Rings of Invariants
and
Inseparable Forms of Algebras
over
the Steenrod Algebra
By Clarence W. Wilkerson
Purdue University
Dedicated to the memory of J.F. Adams
The concept of the maximal torus plays a key role in the classifica-
tion of compact Lie groups. Likewise the role of the cohomology of
the classifying space BT n of the n-torus in the study of characteristic
classes has been long understood via the splitting theorems. With the
work of Adams-Wilkerson (1) it was realized that the cohomology rings
H* (BT n; Fp) as algebras over the Steenrod algebra Ap were universal
in a strong technical sense. This feature was presaged by Serre (13),
Quillen (11), and Wilkerson (14), and is greatly extended by later work
of Carlsson, Miller, Lannes, Zarati, Schwartz, and Dwyer-Wilkerson.
The Adams-Wilkerson work provides an algebraic analogue of the ex-
istence of a maximal torus, in a suitable category of algebras with an
action of the Steenrod algebra. More precisely, it provides for each
graded Fp- algebra R* which is an integral domain of finite transcen-
dence degree n over Fp and equipped with an "unstable" action of the
mod p Steenrod algebra Ap , an embedding
tR* : R* - ! H* (BT n; Fp)
which is an Ap -map of algebras and is an invariant of R* and its Ap -
action. Furthermore, H* (BT n; Fp) is algebraic and normal over R*, via
the morphism tR* .
In this paper, we denote H* (BT n; Fp) together with its Ap -action as
S[V ], the symmetric algebra on a n-dimensional Fp-vector space V;
for V concentrated in degree 2 (or degree 1 for p = 2).
The embedding tR* has additional properties, depending on the prop-
erties of R* as an algebra over the Steenrod algebra:
______________________________________
The author was supported in part by the National Science Foundation, the Wayne
State Fund, and the Institute for Advanced Studies of Hebrew University.
2 Clarence W. Wilkerson
a) S[V ] is integral over R* via tR* if and only R* is a finitely generated
Fp-algebra.
b) tR* is a separable extension on the fraction field level if and only if
the Milnor primitives {P i } in Ap give n linearly independent deriva-
tions on the graded fraction field F R* of R* (as they do on S[V ]). Equiv-
alently, (F S[V ])p \ F R* = (F R*)p, if and only if the field extension is
separable. *
c) There is a subgroup W (R*) of GL(V ) such that R* = S[V ]W (R ) ,
if and only if both conditions a) and b) hold, and, in addition, R* is
integrally closed in its field of fractions.
That is, under some general conditions, R* is a ring of invariants.
Conversely, any ring of invariants has the properties properties a)-c).1
The three conditions required to ensure that R* is a ring of invariants
may seem to lack topological motivation. However, the noetherian and
integral closure requirements together have an intrinsic meaning in terms
of the Steenrod algebra action:
Theorem I. If R* is a graded integral domain of finite transcendence
degree with an unstable Ap -action, then R* is integrally closed and
finitely generated if and only if R* = U n(F R*), the set of unstable
elements in the graded field of fractions of R*. If we consider R* as a
subring of S[V ] via tR* , then U n(F R*) = S[V ] \ F R*.
Interpretation of the separability condition is harder, but algebraic
examples at all primes and topological examples at the prime 2 show that
it is a necessary complication. The inseparable forms portion of the title
refers to a construction given here by which inseparable examples are
made from separable examples. The goal is an algebraic reformulation
of the separability condition so that it can be used more readily with
topological techniques such as "mod p Hopf invariant one", for odd
primes. Roughly stated, our aim in Theorem II below is to show that
any R* with R* = U n(F R*) is a ring of invariants, not necessarily of
S[V ] itself, but of a certain "diagonal" subalgebra D* (R*) of S[V ].
The definition of this D* (R*) is motivated by an analogy to elementary
field theory. Recall that if K ! L is a finite normal field extension, then
there are two ways of decomposing this into sub-extensions. First, one
could find the maximal inseparable extension J of K in L, so K ! J !
______________________________________1
We ask the reader to note that in (1), the statement of Theorem 1.2 should inc*
*lude
the hypothesis that H* is noetherian. This hypothesis was used but was inadvert*
*ently
omitted. There is also an oversight in the statement of Theorem 1.9 - in order *
*to prove
that condition 1.2.2 is necessary, H* should be assumed also to be integrally c*
*losed.
An alternate formulation is to assume that condition 1.2.2 holds for all elemen*
*ts of
the fraction field of H* , as stated above in part b).
Rings of Invariants and Inseparable Forms *
* 3
L: Then J ! L is Galois, with Galois group W = AutK (L) = AutJ (L),
so that J = LW . This is roughly the procedure in (1), with allowances
made there for ambient Steenrod actions. A second method is more
important for this paper - form the maximal separable extension I of
K in L , where K ! I ! L. Here K ! I is separable, and J ! L is
purely inseparable. Notice that K is the intersection of LW with I, or
K = IW , see for example (8),Chapter VII, Section 7, Prop. 12.
Our Theorem II provides a ring level description of this process of tak-
ing a maximal separable extension. Somewhat surprisingly, the analogue
of the maximal separable field extension I above is greatly restricted by
the existence of the Steenrod algebra action. This restricted structure
of D* (R*) is important for applications, (2),(10):.
Theorem II. Let R* be a sub-Ap -algebra of S[V ] such that F R* !
F S[V ] is algebraic. Define a filtration {Ui(R*)} of V by the rule that v
is in Ui(R*) if and only if vpi is separable over F R*: Let D* (R*) be the
"diagonal " subalgebra of S[V ] generated by {Ui(R*)pi}. Let I* be the
maximal separable graded field extension of F R* in F S[V ]. Then
1) R* ! D* (R*) is separable and D* (R*) ! S[V ] is purely insepara-
ble. i
2) D* (R*) = U n(I* ) and Ui(R*) = {v|vp 2 I* }:
3) D* (R*) i(S[Ui(R*)=Ui-1 (R*)])pi as Ap -algebras.
4) AutR* (S[V ]) = AutR* (D* (R*)) = AutF R* (F S[V ])
5) For W (R) =*AutR* (S[V ]); W (R)(Ui(R*)) Ui(R*):
6) D* (R*)W (R ) = U n(F R *):
7) D* (R*) is the smallest Ap -sub-Hopf algebra of S[V ] containing
R*.
Note that although V has a filtration by the W (R*)-vector spaces
{Ui(R*)}, the filtration need not split as W (R*)-modules. That is, V
need not be isomorphic to Ui+1 =Ui as W (R*) vector spaces.
In summary, U n(F R*) is explicitly determined by two pieces of data:
1) The subgroup W (R*) of GL(V )
2) The W (R*) filtration {Ui(R*)} of V; together with the implicit
exponents pi:
This leads to a constructive classification of all noetherian integrally
closed unstable domains. In Section Five we give some applications of
this classification.
These methods also answer a question of Mitchell and Stong:
Theorem III. If the polynomial algebra R* = Fp[y1; : : :; yn ] has an
unstable action of Ap ; then the [A-W] embedding tR* factors through
D* (R*) Fp[z1; : : :; zn ] where |zi| = 2pNi : Since R* ! D* (R*) is
4 Clarence W. Wilkerson
separable, the Jacobian | @yi=@zj | is nonzero for homogeneous algebra
bases {zi} of D* (R*) and {yj} of R*.
Theorem II does not directly answer the problem of realizing these
rings as the cohomology rings of topological spaces, but it does ex-
press the maximum algebraic content obtainable from the Ap -action.
Dwyer-Miller-Wilkerson (2) have used Theorem II together with appli-
cations of the Sullivan Conjecture technology of Miller and Lannes to
prove very strong uniqueness and non-realizability results for classifying
spaces. Theorem IV below from (2) is a sampler that points out that
separability is forced by topological considerations.
Theorem IV. (2) If X is a simply connected CW complex such that
R* = H* (X; Fp) is a integrally closed finitely generated graded integral
domain, then for p > 2
*)
H* (X; Fp) S[V ]W (R :
Furthermore, the homotopy type of X determines a lift of the inclusion
of W (R*) ! GL(V ) up to the general linear group of rank n over the
p-adic integers.
The proof of Theorem IV is beyond the scope of this paper, but we
give here just a brief hint as to the role that Theorem II plays in (2). If
R* = H* (X), then by a fundamental result of Lannes, the embedding
tR* : R* ! S[V ] ! H* (BV ) can be topologically realized by a con-
tinuous function f : BV ! X. The component of the function space
M ap(BV; X)f has cohomology TfV*(R*) = D* (R* ), where TfV*(_) is a
summand of the T -functor of Lannes. Accepting this, "mod p Hopf in-
variant one" shows that the exponents in the definition of D* (R*) are all
zero, since it is the cohomology of a space. Thus M ap(BV; X)f BT n.
That is, (2) carries out the Adams-Wilkerson program on the space level
and obtains new information about separability and lifting of automor-
phism groups to characteristic zero.
There is another approach to the "inseparability" questions treated
in this paper. In [Quillen, (11)], the concept of an F -isomorphism pro-
vided a convenient treatment of p-th powers. Quillen's main theorems
have been treated more abstractly (for unstable algebras rather than
just equivariant cohomology) in work of [Rector , (12)] and [Lam, (7).
We recall the principal technical result of (7), and reinterpret it in terms
of Theorem II.
Theorem V. 1) (7) Let R* be a sub-Ap -algebra of S[V ] such that
S[V ] is algebraic over R*. Suppose in addition R* is p-closed, that is
Rings of Invariants and Inseparable Forms *
* 5
(S[VQ])p \ R* = (R*)p. Then the top Dickson invariant element c0 =
V -{0} v lies in R*.
2) If R* ! S[V ] is algebraic but not necessarily p-closed, then there
exists N 0 such that (c0)pN 2 R*.
N
3) If z = cp0 is inverted to form S-1zR*, then U n(F R*) = U n(S-1zR*):
Part 3) of Theorem V combined with the "localization" invariance
properties of the Lannes T -functor established in (2), gives the com-
putation that each component of Lannes T -functor corresponding to a
monomorphism : R* ! H* (BV ) is D* (R*), even in the absence of
the noetherian or integrally closed requirements on R*. Of course, if we
add the topological input that R* = H* (X) for some space X, then by
(2), in fact D* (R*) = H* (BT n) for some n:
The author would like to thank S. Mitchell and R. Stong for their
interest and correspondence on this problem. The statement of Theorem
III is due to them, and their questions led to the formulation of Theorem
II from an earlier version of the results of Section Two. The author would
also like to thank J.F. Adams for helpful comments on the organization
of the proofs.
x1. The Maximal Separable Extension
Our technique is to apply field level arguments, and then use the
U n-functor of taking the subalgebra of unstable elements to recover
ring level results. A form of U n-functor appeared in [(15),Wilkerson].
It has also found use in (1), (3), and (4).
Proposition 1.1. Let M *be an evenly graded module over Ap . Define
the graded vector space U n(M *) as
U n(M *)2i = {m 2 M2i|P jm = 0; 8j > i}:
Then
a) U n(M *) is closed under the Ap - action on M *, and is an unstable
Ap -module.
b) If M * is a localization of some unstable Ap -algebra R*, then
U n(M *) is an unstable Ap -algebra.
c) U n(F S[V ]) = S[V ]
d) If R* is an unstable integral domain, then u 2 F R* and u integral
over R* implies that u 2 U n(F R*):
Proposition 1.2. If R* is a graded unstable integral domain of finite
transcendence degree with an unstable Ap -action, then R* is integrally
6 Clarence W. Wilkerson
closed and finitely generated as an algebra if and only if R* = U n(F R*);
the set of unstable elements in the graded field of fractions of R*.
Proposition 1.3. Let R* be an Ap -sub-algebra of S[V ] such that S[V]
is algebraic over R*. Define S*R as the set of elements in S[V] which are
separable over R*. Then S*R is a Ap -sub-algebra of S[V] which contains
R*. In addition,
1) The fraction field F S*R of S*R is the maximal subfield I* of the
fraction field of S[V] which is separable over R*:
2) R* ! S*R is separable, and S*R is maximal for this property.
3) S*R = U n(F S*R) = U n(I* ) and hence is integrally closed and
noetherian.
4) S*R ! S[V ]sis purely inseparable ( for each x in S[V] there exists
an s such that xp 2 S*R ).
Proof of Prop. 1.3.
Let I* be the maximal separable extension of F R* in F S[V ]. Then
S*R = U n(I* ), since U n(F S[V ]) = S[V ]. It follows that S*R is maximal
among subalgebras of S[V ] separable over R*. By I.1, S*R is integrally
closed and noetherian, since S*R = U n(F S*R). Now if u 2 S[V ], there
exists N 0 so that upN 2 I* . Since this power of u is unstable, it is
actually in S*R.
It remains to show that F S*R = I* . Let u in F S[V ] be separable over
R* , with minimal separable equation g(u) = 0 = a0uN +: : :+aN with ai
in R*: Multiply by aN-10 to obtain an integral equation for a0u = x over
R*: Then x is in S[V ], since it is integral over R*. Also, x is separable
over R*. Hence x 2 U n(I* ) = S*R. Thus putting y = a0, we obtain
u = x=y, with x and y in S*R. That is, F S*R = I* .
Proof of Lemma 1.2.
The favor of these arguments is similar to those in (15). From (1),
R* is noetherian if and only if S[V ] is integral over R* via tR . Also,
since U n(F S[V ]) = S[V ], for any subfield L*, U n(L*) = L* \ S[V ].
Now assume R* = U n(F R*). If u 2 F R* is integral over R*, then
u 2 U n(F R*) = R*. That is, R* is integrally closed. Next if v 2 V has
x = vpN separable over F R*, x satisfies a minimal equation of the form
Xr + c1Xr-1 : :+:cr = 0
with the coefficients {cj} in F R*. But since the extension is normal,
and has one root x in S[V ] , it factors completely, with all roots in S[V ].
Hence the coefficients {ci} are unstable, since they are polynomials in
Rings of Invariants and Inseparable Forms *
* 7
the roots. Thus x is integral over U n(F R*) = R*, V is integral over
R*, and so is S[V ]. From property a) of the introduction, R* is finitely
generated.
Conversely, assume that R* is noetherian and integrally closed. We
need to show that R* = U n(F R*). Let u 2 U n(F R*). Then u 2 S[V ],
and hence is integral over R* since R* is noetherian. Hence u is in the
integral closure of R* in its field of fractions. But we assumed that R*
is integrally closed, so u 2 R*. That is, U n(R*) R*.
Proof of Proposition 1.1.
If the action of the Bockstein were non-zero, the appropriate definition
of an "unstable" element would be more complicated. The concept of
"unstable" involves recursion, since if u is unstable, one also wants u to
be unstable, for any 2 Ap . However, Lemma 2.6 from (1), shows that
this recursion is automatic if only the reduced power Steenrod operations
are considered. If M * is an algebra over Ap , then U n(M *) is also, and
of course, U n(M *) is also an unstable module. However, it need not be
an unstable algebra, since this requires also the condition P im2i = mp
for each m. Although this is true for case b) by (15), it is not true in
general.
Finally, let u = x=y in F R*. Let
uN + c1uN-1 : : :c0 = 0
be the monic equation of integrality, so that ci 2 R*: Apply the total
Steenrod operation PT to the equation. Multiply through by PT (y)N ,
and compare coefficients of powers of T . One sees that the coefficients
for PT (u) vanish above half the dimension of u. A form of this argument
appears in (15).
x2. Jacobson Differential Correspondence
The proof of Theorem II requires a study of the subfields intermediate
between the fraction field F S[V ] of S[V ] and some ps-th ower of this
field. The setting of inseparable field extensions has has a rich algebraic
structure, and there is a large established theory, see [(16), Chapters
5 and 6] for a survey. Let K ! L be an inseparable extension - the
theory revolves around the correspondence between the subfield K and
the subring EndK (L) of additive functions from L into itself which are
K-linear. In the case of our interest L = F S[V ]; the L-span of the
Steenrod operations form a large part of the endomorphism ring. One
in principle could work through the general theory, and prove Theorem
II by characterizing the endomorphisms of S[V ] which are linear over R*
8 Clarence W. Wilkerson
as those which are linear over the diagonal algebra D* (R*): This would
be in the spirit of (1), section 5, and, indeed was the original intent.
However, we can apply a simpler theory. The general approach to
inseparable Galois theory sketched above was designed to generalize to
larger exponents the exponent 1 correspondence of Jacobson, [(5), Vol-
ume III, Chapter 4, page 186]:
Jacobson Differential Correspondence. Let L be a field of char-
acteristic p. There is a 1-1 correspondence between
1) subfields K of L which contain Lp and which have finite codimension
in L ( dimK (L) < 1 ),
and
2) finite dimensional L-subspaces of Der(L) which are closed under
commutators and p-th powers (L-restricted Lie subalgebras of Der(L)).
The correspondence is
K DerK (L)
and
B LB ;
the constant field of B. Here dimK (L) = pdimL DerK (L)
The present proof of Theorem II uses an induction step provided by
the treatment of p-th powers by the Steenrod algebra, together with a
version of the results of section 5 of (1), interpreted by the Jacobson
Differential Correspondence. Recall that the Milnor primitive {P (i) }
from Ap acts as a derivation of degree 2(pi - 1) on any algebra over Ap
e.g. (1).
Proposition 2.1. Suppose S[V ]pN ! R* ! S[V ], are monomorphisms
of unstable Ap -algebras, for some N 0, and that R* = U n(F R*):
Define M *= R* \ (S[V ])p, so (F R*)p ! F M *! F R*. Then
1) The Milnor primitives {P (i) } span the graded derivations of F R*
into F R* which vanish on F M *, DerF M* (F R*).
2) RankF R* SpanF R* {P (i) } = rankFp (V \ R*) = nR .
3) dimF M* (F R*) = pnR
The Ap -action is crucial at this point. For example, consider the
elements x = t21; y = t22; u = t1t2 in F2[t1; t2]. Let R* be the sub-algebra
generated by these elements. Then R* is not closed under the Steenrod
operations, but it is finitely generated and integrally closed. However,
V \ R* = 0 even though rankF R* DerF M* (F R*) = 1. In general, there
are many more intermediate rings between S[V ]p and S[V ] than those
predicted by Proposition 2.1. The Steenrod action forces a more "linear"
structure than the general theory can see.
Rings of Invariants and Inseparable Forms *
* 9
Lemma 2.2. If p = 2 or we restrict to evenly graded objects, then
given an Fp-restricted graded Lie algebra L and an action of L on a
commutative graded field K, the tensor product K Fp L is a graded
K-Lie algebra closed under brackets and p-th powers, and it inherits
an action on K.
Proof of Prop. 2.1.1.
By 2.2, the F R*-span of {P (i) } in Der(F R*) is a F R*-restricted Lie
subalgebra. Since by (1) the Milnor primitives span the derivations of
F S[V ], an element of F S[V ] is a p-th power in F S[V ] if and only if it is
annihilated by each P (i) . That is, the constant field of the F R*-span
of {P (i) } acting on F R* is F M *, the intersection of F R* with the
p-th powers from F S[V ]. By the Jacobson Correspondence, the Milnor
primitives span DerF M* (F R*).
Proof of Proprositions 2.1.2 and 2.1.3.
By 5.1 of (1), there exists for R* an nR such that any distinct nR
of the {P (i) } are linearly independent over F R* in Der(F R*), and
any nR + 1 are linearly dependent. This holds even if the "grading"
derivation P"0 is included. Here P"0x2n = nx. Hence
RankF R* SpanF R* {P (i) } = nR :
For any non-trivial equation of linear dependence with nR + 1 terms
ffi = c0P"0 + : :+:crP (nR )
all of the coefficients {ci} are non-zero, since otherwise the choice ofrnR
would be contradicted. If v 2 V0 = V \R*, then ffiv = 0 = c0v+: :+:crvp
for r = nR . Hence dimFp V0 rnR . However, by 5.2.(i-ii) of [A-W], all
solutions of {c0X + : :+:crXp } are in S[V]. Hence dimFp V0 = nR , since
the equation has no repeated roots.
Proof of 2.2.
One takes the natural bracket definition :
[aD; bffi] = ab[D; ffi] + aD(b)ffi - bffi(a)D
and extends by bilinearity, keeping the action of L on K in mind. The
p-th power operation requires more work. In fact there are two non-
trivial formulas which are needed:
1) Given a 2 K and D 2 L, there exist b; c 2 K so that (aD)p =
bD + cDp .
10 Clarence W. Wilkerson
This is attributed to Hochshild in exercise E.5.10, page 125 of (16).
This allows us to define the p-th power on monomials from K L.
2) If a; b 2 K and D; ffi 2 L; we need a formula for (aD + bffi)p in
terms of brackets and p-th powers. Then induction on the number of
summands in aiDi provides a general definition of the p-th power map
on K L. The needed formula is provided in [ Jacobson, (6), page 187]
:
In the free associative algebra Fp < X; Y >,
p-1X
(X + Y )p = Xp + Y p + Si(X; Y )
1
where the Si are (non-commutative) polynomials expressible in terms of
iterated commutators in X and Y .
Notice that even if the original Lie algebra L had zero p-th powers
and zero Lie brackets ( as in the case of application to the Lie algebra
spanned by the {P (i) }, the same is not necessarily true for the semi-
tensor product Lie algebra constructed via these formulas.
Comment.
If the reader prefers an absolutely minimal path to the proof of The-
orem B, the crucial fact established in this section is that
*)
dimF M* (F R*) = pdimFp (V \R :
This can be deduced in the case above from a result of Gersten-
haber and Zaromp quoted in exercise E.5.9 of (16), using the derivations
{P (i) } for their {Di}. The critical property needed is closure under the
p-th power map, and this is clear for the {P (i) }. Note that there is an
slight error of the statement of the result of Gertenhaber and Zaromp
in both their paper and in (16).
We now give a more precise statement of what should be the outcome
of working through the Hopf-algebra treatment of modern Galois theory
for this case:
Conjecture 2.3. Let be the left F S[V ]- span of Ap . Then is a
Hopf-algebra in the sense of [Winter, Chapter 6 ]. There is a natural
representation into End(F S[V ]), with image . Let R* denote the
subset of elements which are R*-linear. Then
1) R* is a Hopf-algebra in the sense of (16).
2) F D* (R*) is the field of constants for R* , and
R* = EndF D*(R*) (F S[V ]):
Rings of Invariants and Inseparable Forms *
*11
x3 Proof of Theorem II in the purely inseparable case
By the results of Section One, we can replace the arbitrary R* by
its maximal separable closure SR* . This leaves some details about the
automorphism groups to be sorted out in Section Four, but the harder
work is here, to show that if the extension R* ! S[V ] is purely insepa-
rable and R* = U n(F R*), then R* = D* (R*). The method of proof is
an induction using the results of Section Two.
Recall that the exponent of an inseparableeextension is the smallest
integer e such that for any x in F S[V ], xp is in F R*:
The induction will be on the exponent e(R) of the inseparable exten-
sion F R* ! F S[V ]: The exponent 0 case is covered by the hypothesis
R* = U n(F R*), while the exponent 1 case is essentially Proposition 2.1.
Exponent 1 Case: (F S[V ])p ! F R* ! F S[V ]. The filtration is
U0(R*) = V \ R* and U1(R*) = V . We have
(S[V ])p ! D* (R*) ! R* ! S[V ]:
But by Proposition 2.1, the dimension of F *R* and that of F D* (R*)
over F S[V ]p are each pnR . Hence the two fields coincide.
Special Case: If R* is entirely containedpin_S[V_]p,pwe_can_replace R*
by thepalgebra_of itspp-th_roots in S[V ], p R* : e( p R* ) =p e(R)_- 1
and pp R*__=_U nF p R*0p_so_the induction hypothesis gives thatp p_R*_ =
D* ( p R*) . Since ( p R* )p = R* as Ap -algebras, R* = (D* ( p R* ))p =
D* (R*).
Induction Step: If V0 = V \R* 6= 0; then we use R*00= R*\S[V ]p: We
need to show first that R* 00! D* (R* ): Now 1 e(R00) max(e(R); 1).
If it is strictly0less0than0e(R),0we appeal to the induction hypothesis to
conclude that R* = D* (R* ) If the exponent is still e(R), we obtain
the0same0isomorphism from the special case conclusion. Thus we obtain
R* ! D* (R*). Then we have
p *00 * * *
R* ! R ! D (R ) ! R ;
and we must show that D* (R*) = R*:
We again appeal to Proposition 2.1. Since U0 = V \ R* = V \ D* (R*)
by definition of D* (R*), we have
dimF R*00(F D* (R*)) = dimF R*00(F D* (R*)):
Thus F D* (R*) = F R* and R* = D* (R*).
12 Clarence W. Wilkerson
Comments.
The use of the Jacobson Correspondence in Proposition 2.1 to compute
the relative derivations DerF R*00(F R*) is very helpful. If the full Lie
algebra of derivations Der(F R*) were known, one could seek to explicitly
compute DerF R*00(F R*): However, while it is true that Der(F R*) can
be spanned by linear combinations of Steenrod operations, more than
just the Milnor primitives may be required in general.
For example, if
R* = F2[t1; t22] = F2[x; y]
then the standard partial derivative basis for Der(F R*) is given by
@=@x x-2 Sq1
@=@y = (y4 + x4y2)-1 (Sq(0;2)- x4Sq2) :
in terms0of0Steenrod operations. In this case, the constant field is
(F R*)p; R* = F2[x2; y], and
DerF R*00(F R*) = F R*{Sq1} = F R*{@=@x};
by direct computation. However, by Proposition 2.1 one achieves the
same result by observing that with V \ R* = {0; x}, any single Milnor
primitive spans the relative derivations, since it is non-zero on x. Such
direct computation has drawbacks for more complicated examples.
x4 Proofs of Theorems II, III and V
We are left to provide proofs for those statements in Theorem II
involving the automorphism groups and the Hopf-algebra structure of
D* (R*).
It is not obvious that the automorphisms computed in various possible
ambient categories should agree. The essential point is that these are
relative automorphisms that do preserve all structures on a large sub-
object. Again these arguments are similar to those in (15).
Proposition 4.1. Let R* be a sub-Ap -algebra of S[V ] such that S[V ]
is algebraic over R*. Let L* be a graded field in F S[V ]. Then
1) If i : F R* ! L is a separable extension of graded fields, then L
has a unique Ap action respecting the Cartan formula and such that i
is a map of Ap -algebras. Thus any automorphism of L* which is the
identity on R* respects this Ap -structure.
2) If j : F R* ! F S[V ] is purely inseparable, then any graded field
automorphism of F S[V ] which restricts to an Ap -automorphism of F R*
is itself already an Ap -automorphism of F S[V ].
Rings of Invariants and Inseparable Forms *
*13
Hence, for SR* = D* (R*) as in Section Three
3) AutR* (D* (R*)) = AutF R* (F D* (R*)).
4) AutD*(R*) (S[V ]) = AutF D*(R*) (F S[V ]) = Id and
5) AutR* (S[V ]) = AutF R* (F S[V ]) = AutR* (D* (R*)) and these pre-
serve Steenrod operations.
Proposition 4.2. Let U0 U1 : : : V be an increasing filtration of
V , and D* the associated diagonal algebra. Then the following groups
coincide:
1) G1 : the subgroup of GL(V ) which respects the filtration.
2) G2 : the gradation respecting, Steenrod action preserving algebra
automorphisms of D* .
3) G3 : the gradation respecting, Steenrod action preserving field
automorphisms of F D* .
We remark that for S[V ] itself the automorphism groups in several
different plausible categories coincide. However this property is not in-
herited for the "diagonal" subalgebras D* :
Example 4.3. A gradation preserving algebra automorphism of D*
need not respect the Steenrod algebra action.
Details of 4.3.
Let D* = F2[t1; t2; t23], and define (t1) = t1; (t2) = t2, but (t23) =
t23+ t1t2. Extend this to all of D* . Then does not commute with Sq1
on t23.
Thus Theorem 4.1.1 has non-trivial content. Theorem II.7 observes
that D* (R*) is also a Hopf-algebra. The correct version of 4.3 is that
Hopf-algebra automorphisms of D* (R*) respect the Steenrod algebra
action.
Proof of Theorem II.
We first observe that if R* is replaced by its maximal separable ex-
tension in S[V ], S*R*, then Section II proves Theorem II in this special
case, with W (S*R*) = {Id} . However, it is clear that the filtrations
and diagonal subalgebras defined by R* and S*R* agree. It remains to
check that the action of W (R*) preserves the filtration on V . But if
vpi is separable over R*, then any Galois conjugate of it is also separa-
ble over R*. The identification of the various possible definitions of the
automorphism groups is done in 4.1.
For II.7, the Hopf-algebra structure of D* (R*) is clear. It remains to
show that if H* is an Ap -sub-Hopf-algebra of S[V ], then H* = D* (H* ).
14 Clarence W. Wilkerson
From the Borel decomposition theorem, as an algebra, H* is a finitely
generated polynomial algebra, and hence integrally closed. By II.6,
*)
H* = (D* (H* ))W (H :
However, the Hopf algebra quotient S[V ]==H* is a finite evenly graded
abelian Hopf-algebra, so each algebra generator of the quotient has
height a power of p. By induction, one sees that S[V ] is purely in-
separable over H* . Hence W (H* ) = {id} and H* = D* (H* ).
Proof of Theorem III.
Let R* = Fp[y1; : : :; yn ] be the polynomial algebra, and D* (R*) its
maximal separable extension in S[V ]. Any derivation of R* or its fraction
field F R* into F D* (R*) extends uniquely to a derivation on F D* (R*),
since the extension is separable. More explicitly, if ffi 2 Der(F R*) and
ff 2 F D* (R*) has minimal separable equation over F R* of
f (ff) = ffN + b1ffN-1 + : :+:b0
then X
ffiff = -( ffN-i ffibi)=f 0(ff):
Since f is separable, there are no repeated roots and f (ff) 6= 0 . In
the case at hand of polynomial algebras, the partial derivatives in {yi}
and {zj} form bases respectively for Der(F R*) and Der(F D* (R*)). But
from the above {@=@yi} is also a basis for Der(F D* (R*)) over F D* (R*).
The Jacobian |@yi=@zj| is the determinant of the change of basis matrix
for these two bases of Der(F D* (R*)) over F D* (R*), and hence must
be non-zero.
Proof of Prop.4.2.
Certainly, G1 G2 G3 We know that D* = U n(F R*). Since G3
induces an automorphism of D* , evidently G3 G2. But the induced N
automorphism on D* extends uniquely to S[V ], by the rule that if vp 2
D* , then (v) = ( vpN )1=pN . Since p-th roots are unique if they exist,
this extension is well defined, since the question of whether an element
is a p-th power is detectable by the Steenrod action. Hence G1 G2.
Since U n(F D* ) = D* , one has G2 = G3 immediately.
Proof of Theorem V.
Statement 1) is directly from [Lam, (7)].
For 2) consider the p-th root closure of (R*)00 in S[V ]. That i*
*s,
y 2 (R*)00 if and only if there exists some N 0 with ypN 2 R*.
Rings of Invariants and Inseparable Forms *
*15
Then by definition, (S[V ])p \ (R*)00 ((R*)00)p and 1) applies. That is,
N
c0 2 (R*)00, and there exists N 0 with cp0 2 R*. This argument holds
for any non-zero ideal of R* which is closed under the Steenrod algebra
action also. ____ ____
To prove 3), let R* = U n(F R*) . If y 2 R* consider the conductor
ideal C(y; R*) R* consisting of those elements r 2 R* for which
ry 2 R*. Let S(y; R*) be the radical of this ideal, that is, the set of all
elements which have a power in C(y; R*). Finally, define
____ * " *
S(R* ; R ) = y2___R*S(y; R ):
____
Then we claim that S(R* ; R*) is a non-zero ideal in R* which is closed
under the action of the Steenrod algebra._ If so, then by part 2) of this
theorem, some power of c0 falls in S(R* ; R*). That is,
____ -1 *
S-1zR* = Sz R :
____
It remains to check that S(R* ; R*) is closed under the Steenrod alge-
bra action. It suffices to check this for each S(y; R*). (this need not be
true for each C(y; R*)). Let |y| = 2M , and r 2 C(y; R*). Now use the
Cartan formula to compute
M pM ipM pM X jpM pM (i-j)pM i pM
P ip r y = (P r )y + P r P y = (P r) y:
j>0
The sparseness of the first expansion is due to the treatment of p-th
powers by the Steenrod algebra. But moreover, since y is unstable of
dimension 2M , each term on the right hand side except the first vanishes,
since (i - j)pM > M . This demonstrates that if r 2 C(y; R*), then
P ir 2 S(y; R*), and by replacing r by a suitable p-th power, that
P iS(y; R*) S(y; R*). ____ ____
Finally, to show that S(R* ; R*) 6= 0, since R* _is_notherian, we can
choose a finite set {yk } of algebra generators for R* . Then for each k,
there exist wk ; xk 2 R* such that yk = xk =wk . Then the element
Y ____
= wk 2 S(R* ; R*):
x5 Discussion and Examples
In this section, we make some comments on the algebraic classifica-
tion of unstable domains, and give some illustrations of the filtration
structure pointed out by Theorem II.
16 Clarence W. Wilkerson
The special case of polynomial algebras has great historical interest.
The application of Theorem II to this case is a success, since Theorem
IV from (2) gives very strong restrictions on such algebras to be topo-
logically realizable. In particular, the filtration produced in Theorem II
must degenerate to U0(R*) = V for odd primes p.
However from a strictly algebraic viewpoint, there is still a nagging
question as to whether the property of the W -action having a polynomial
algebra as its ring of invariants is inherited by the action on the diagonal
algebras D* (R*), and vice versa. More precisely,
Conjecture 5.1. Let W be a subgroup of GL(V ) such that W restricts
to an action on D* , a diagonal subalgebra of S[V ]. Then (D* )W is a
polynomial algebra if and only if S[V ]W is a polynomial algebra.
This conjecture is easily verified if the order of W is prime to p, since
in that case the ring of invariants is a polynomial algebra if and only if
W is generated by generalized reflections.
Without attempting to restrict to just polynomial algebras, Theorem
II provides give a construction and classification of all Ap - integral
domains which are noetherian and integrally closed. One can view this
in two steps:
1) Choose a particular subgroup W GL(V ). This generates the
separable example as S[V ]W :
2) The inseparable forms of this algebra W are then parameterized by
W -invariant filtrations of V and choices of exponents, or equivalently, a
"diagonal" algebra D* such that the action of W on S[V ] restricts to
D* . So the inseparable form is R* = (D* )W .
Notice that if the representation of W has no proper invariantNsub-
spaces, then there are always only the "standard" copies ((S[V ])p )W
associated with W: Therefore, non-trivial examples of Theorem II must
use non-trivial W - filtrations on V .
Example 5.2.
Our first example is the Weyl group of the simple Lie group G2. For
p = 3, this Weyl group provides an example of proper invariant sub-
spaces. The Weyl group of G2 is the dihedral group D12 of order 12.
The mod 3 cohomology of BG2 provides a good illustration of two pos-
sible viewpoints of the Steenrod algebra action. One is the internal view
provided by the explicit Steenrod algebra action, and the other is the
external view provided by the (1) embedding and associated Weyl group
action. The filtration is easiest to see in terms of the W - action, but
is also visible indirectly in the formulas detailing the Steenrod algebra
action.
Rings of Invariants and Inseparable Forms *
*17
We take the repesentation of D12 to be determined by the two elements
ff = -10 11 ; fi = 10 -10
of GL(Z; 2):
So over F3; there are generators for the invariants
x4 = t21;
and
< y12 = (t2(t1 + t2)(t1 - t2))2 > :
There is only one nontrivial stable subspace for D12; {0; t1; -t1}; and
it has no D12 complement. This filtration corresponds to the family of
A3-polynomial algebras parametrized by N; M 0
N 3M
(F3[x4; y312])
in
F3[x4; y12] = H* (BG2; F3):
Here the A3 -actions can be specified as
N 3N 3N 3N 3N+1 3N 3N 3N+1 3N
P 1x = -x2; P 3 y = x y ; P y = y (x - y )
and analogues for M > 0: On the otherMhand one sees from these for-
mulas that oneMcan not introduce a x3 instead of x without also sub-
stituting y3 for y: That is,
M 3N
F3[x3 ; y ]
is not closed under the A3 -action if N < M . This is reflected on the
vector space level by the fact that there is only one proper invariant
subspace. In particular, there is no W -splitting of the filtration in this
case.
For G2 for p > 3; any proper invariant subspace would also be a direct
summand, but in fact, there are no invariant 1 dimensional subspaces
for p > 3: Hence for p > 3 the only inseparable forms of Fp[x4; y12] have
the form Fp[x4; y12]pN ; for N non-negative.
18 Clarence W. Wilkerson
Example 5.3.
We now sketch the classification of noetherian integrally closed unsta-
ble domains of rank 2 in F2[t1; t2]. We define some useful classes in this
ambient polynomial ring in order to ease the notation later.
w1 = t1 + t2; w2 = t1t2
are the Steifel-Whitney classes, and the Dickson invariants are
c0 = t1t2(t1 + t2); c1 = t21+ t22+ t1t2
Finally
u = t31+ t32+ t21t2:
We now provide the classification by listing the Weyl groups, filtra-
tions, and exponents that are possible for a given group.
1) W = (id). All subspaces are invariant. The algebras contructed by
choices of filtrations and exponents are isomorphic to F2[t2i1; t2j2].
2) W = Z=2Z . Up to conjugation, we can use the representation
o = 01 10 :
There is one non-trivial subspace, spanned by w1 . Hence the examples
corresponding to 0 V0 V and a choice of exponents N and M give
rise to algebras isomorphic to
N 2M
F2[w21 ; w2 ] F2[w1; w2]
with M N .
3) W = Z=3Z. Up to conjugation, we can use the representation
ae = 01 11 :
There are no non-trivial W -invariant subspaces. In this case the invari-
ants are not polynomial. In fact,
R* = S[V ]W = F2[c0; c1; u]=(u2 + c0u + c20+ c31):
Any inseparable examples have the form (R*)2N .
4) W = GL(V ) and we can use as generators for the representation
the matrices for ae and o in 2) and 3) above. Again, there are no proper
invariant subspaces, so the only inseparable forms are F2[c0; c1]2N .
From the inseparability viewpoint, only 1) and 2) are interesting. The
reader might wish to attempt this classification of rank 2 unstable poly-
nomial algebras over F2 by direct computation instead of this contruc-
tion suggested by Theorem II.
One special case of 1) and 2) that arises in the classification of the
equivariant cohomology rings associated to involutions on cohomology
projective planes is
Rings of Invariants and Inseparable Forms *
*19
Proposition 5.4. If the polynomial algebra F2[x1; ym ] with |x1| =
1; |ym | = m has an unstable A2-action then m = 2N and either
a)
N
F [x1; y2N ] F2[t1; (t2)2 ]; |tj| = 1
or
b)
N-1 *
F2[x1; y2N ] F2[w1; (w2)2 ] H (BO(2)):
In case a) Sqk y = 0; 0 < k < 2N
In case b) Sqk y = 0; 0 < k < 2N-1 and Sq2N-1 y = x2N-1 y
Of course, the formulas for the Steenrod operations in case b) can be
perturbed slightly if one replaces the generator y with y + xm .
References
(1) Adams, J.F., Wilkerson, C.W., Finite H-spaces and Algebras over the Stee*
*nrod
Algebra, Annals of Mathematics 111 (1980), 95-143.
(2) Dwyer, W.G., Miller,H., Wilkerson, C.W., The Homotopical Uniqueness of
Classifying Spaces, to appear in LMS memorial volume for J.F. Adams, see
also talk by Wilkerson at BCAT 1986.
(3) Dwyer, W.G., Wilkerson, C.W., Smith Theory revisited, Annals of Mathemat-
ics (1988).
(4) Dwyer, W. G., Wilkerson, C. W., Smith Theory and the Functor T, preprint
(1988).
(5) Jacobson, N., "Lectures in Abstract Algebra, Vol. III.," Van Nostrand, P*
*rince-
ton, 1964.
(6) Jacobson, N., "Lie Algebras," Wiley (Interscience), New York, 1962..
(7) Lam, S.P., Thesis, Trinity College, Cambridge, 1982..
(8) Lang, S., "Algebra," Addison-Wesley, Reading, 1965.
(9) J. Lannes, Sur la cohomologie modulo p des p-groupes Abelienselementaire*
*s,
in "Homotopy Theory, Proc. Durham Symp. 1985," edited by E. Rees and
J.D.S. Jones, Cambridge Univ. Press, Cambridge, 1987.
(10) Mitchell, S., Stong, R. E., An adjoint representation for polynomial al*
*gebras,
Proceedings of the American Mathematics Society 101(1) (1987), 161-167.
(11) Quillen, D., The spectrum of an equivalent cohomology ring I, II, Annal*
*s of
Mathematics 94 (1971), 573-602.
(12) Rector, D. L., Journal of Pure and Applied Algebra (1984).
(13) Serre, J.-P., Sur la dimension cohomologique des groupes profinis, Topo*
*logy
3 (1965), 413-420.
(14) Wilkerson, C.W., Classifying Spaces, Steenrod Operations and Algebraic *
*Clo-
sure, Topology 16 (1977), 227-237.
(15) Wilkerson, C. W., Integral closure for algebras over the Steenrod Algeb*
*ra, J.
of Pure and Applied Algebra (1979).
(16) Winter, D., "The Structure of Fields," Graduate Texts in Mathematics. V*
*ol-
ume 16, Springer-Verlag, New York, 1974.
Purdue University, West Lafayette, Indiana 47907