1 ].
Since 1 , . . . , k act trivially on H t (Z/p ; F[V ]) for t > 0 (Lemma 3.1) it follows that
E *,t
2 = Tor *
F[ 1 ,..., k ]
F , H t (Z/p ; F[V ]) @ H t (Z/p ; F[V ]) Ä E[s -1 1 , . . . , s -1 k ].
In particular E -k,1
2
Q= 0 since it contains the nonzero element 1 Ä s -1 1 · · · s -1 k . The total
degree of E -k,1
2 is negative and E -k,1 O = 0 since the target of the spectral sequence is zero in
negative gradings. As the following picture
t
s
0
-k + 2
-k + 1
-k
0
1
2
3
. . . · · ·
. . . · · ·
. . . · · ·
. . . · · ·
0
E -k,1
2
E -k+2,0
2
d 2
shows, the only possible nonzero differential that can either originate or terminate at E -k,1
2 is
d 2 : E -k,1
2
& E -k+2,0
2 , which must therefore be an isomorphism, so
0 Q= E -k+2,0
2 = Tor -k+2
F[ 1 ,..., k ]
F , F[V ] Z/p
.
In a similar manner we see that
0 = E -k+1,0
2 = Tor -k+1
F[ 1 ,..., k ] F , F[V ] Z/p
,
and hence hom--dim F[ 1 ,..., k ] (F[V ] Z/p ) = k - 2. The AuslanderBuchsbaum equality, loc.cit.,
then implies hom--codim F[ 1 ,..., k ] (F[V ] Z/p ) = 2, so at most two of 1 , . . . , k can form a regular
sequence. The proof of [30] Proposition 6.7.9 shows that indeed any two distinct ones do so.
7
LARRY SMITH
NOTATION NOTATION NOTATION NOTATION NOTATION: For i > 0, H i (Z/p ; F[V ]) will be denoted by H ev (n 1 , . . . , n k ) when i is even,
and by H od (n 1 , . . . , n k ) when i is odd, while if the parity of i is unimportant H(n 1 , . . . , n k )
denotes either of the two.
The subgroup
GL(n 1 , F) × · · · × GL(n k , F) H GL(n, F)
consisting of the block matrices
2 4
T 1
0 . . . 0
T k
3 5 T i V GL(n i , F) i = 1 , . . . , k
has as ring of invariants the tensor product
D * (n 1 ) Ä · · · Ä D * (n k ) =: D * (n 1 , . . . , n k )
of Dickson algebras, and H(n 1 , . . . , n k ) is a module over D(n 1 , . . . , n k ).
PROPOSITION 3.3
PROPOSITION 3.3
PROPOSITION 3.3
PROPOSITION 3.3
PROPOSITION 3.3: With the preceeding notations, the k elements in D * (n 1 , . . . , n k )
d n 1 ,n 1 -1 Ä 1 Ä · · · Ä 1
1 Ä · · · Ä d nm ,n m -1 Ä 1 Ä · · · Ä 1
1 Ä · · · Ä 1 Ä d n k ,n k -1
form a regular sequence on H(n 1 , . . . , n k ), and the remaining Dickson polynomials
d n 1 ,n 1 -j 1
Ä 1 Ä · · · Ä 1 , j 1 = 0 , . . . , n 1 - 2
1 Ä · · · Ä d nm ,n m -j m
Ä 1 Ä · · · Ä 1 , j m = 0 , . . . , nm - 2
1 Ä · · · Ä 1 Ä d n k ,n k -j k
, j k = 0 , . . . , n k - 2
act nilpotently.
PROOF PROOF PROOF PROOF PROOF: From Lemma 3.1 we have an exact sequence
0 & H(n 1 , . . . , nm , . . . , n k ) & H(n 1 , . . . , nm - 1 , . . . , n k ) & H(n 1 , . . . , nm , . . . , n k ) & 0.
By Proposition 4.4 of [33] the desired conclusion holds for n 1 = n 2 = · · · = n k = p, since in this
case N : Z/p GL(n , F) is conjugate to a permutation representation. The result follows by
downward induction from this case.
COROLLARY 3.4
COROLLARY 3.4
COROLLARY 3.4
COROLLARY 3.4
COROLLARY 3.4: With the preceeding notations we have
hom--dim D * (n 1 ,..., n k ) H(n 1 , . . . , n k ) = n - k .
PROOF PROOF PROOF PROOF PROOF: The proof is completely analagous to that of Theorem 1.5.
COROLLARY 3.5
COROLLARY 3.5
COROLLARY 3.5
COROLLARY 3.5
COROLLARY 3.5: With the preceeding notations the Dickson polynomials d n,n-1 , . . . , d n,n-k
VD * (n) are a regular sequence on H(n 1 , . . . , n k ), and d n,n-k+1 , . . . , d n,0 VD * (n) act nilpotently.
PROOF PROOF PROOF PROOF PROOF: This follows from Proposition 3.4, the AuslanderBuchsbaum equality (loc. cit.),
and the main result of [7].
THEOREM 3.6
THEOREM 3.6
THEOREM 3.6
THEOREM 3.6
THEOREM 3.6(EllingsrudSkjelbred [13]): Let p be an odd prime and N : Z/p GL(n , F)
a faithful representation of the cyclic group Z/p over the field F of characteristic p. Then
hom--codim(F[V ] Z/p ) = 2 + dim F (V Z/p ).
8
INVARIANT THEORY AND THE KOSZUL COMPLEX
PROOF PROOF PROOF PROOF PROOF: Write V = V n 1
Å · · · Å V n k
as a sum of indecomposable representations of Z/p. We
consider the spectral sequence {E r , d r } of [33] Proposition 2.2 with
E r Þ H * (Z/p ; F[V ] GL(n 1 ;F)×···×GL(n k ;F) )
E s,t
2 = Tor s
D * (n 1 ,..., n k ) F , H t (Z/p , F[V ])
.
From Corollary 3.4 we have that E -(n-k),1
2 = Tor -(n-k)
D * (n 1 ,..., n k ) F , H 1 (Z/p ; F[V ])
Q= 0. The total
degree of this term is 1 + k - n, which is negative. The target of the spectral sequence
H * (Z/p ; F[V ] GL(n 1 ;F)×···×GL(n k ;F) ) is zero in negative degrees, and, as the following picture
t
s
0
-(n - k) + 2
-(n - k) + 1
-(n-)k
0
1
2
3
. . . · · ·
. . . · · ·
. . . · · ·
. . . · · ·
0
E -(n-k),1
2
E -(n-k)+2,0
2
d 2
shows, the only possible nonzero differential either originating or terminating at E -(n-k),1
2 is
d 2 : E -(n-k),1
2
& E -(n-k)+2,0
2 , so we must have
0 Q= E -(n-k)+2,0
2 = Tor -(n-k)+2
D * (n 1 ,..., n k ) F , F[V ] Z/p
.
The same picture shows
0 = E -(n-k)+1,0
2 = Tor -(n-k)+1
D * (n 1 ,..., n k ) F , F[V ] Z/p
.
Hence
hom--dim D * (n 1 ,..., n k ) F[V ] Z/p
= n - k - 2
whence the result follows from the AuslanderBuchsbaum (loc.cit.) equality.
REMARK REMARK REMARK REMARK REMARK: For n = sp + r where r < p the Jordan block matrix
J n =
2 6 6 6 6 4
1 1
0 1 1
0 . . . . . . 0
1 1
1
3 7 7 7 7 5
V GL(n , F) 2 £ n £ p ,
implements a representation of the cyclic group Z/p s+1 of order p s+1 and a completely analagous
analysis of the corresponding representations can be made.
PROPOSITION 3.7
PROPOSITION 3.7
PROPOSITION 3.7
PROPOSITION 3.7
PROPOSITION 3.7: If p is a prime and N : Z/p GL(n , F) a faithful representation of the
cyclic group Z/p over a field F of characteristic p, then the quotient algebra F[V ] Z/p /Im(Tr Z/p )
is CohenMacaulay of Krull dimension dim F (V Z/p ).
PROOF PROOF PROOF PROOF PROOF: For p = 2 this is shown in [34] Theorem 1, so we suppose p is odd, and without
loss of generality that F = F p . Let
V = V n 1 Å · · · Å V n k
be a decomposition of V into irreducible Z/prepresentations. We know from [18] that
ht(Im(Tr Z/p )) = n - k
9
LARRY SMITH
so for the Krull dimension of the quotient algebra we have
dim F[V ] Z/p /Im(Tr Z/p ) = k .
By Proposition 3.3 we know the k elements
n 1 Ä · · · Ä 1 Ä d n i ,n i -1 1 Ä · · · Ä 1i = 1 , . . . , k o
are a regular sequence on H ev (n 1 , . . . , n k ) = F[V ] Z/p /Im(Tr Z/p ), so
hom--codim D * (n 1 ,..., n k ) F[V ] Z/p /Im(Tr Z/p ) ³ k .
Since
D * (n 1 , . . . , n k )/Im(Tr Z/p ) Ç D * (n 1 , . . . , n k ) F F[V ] Z/p /Im(Tr Z/p )
is a finite extension it follows from [3] (or the nonpreprint [32] which is available on request))
that
hom--codim F[V ] Z/p /Im(Tr Z/p ) ³ k .
Combining this with the general inequality [8] or [32]
k = dim F[V ] Z/p /Im(Tr Z/p ) ³ hom--codim F[V ] Z/p /Im(Tr Z/p )
yields the desired conclusion.
§4. Degree Bounds
In this section we combine the cohomological computations of Section 1 with results of [34] on
permutation representations to provide an upper bound on the degrees of algebra generators
for rings of invariants of Z/p in characteristic p.
To set the stage we suppose that F is a field of characteristic p and N : Z/p GL(n , F) a
faithful representation. If
V = V n 1 Å · · · Å V n k
dim F (V n i
) = n i , for i = 1 , . . . , k
is a decomposition of V into irreducible Z/prepresentations, and z 1 (i) , . . . , z n i
(i) is a Jordan
basis for V *
n i
for i = 1 , . . . , k, then n z i (j)i = 1 , . . . , n i , j = 1 , . . . , k o is a Dade basis for V * .
This means that the top Chern classes n c top z i (j)
i = 1 , . . . , n i , j = 1 , . . . , k o are a system
of parameters for F[V ], and since they are invariant, also for F[V ] Z/p . This may be verified
directly from [30] Proposition 5.3.7 (see also [25]).
A Jordan block of dimension m contributes 1 linear top Chern class (that of the fixed basis
vector) and m - 1 Chern classes of degree p (those of the remaining basis vectors, whose
Z/porbits have p elements) to a system of parameters. Let
D * (n 1 , . . . , n k ) := F h c top z i (j)
i = 1 , . . . , n i , j = 1 , . . . , k i .
This is a subalgebra of F[V ] Z/p which we will refer to as the Dade subalgebra associated
to the Dade basis (in this case a Jordan basis) n (z i (j)i = 1 , . . . , n i , j = 1 , . . . , k o . Note that
as an algebra D * (n 1 , . . . , n k ) is a polynomial algebra on linear forms and forms of degree p.
Since F[V ] is CohenMacaulay a short computation with Poincar’ e series shows that F[V ] is
generated as a D * (n 1 , . . . , n k )module by forms of degree at most (n - k) · (p - 1). The transfer
homomorphism Tr Z/p : F[V ] & F[V ] Z/p is a map of D * (n 1 , . . . , n k )modules, and therefore we
have shown:
10
INVARIANT THEORY AND THE KOSZUL COMPLEX
LEMMA 4.1
LEMMA 4.1
LEMMA 4.1
LEMMA 4.1
LEMMA 4.1: Let N : Z/p GL(n , F) be a faithful representation of Z/p over a field F of
characteristic p, V = F n , and D * H F[V ] Z/p the Dade subalgebra associated to a Jordan basis
for V * . Then Im(Tr Z/p ) F F[V ] Z/p is generated as a D * module, and hence also as an ideal, by
forms of degree at most (n - k) · (p - 1), where k = dim F (V Z/p ) is the number of Jordan blocks.
Recall from Lemma 3.1 we have an exact sequence
(C) 0 & H(n 1 , . . . , nm , . . . , n k ) i *
& H(n 1 , . . . , nm - 1 , . . . , n k ) d
& H(n 1 , . . . , nm , . . . , n k ) & 0 ,
where n 1 , . . . , n k are the dimensions of the Jordan blocks in a decomposition of V into irre
ducible Z/prepresentations. The map i * has degree 0 and the map d degree 1. Recall we
also have an exact sequence
(@) 0 & Im(Tr Z/p ) & F[V ] Z/p & H ev (n 1 , . . . , n k ) & 0
of D * modules, and know from [34] and [33] Proposition 4.4 that
H(p , . . . , p
!&&& k &&& ) @ F[c top (z 1 (j)j = 1 , . . . , k].
Hence H(p , . . . , p
!&&& k &&& ) has a single generator of degree 0 as a D * module. By downward induction
from this case we therefore obtain:
LEMMA 4.2
LEMMA 4.2
LEMMA 4.2
LEMMA 4.2
LEMMA 4.2: Let N : Z/p GL(n , F) be a faithful representation of Z/p over a field F of
characteristic p, V = F n , and D * H F[V ] Z/p the Dade subalgebra associated to a Jordan basis
n (z i (j)i = 1 , . . . , n i , j = 1 , . . . , k o for V * . Then H(n 1 , . . . , n k ) is generated as a D * module
by elements of degree at most k · (p - 1), and hence as an algebra by forms of degree at most
max p, k · (p - 1)
.
From these two lemmas we obtain our first degree bound.
THEOREM 4.3
THEOREM 4.3
THEOREM 4.3
THEOREM 4.3
THEOREM 4.3: Let N : Z/p GL(n , F) be a faithful representation of Z/p over a field F of
characteristic p, V = F n , and D * H F[V ] Z/p the Dade subalgebra associated to a Jordan basis
n (z i (j)i = 1 , . . . , n i , j = 1 , . . . , k o for V * . Then F[V ] Z/p is generated by forms of degree at
most (n - k) · (p - 1) as D * module and hence by forms of degree at most max p, (n - k) · (p - 1)
as Falgebra.
PROOF PROOF PROOF PROOF PROOF: We can certainly assume that none of the Jordan blocks are trivial, so for a Jordan
block of dimension m we always have m ³ 2 with the dimension of the fixed point set being 1.
Therefore
(n - k) · (p - 1) = (n 1 - 1) · (p - 1) + · · · + (n k - 1) · (p - 1)
³ (p - 1) + · · · + (p - 1) = k(p - 1)
and the result follows from the exact sequence (@) and the Lemmas 4.1 and 4.2.
The discussion provides a strategy for computing a set of algebra generators for F[V ] Z/p ,
namely, first by hook or by crook, compute the invariants up to degree k · (p - 1), where k is the
number of Jordan blocks, and then the image of the transfer through dimension (n - k) · (p - 1).
Among a vector space basis for this will be a set of algebra generators. Again, similar results
may be obtained for Z/p s , s V N, by working with Jordan blocks of size (s + 1) · p + r, where
0 £ r < p.
Here are some examples to illustrate how sharp the bound of 4.3 is in comparison to other
known upper bounds.
11
LARRY SMITH
EXAMPLE 1
EXAMPLE 1
EXAMPLE 1
EXAMPLE 1
EXAMPLE 1: Permutation Representations:
We suppose that X is a finite Z/pset, and for simplicity, this is no loss of generality, that Z/p
acts freely on X . Then ïXï = kp, for some k VN, and the upper bound given by the preceeding
theorem is k · (p - 1) 2 (since the dimension of the corresponding linear representation is kp
and that of the fixed point set k). For p = 2 this gives k as upper bound, which is sharp by
[26] and [27]. By comparison G˜ obel's bound for permutation representations [16] gives kp
2
which generally is larger than k · (p - 1) 2 .
EXAMPLE 2
EXAMPLE 2
EXAMPLE 2
EXAMPLE 2
EXAMPLE 2: A single Jordan block:
In this case 2 £ n £ p and, apart from some low dimensional cases, Theorem 4.3 gives (n - 1) ·
(p - 1) as an upper bound for algebra generators. For example, in the well studied example of
the submaximal Jordan block (i.e., n = p - 1) (see e.g., [4], [5], [1], [2], [19], and [13], to name
just a few references) where p = 5 and n = 4, we get 12 as an upper bound. Moreover, the
generators in degrees above 4 all lie in the image of the transfer. However, we will see shortly
that this bound can be reduced to 8 if we exercise a bit more care in how we choose a system
of parameters in the proof of Lemma 4.1.
The dominant factor in the upperbound estimate of Theorem 4.3 is the degree bound contained
in Lemma 4.1 for the ideal generators of Im(Tr Z/p ), which in turn depends on the obvious
upperbound for the degree of module generators of F[V ] as a module over the Dade subalgebra
D * (n 1 , . . . , n k ) H F[V ] Z/p . But, as the proof shows, we are free to replace D * (n 1 , . . . , n k ) by
any subalgebra S * H F[V ] Z/p that contains a system of parameters. So if we can find a better
system of parameters, then the proof of Theorem 4.3 will give a better estimate. Finding such
a better system of paramaters is facilitated by computations in [17] and [29].
We begin by considering the irreducible representation of Z/p over F = F p , V n , of dimension
n , 2 £ n £ p.
LEMMA 4.4
LEMMA 4.4
LEMMA 4.4
LEMMA 4.4
LEMMA 4.4: Let V n be the irreducible representation of Z/p over the field F of characteris
tic p > 2, implemented by the matrix J n V GL(n , F), and z 1 , . . . , z n V V * the dual to the Jordan
basis for V . Then the quadratic forms
Q i = z 2
i + 2 · 2 4
n-i
X j=1
(-1) j z i-j z i+j
3 5 n+1
2 £ i < n
are invariant, i.e., Q n+1
2
, . . . , Q n-1 V F[V n ] Z/p . (N.b. The condition on i is what assures that
the indices in the sum stay in the range 1 , . . . , n.)
PROOF PROOF PROOF PROOF PROOF: By direct computation we have
. (Q i ) = -2z i z i+1 + 2 2 4
n-i-1
X j=1
(-1) j+1 z i-j+1 z i-j + z i-j z i+ j +1 3 5 + 2(-1) n-i z 2i-n+1 z n ,
which upon rearranging terms gives
.(Q i ) = -2z i z i+1 + 2z i z i+1 +
2z i-1 z i+2 - 2z i-1 z i+2 +
. . .
2(-1) n-i+1 z 2i-n+i z n + 2(-1) n-i z 2i-n+1 z n
= 0
and the result follows.
The number of invariant quadratic forms found in Lemma 4.4 will be denoted by b(n). One
finds by counting that
b(n) = ( n
2 - 1 if n is even
n-1
2
if n is odd.
12
INVARIANT THEORY AND THE KOSZUL COMPLEX
According to [29] Theorem 3.2 there are elements of degree p - 1, f 2 , . . . , f n-b(n)-1 V F[V n ],
lying in the image of the transfer, and having leading monomials
z p-1
2 , . . . , z p-1
n-b(n)-1
respectively, with respect to the lexicographic order of monomials. (N.b. The indexing in [29]
is inverse to that employed here, i.e., if x 1 , . . . , x n are the variables from [29], the z i = x n-i+1
for i = 1 , . . . , n. So the use of reverse lexicographic ordering in [29] turns into lexicographic
ordering here.)
LEMMA 4.5
LEMMA 4.5
LEMMA 4.5
LEMMA 4.5
LEMMA 4.5: Let V n be the irredicible representation of Z/p over F = F p of dimension n,
2 £ n £ p. Then, the invariants
c t op (z 1 ) , f 2 , . . . , f n-b(n)-1 , Q n-b(n) , . . . , Q n-1 , z n V F[V n ] Z/p
form a system of parameters.
PROOF PROOF PROOF PROOF PROOF: If we look at the ideal of leading terms, I , of the polynomials
c t op (z 1 ) , f 2 , . . . , f n-b(n)-1 , Q n-b(n) , . . . , Q n-1 , z n V F[V n ] Z/p ,
we see that it contains
z p
1 , z p-1
2 , . . . , z p-1
n-b(n)-1 , z 2
n-b(n) , . . . , z 2
n-1 , z n ,
so the quotient algebra F[V ]/I is finite dimensional, since
F[z 1 , . . . , z n ] . z p
1 , z p-1
2 , . . . , z p-1
n-b(n)-1 , z 2
n-b(n) , . . . , z 2
n-1 , z n ,
already is. By Gr˜ obner basis theory [11], Chapter 5 Section 3 Proposition 4, the quotient
algebra
F[V ] . c t op (z 1 ) , f 2 , . . . , f n-b(n)-1 , Q n-b(n) , . . . , Q n-1 , z n
is also finite dimensional, and the result follows.
Let C be the subalgebra generated by
c t op (z 1 ) , f 2 , . . . , f n-b(n)-1 , Q n-b(n) , . . . , Q n-1 , z n V F[V n ] Z/p
A short Poincar’ e series computation shows that F[V ] is generated as a module over C by
forms of degree at most
(H) b p (n) := ( n
2
p - 1 if n is even
1 + n-1
2
p - 1 if n is odd.
Hence using the system of parameters
c t op (z 1 ) , f 2 , . . . , f n-b(n)-1 , Q n-b(n) , . . . , Q n-1 , z n V F[V n ] Z/p
in the proof of 4.3 leads to:
THEOREM 4.6
THEOREM 4.6
THEOREM 4.6
THEOREM 4.6
THEOREM 4.6: Let N : Z/p GL(n , F) be a faithful irreducible representation of Z/p over
a field F of characteristic p, 2 £ n £ p. Set V = F n . Then F[V ] Z/p is generated by forms of
degree at most
max p , ( n
2 p - 1 if n is even
1 + n-1
2
p - 1 if n is odd
! .
This result is indeed sharp, as comparison with the results of [2], [12], [29], and the references
to be found there, show. It extends to all finite dimensional Z/p representations in the same
way that the results of Section 3 follow from those of Section 2. The proof is left to the reader.
13
LARRY SMITH
COROLLARY 4.7
COROLLARY 4.7
COROLLARY 4.7
COROLLARY 4.7
COROLLARY 4.7: Let N : Z/p GL(n , F) be a representation of the group Z/p of prime
order p over the field F of characteristic p. Let V = V n 1
Å · · · Å V n k
be a decomposition of V as
Z/pmodule into a sum of irreducible Z/pmodules. Then F[V ] Z/p is generated as an algebra
by forms of degree at most
max 0 @ p ,
k
X i=1
b p (n i ) 1 A ,
where b p (n) is given by formula (H).
§5. Closing Comments
Recall that the Dickson algebra D * (n) over the Galois field F q with q = p n elements has a
fractal property:
F D * (n) = F(F[V ]) GL(n ,F q )
where V = F n
q and F : F q [V ] K is the Frobenius homomorphism f ) f q . From Proposition
3.3 and Corollary 3.5 we obtain the following partial answer to the question posed in the
introduction:
PROPOSITION 5.1
PROPOSITION 5.1
PROPOSITION 5.1
PROPOSITION 5.1
PROPOSITION 5.1: Let N : Z/p & GL(n , F p ) be a representation of the cyclic group of prime
order over the Galois field F = F q . Then there exists an integer s > 0 such that H 1 (Z/p ; F[V ]) @
H 2 (Z/p ; F[V ]) as modules over F s D * (n)
.
The bound of Theorem 4.6 can be used to start an iterative procedure that leads to a bound
for all finite p groups, and hence for all finite groups.
PROPOSITION 5.2
PROPOSITION 5.2
PROPOSITION 5.2
PROPOSITION 5.2
PROPOSITION 5.2: Fix an odd prime p, and let F be a field of characteristic p. There is an
integer valued function m p (n, a) of two integer variables n , a V N, such that: for any finite p
group P of order p a and any faithful representation N : P GL(n , F) the ring of invariants
F[V ] P is generated as an algebra by forms of degree at most m p (n , a).
The proof of the theorem will yield a rate of growth for m p (n , a) that is far too large to be of
any pratical use (ca. (n!) a-1 ). However, since the proof is not hard, here are the details.
NOTATION NOTATION NOTATION NOTATION NOTATION: If A is a graded, connected, commutative Noetherean algebra over F then
b(A) denotes the maximal degree of a generator in a minimal algebra generating set for A.
This is just the degree of the Poincar’ e series P(QA , t), where QA = F Ä A
•
A and •
A is the
augmentation ideal of A. (See e.g., [30] Chapter 4.)
LEMMA 5.3
LEMMA 5.3
LEMMA 5.3
LEMMA 5.3
LEMMA 5.3: Let A be a graded, connected, commutative, Noetherean algebra over the
field F and N : G Aut * (A) a faithful representation of the group G by grading preserving
automorphisms of A. Let W i denote the graded vector space with W i = A i and all other homo
geneous components 0. Then the natural map v i : S(W i ) & A is G equivariant, where S(W)
denotes the symmetric algebra on W , and, the induced map
b(A G )
Ä
i=1
v G
i :
b(A G )
Ä
i=1
S(W i ) & A
is an epimorphism.
PROOF PROOF PROOF PROOF PROOF: A G is finitely generated by [30] Theorem 2.3.1. Let b = b(A G ). Since A G is gen
erated by homogeneous elements of degree at most b it will be enough to show that
v G
i : S(W i ) G
i
& A i
is an epimorphism for i £ b. By the definition of the symmetric algebra functor S( ) we have
14
INVARIANT THEORY AND THE KOSZUL COMPLEX
an inclusion W i H S(W i ) i and by definition of v i the triangle
W i A
S i
S(W i )
commutes. Recall, per definition W i = A i , so taking homogeneous components of degree i and
passing to fixed point sets gives
W G
i = A G
i
S i
S(W i ) G
i
from which the lemma follows.
LEMMA 5.4
LEMMA 5.4
LEMMA 5.4
LEMMA 5.4
LEMMA 5.4: Let A be a graded, connected, commutative, Noetherean algebra over the field
F and N : Z/p Aut * (A) a faithful representation of the group Z/p by grading preserving
automorphisms of A. Then b(A G ) £ max d i i £ b(A) · p where d i = dim F (A i ).
PROOF PROOF PROOF PROOF PROOF: This follows from Lemma 5.3 and Theorem 4.3.
PROOF OF THEOREM 5.2
PROOF OF THEOREM 5.2
PROOF OF THEOREM 5.2
PROOF OF THEOREM 5.2
PROOF OF THEOREM 5.2: By induction on a V N. For a = 1 this is contained in Theorem
4.3. Assume the result has been established for a - 1 and that P has order p a . Choose a
maximal normal subgroup Q P of index p, so P/Q @ Z/p. Then F[V ] P = F[V ] Q
Z/p
. From
the induction hypothesis we then have b(A Q ) £ m p (n , a - 1). Since F[V ] Q
b
H F[V ] b , b VN, we
certainly have
dim F (F[V ] Q
b ) £ dim F (F[V ] b ) = n + b - 1
n - 1 b V N ,
so if we apply Lemma 5.4 to the action of Z/p on F[V ] Q we obtain
b(F[V ] P ) £ max n + b - 1
n - 1 b £ b F[V ] Q
· p £ n + m p (n , a - 1) - 1
n - 1 · p
and the result follows.
15
LARRY SMITH
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[26] D. R. Richman, Invariants of Finite Groups over Fields of Characteristic p, Adv. in Math. 124 (1996), 25--48.
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INVARIANT THEORY AND THE KOSZUL COMPLEX
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Larry Smith
Yale University
New Haven, CT, USA
and
# G˜ottingen, Germany
LARRY@SUNRISE.UNIMATH.GWDG.DE
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