Unspecified Journal
Volume 00, Number 0, Pages 000-000
S ????-????(XX)0000-0
INSEPARABLE EXTENSIONS OF ALGEBRAS OVER THE
STEENROD ALGEBRA WITH APPLICATIONS TO MODULAR
INVARIANT THEORY OF FINITE GROUPS II
MARA D. NEUSEL
Abstract.We continue our study*ofpthe_homological properties of the pure*
*ly
inseparable extensions H ,! P H of integrally closed unstable Noetherian
integral domains over the Steenrod algebra. It turns out that the*projec*
*tivep_
dimension of H is a lower*boundpfor_the projective dimension of P H . Fu*
*rther-
more, depth(H) depth( P H), where depthdenotes the depth. Moreover,
both algebras have the same global dimension. We apply these results to
extension F[Vo]G ,! F[V ]G of rings of invariants.
1.Introduction
Let H be a unstable reduced algebra over the Steenrod algebra of reduced powe*
*rs
P*. We denote the characteristic by p, and the order of the ground field F by
q. Recall that the Steenrod algebra contains an infinite sequence of derivatio*
*ns
iteratively defined as
P 1= P 1,
i-1 qi-1
P i= P i-1P q - P P i-1 for i 2.
We set
P 0(h) = deg(h)h 8h 2 H.
Note that P 0is not an element of the Steenrod algebra.
The algebra H is called P*-inseparably closed, if whenever h 2 H and
P i(h) = 0 8i 0,
then there exists an element h02 H such that
(h0)p = h.
p*__
The P*-inseparable closure of H is a P*-inseparably closed algebra P H con-
taining H such that the following universal property holds: Whenever we*havepa_*
*P*-
inseparably closed algebra H0containing H there exists an embedding P H ,! H0.
____________
Received by the editors May 24, 2007.
2000 Mathematics Subject Classification. Primary 55S10 Steenrod Algebra, Sec*
*ondary 13A50
Invariant Theory.
Key words and phrases. Inseparable Extensions, Inseparable Closure, Cohen-Ma*
*caulay, Pro-
jective Dimension, Depth, Steenrod Algebra, Invariant Theory of Finite Groups.
I would like to thank Lucho Avramov, Lars Christensen and Arne Ledet for man*
*y good
discussions.
Oc0000 (copyright holder)
1
2 MARA D. NEUSEL
In Section 4.1 of [5] an explicit algorithm to construct the inseparable clos*
*ure is
given. We collect known results in the following proposition.
Proposition 1.1. Consider the natural inclusion
*p__
OE : H ,! P H
of unstable reduced algebras over the Steenrod algebra. Then the following sta*
*te-
ments are valid:
*p__
(1) H is an*integralpdomain_if and only if P H is an integral domain.
(2) H ,! P H is an integral extension, and both algebras have the same Kru*
*ll
dimension. *p__
(3) If H is integrally closed, then so*ispP_ H.
(4) H is Noetherian if and only if P H is Noetherian.
If in addition H is Noetherian, then
(5) The_extension OE is finite. _____
*p_ __
(6) H is Cohen-Macaulay if and only if P H is Cohen-Macaulay, where -
denotes_the integral closure of -.___
*p_
(7) H is polynomial if and only if P H is polynomial.
Proof.For (1) - (3) see Proposition 4.2.1 in [5], for (4) see part (2) of Lemma*
* 4.1.3,
Lemma 4.2.2, Proposition 4.2.4, and Theorem 6.3.1 loc.cit., for (5) see Proposi*
*tion
4.2.4 [5] and [12]. Statement (6) was proven in1 [8]. Statement (7) was conject*
*ured
by C. W. Wilkerson around 1980, [12], and proven in [8].
In this paper we proceed with the investigation of the similarities of an uns*
*table
integrally closed algebra over the Steenrod algebra and its inseparable closure*
*. The
proofs of statements (6) and (7)*inpthe_above proposition led to the conjecture*
* that
H and its inseparable closure P H share all properties that have homological c*
*har-
acterizations, like, e.g., the depth, the projective dimension, the global dime*
*nsion,
and the Gorenstein property. In this generality this is not true. We illustrate*
* this
with the following two examples.
Example 1.2. Let F be a finite field of characteristic two. Consider the P*-pur*
*ely
inseparable extensions
F[x2, y2] ,! F[x2, y2, y3, x2y] ,! F[x2, y] ,! F[x, y].
All four algebras have Krull dimension two. Moreover F[x, y] is the P*-insepara*
*ble
closure of the other three. Their respective fields of fractions are
F(x2, y2) ,! F(x2, y) = F(x2, y) ,! F(x, y).
Thus all of them are integrally closed except for F[x2, y2, y3, x2y]. Observe t*
*hat all
of them have depth two, except F[x2, y2, y3, x2y] which has depth one. Thus we
see that the three integrally closed algebras are isomorphic as ungraded F-alge*
*bras
(even though not as algebras over the Steenrod algebra). However, we note that
F[x2, y2, y3, x2y] is not only not isomorphic to F[x2, y2], nor is it isomorphi*
*c to
F[x2, y], but also they do not have the same depth either.
Here is another example illustrating that we cannot expect good results for
algebras that are not integrally closed.
____________
1Note that the necessary assumption on H being integrally closed is missing *
*in that reference.
INSEPARABLE EXTENSIONS 3
Example 1.3. Consider the P*-purely inseparable extension
K = F[x2, y, xy] ,! H = F[x, y],
__
where |F| = 2. Then K = H, and its global dimension is
gl - dim(H) = 2.
However
gl - dim(K) = proj - dimK(F) = 1,
where proj - dimdenotes the projective dimension.
2. An Unstable Algebra and its Inseparable Closure
We assume from now on that H is an integral domain.
Proposition 2.1. Let H be an integrally closed unstable algebra over the Steenr*
*od
algebra. Then *p_
gl - dim(H) = gl - dim( P H ).
Proof.The global dimension of H is finite if and only if H is*apNoetherian_poly*
*no-
mial algebra. By Theorem 7.4 in [8] this is equivalent to P H being Noetherian
and polynomial. Thus the global dimensions of H and its inseparable closure are
simultaneously finite and equal to their common Krull dimension by Theorem 6.3.1
in [5].
We denote by H[p] H the subalgebra generated by the pth powers of elements
in H. The classical Frobenius map
H -! H[p], h 7! hp
provides us with an (ungraded) isomorphism between the two F-algebras.
Proposition 2.2. Let H be an integrally closed Noetherian integral domain. Then
the extension H[p],! H splits as a modules2 over H[p] P*.
Proof.Since H is Noetherian the extension H[p],! H is finite. Thus we can pick a
set of generators of H as a module over H[p], say t1, . .,.tk, and obtain
Xk
(?) H = H[p]ti.
i=1
By Proposition 5.1 in [7] we can choose the ti's to be Thom classes, i.e., for *
*all
j = 1, . .,.k
Xj j-1X j-1X
H[p]ti= H[p]ti= H[p]tj=( H[p]ti) \ H[p]tj)
i=1 i=1 i=1
is isomorphic to a suspension of an unstable cyclic module over H[p]. Without l*
*oss
of generality we can assume that t1 = 1. Consider the extension
Xk
IFIF(H[p]) ,! IFIF(H[p])ti,! IFIF(H).
i=1
____________
2The notation H[p] P*-module means that we are looking at modules over H[p]*
*that carry a
Steenrod algebra action, that is compatible with the Steenrod algebra action of*
* H[p].
4 MARA D. NEUSEL
Pk
We claim that IFIF(H) = IFIF(H[p])ti. To that end take an element h_k2 IFIF(*
*H)
i=1
with h, k 2 H. Then
h_= _1_hkp-1 = _1_kX h t = Xk hi_t 2 Xk IFIF(H[p])t ,
k kp kp i=1 i i i=1kp i i=1 i
for suitable hi2 H[p]. Since IFIF(H) is a finite dimensional vector space over *
*IFIF(H[p])
and {t1, . .,.tk} forms a spanning set we find a basis among it and obtain
Ml
IFIF(H[p]) ,! IFIF(H[p])ti= IFIF(H)
i=1
for some l k. By choice of the ti's we can rewrite this and obtain a direct s*
*um
decomposition as IFIF(H[p]) P*-modules as follows
_ l !
M Ml
IFIF(H) = IFIF(H[p])t1 IFIF(H[p])ti =IFIF(H[p])t1 \ IFIF(H[p])ti.
i=2 i=2
We take the unstable part of IFIF(H). By [4] we have that
H = Un(IFIF(H))
because H is assume to be integrally closed. Since Un commutes with direct sums
(see [9]) we obtain
_ l !
M
H = Un(IFIF(H)) = Un(IFIF(H[p])t1) Un IFIF(H[p])ti =IFIF(H[p])t1.
i=2
Since t1 = 1 and H[p]is integrally closed, we find Un(IFIF(H[p])t1) = H[p]Thus
_ l !
M
H = H[p]t1 Un IFIF(H[p])ti =IFIF(H[p])t1
i=2
as desired.
In Chapter 4 of [5] an explicite algorithm to construct the inseparable closu*
*re is
given. We recollect the few steps we need in what follows:
Denote by C(H) H the subalgebra consisting of the so-called P*-constants:
H[p] C(H) = {h 2 H|P i(h) = 0 8i 0} H.
Let {si, i 2 I} be a set of generators of C(H) as a module over H[p]. We adjoin*
* the
p-th roots of the si's and obtain
q __________________
H0= H[fl1, fl2, . .].= (flpi- si, i = 1, 2,.. .).
Set H = H0 and H0= H1. Then we define Hi= (Hi-1)0and we obtain an ascending
chain of unstable algebras
H = H0 ,! H1 ,! H2 ,! . ...
The P*-inseparable closure is then the colimit
P*p_H= colim
i{Hi};
INSEPARABLE EXTENSIONS 5
see Proposition 4.1.5 in [5]. Furthermore, for the corresponding fields of frac*
*tions
we have the following:
q __________________
IFIF(Hi+1) = IFIF(Hi)[fl1, fl2, . .].= (flpi- si, i = 1, 2,;. .).
see Proposition 2.4 in [8].
We note that C(H) is itself an unstable Noetherian integral domain over the
Steenrod algebra, see Lemmata 4.1.1 and 4.1.2 in [5] if H is. We need another
property of C(H):
Lemma 2.3. If H is an integrally closed integral domain then so is C(H).
Proof.Consider the commutative diagram
IFIF(C(H)),! IFIF(H)
[ [
C(H) ,! H.
Let c_d2 IFIF(C(H)), with c, d 2 C(H), be integral over C(H). Thus c_d2 IFIF(H*
*) is
integral over H. Since H is integrally closed we find that c_d= h 2 H. Thus c =*
* dh
and we have for all i that
0 = P i(c) = P i(dh) = P i(d)h + dP i(h) = dP i(h).
Since H is an integral domain we have P i(h) = 0 and thus c_d= h 2 C(H) as
desired.
Lemma 2.4. Let H be an unstable algebra over the Steenrod algebra. Then
C(H) = (H1)[p].
Proof.By construction the extension H ,! H1 is purely inseparable of exponent
one. Thus (H1)[p] H, and since this algebra consists of P*-constants we have
(H1)[p] C(H).
To prove the reverse inclusion note that every element in C(H) has a pth root in
H1, thus is contained in (H1)[p].
Theorem 2.5. Let H be an integrally closed Noetherian integral domain. Then
proj - dim(Hi-1) proj - dim(Hi) 8i,
where proj - dimdenotes the projective dimension.
Proof.Since Hi is an integrally closed integral domain whenever H is (see Lemma
2.2 in [8]) it is enough to show the statement for i = 1. We note that the proj*
*ective
dimension of H can be calculated by finding the projective dimension as a module
over a system of parameters, say S. Since H ,! H1 is finite S H1 is a Noether
normalization as well. Consider the following commutative diagram of S-module
6 MARA D. NEUSEL
homomorphisms and exact rows and columns
0 0
" "
0 ,! H=C(H) -! H=C(H) -! 0
" " "
0 -! H[p] ,! H -! H=H[p] -! 0
|| " "
0 -! H[p] ,! C(H) -! C(H)=H[p] -! 0
" " "
0 0 0
By Lemma 6.3 in [8] the algebras H and H[p]have the same projective dimension.
Set proj - dim(H) = proj - dim(H[p]) = d and proj - dim(C(H)) = t. Since H[p],!
H splits by Proposition 2.2, we read off the second exact row that proj - dim(H*
*=H[p])
d.
We want to show that d t. Assume to the contrary that d > t. We proceed by
depth chasing: The last row tells us that proj - dim(C(H))=H[p]= d + 1. Thus the
last column gives that proj - dim(H=C(H)) = d + 2. However the middle column
says proj - dim(H=C(H)) = d. This is the desired contradiction. Thus we have
proj - dim(H[p]) = proj - dim(H) proj - dim(C(H)).
To conclude the proof note that C(H) = (H1)[p]by Lemma 2.4 and thus
proj - dim(H) proj - dim((H1)[p]) = proj - dim(H1)
as claimed.
Corollary 2.6. Let H be a Noetherian integrally closed integral domain. Then
*p_
proj - dim(H) proj - dim( P H ).
Proof.Since H is Noetherian the chain of algebras
*p__
H = H0 ,! H1 ,! H2 ,! . .,.! Hr = P H
stabilizes at some r 2 N; see Theorem 6.3.1 in [5]. Furthermore, if H is integr*
*ally
closed, then so is Hifor all i; see Proposition 4.2.1 (5) in [5]. Thus the resu*
*lt follows
from the preceding by induction on r.
We have the following immediate corollary:
Corollary 2.7. Let H be a Noetherian*integrallypclosed_unstable integral domain
over the Steenrod algebra and let P H be its P*-inseparable closure. Then
*p_
depth(H) depth(Hi) depth( P H )
for all i.
p*__
Proof.Since H is Noetherian, the extensions H ,! Hi ,! P H is finite. Thus*ap_
Noether normalization S H of H is a Noether normalization for Hi and P H as
well. Thus the statement follows from the Auslander-Buchsbaum formula.
Remark 2.8. Note that the above result contains as a special case*Corollaryp2.2*
*_in
[6], where the above inequality was proven for Cohen-Macaulay P H.
INSEPARABLE EXTENSIONS 7
3.Applications to Modular Invariant Theory
Let ae : G ,! GL(n, F) be a faithful representation of a finite group over a *
*finite
field F. Denote by V = Fn the n-dimensional vector space over F, and by F[V ] t*
*he
symmetric algebra on the dual V *. The representation ae induces a linear actio*
*n of
G on F[V ]. Denote by F[V ]G F[V ] the subring of G-invariant polynomials. By*
* the
Galois Embedding Theorem an integrally closed P*-inseparably closed Noetherian
unstable integral domain over the Steenrod algebra is such a ring of invariants
F[V ]G for a suitable representation ae of some group G, see [1] and Theorem 7.*
*1.1
in [5].
Let V = W0 . . .We be a vector space decomposition. Set
e]
F[Vo] = F[W0] F F[W1][p] F . . .FF[We][p .
The generalized Galois Embedding Theorem states that H is isomorphic to F[Vo]G
as an algebra over the Steenrod algebra for some suitable flag Vo, group G, and
representation ae if and only if H is an integrally closed Noetherian unstable *
*integral
domain over the Steenrod algebra, see [12] and Theorem 5.2 in [8]. Furthermore
we have a commutative diagram
H = F[Vo]G ,! F[Vo]
# # ,
P*p_H= F[V ]G,! F[V ]
where the horizontal inclusions are Galois and the vertical inclusions are pure*
*ly
P*-inseparable.
In Proposition 6.4 and Theorem 7.4 in [8] we saw that F[Vo]G and F[V ]G are
simultaneously Cohen-Macaulay, or polynomial. Based on the results of the pre-
ceding section we can add to that list the following properties.
Theorem 3.1. Let F[Vo]G ,! F[V ]G be an extension of rings of invariants, where
Vo is a G-flag in V , and F is the prime field of characteristic p. Then
(1) F[Vo]G and F[V ]G have the same global dimension,
(2) proj - dim(F[Vo]G ) proj - dim(F[V ]G ), and
(3) depth(F[Vo]G ) depth(F[V ]G ).
Proof.The first statement follows from Proposition 2.1. The second statement
follows from Corollary 2.6 and the last from Corollary 2.7.
Remark 3.2. We note that the preceding result can be refined for the extension
F[V?]G ,! F[Vo]G ,
where V? Vo denotes a subflag. We find
(1) F[Vo]G and F[V?]G have the same global dimension by Proposition 2.1,
(2) proj - dim(F[V?]G ) proj - dim(F[Vo]G ) by Theorem 2.5, since F[Vo]G =
(F[V?]G )i for some i 0, and similarly
(3) depth(F[V?]G ) depth(F[Vo]G ) by Corollary 2.7.
8 MARA D. NEUSEL
We note that in the case of the preceding result the image of G under ae nece*
*ssarily
consists of matrices of the form
2 3
A0 0 . . .0
66* A1 0 . . .0 7
66 .. ..77
66 * . . 777
4 . . . ... 0 5
* . . . * Ae
where Ai is an invertible nix ni-matrix with ni= dim(Wi).
Proposition 3.3. If ae(G) consists of matrices of block diagonal form
2 3
A0 0 . . .0
660 A1 0 . . .0 7
66 .. ..77
66 0 . .777,
4. . . ... 0 5
0 . . .0 Ae
*p__
then H and P H are ungraded isomorphic.
Proof.Consider the (ungraded) isomorphism
j
OE : F[V ] -! F[Vo], xi7! xpi
for xi 2 F[Wj] as basis element. Since G acts on F[Wj] for all j, the map OE
commutes with the group action. Thus the result follows.
Remark 3.4. Obviously the preceding result remains true over any field of finite
characteristic.
We want to illustrate these results with an example taken from [8]; see Examp*
*le
7.6 loc.cit.
Example 3.5. Let p be odd, and F a field of characteristic p. Consider the four
dimensional modular representation Z=p ,! GL(4, F) afforded by the matrix
2 3
1 1 0 0
660 1 0 077
40 0 1 15 .
0 0 0 1
Its ring of invariants turns out to be a hypersurface
F[x1, y1, x2, y2]Z=p= F[c1, y1, c2, y2, q]=(r),
where ci = xpi- xiyp-1iare the top orbit Chern classes of xi, i = 1, 2, and
q = x1y2 - x2y1 is an invariant quadratic form. The relation is given by
r = qp - c1yp2+ c2yp1+ qyp-11yp-12,
see Theorem 2.1 in [3]. Certainly, Z=p acts also on F[x1, y1] F[xp2, yp2] and*
* we find
that
F[x1, y1] F[xp2, yp2] Z=p= F[c1, y1, cp2, yp2, q0]=(r0),
INSEPARABLE EXTENSIONS 9
2 p p p-1 p(p-1)
where q0= x1yp2- xp2y1 and r0= (q0)p- c1yp2+ c2y1 - q0y1 y2 . We note that
the two rings are isomorphic, but not graded isomorphic, nor (in the case of a *
*finite
ground field F) isomorphic as algebras over the Steenrod algebra.
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Department of Mathematics and Statistics, MS 1042, Texas Tech University, Lub-
bock, Texas 79409
E-mail address: Mara.D.Neusel@ttu.edu