KA"HLER DIFFERENTIALS, THE T -FUNCTOR,
AND A THEOREM OF STEINBERG
W.G. Dwyer and C.W. Wilkerson
University of Notre Dame
Purdue University
Abstract. Let T be the functor on the category of unstable algebras
over the Steenrod algebra constructed by Lannes. We use an argument
involving K"ahler differentials to show that T preserves polynomial alge-
bras. This leads to new and relatively simple proofs of some topological
and algebraic theorems.
x1. Introduction
Fix a prime number p, and let K denote the category of unstable
algebras over the mod p Steenrod algebra Ap [19, x1]. Recall that an el-
ementary abelian p-group is a finite abelian group which is a vector space
over Fp. Given an object R of K and an elementary abelian p-group E,
Lannes [16] constructs an associated "function object" TE (R); if X is a
space and R = H *(X; Fp), then TE (R) is an algebraic approximation to
the mod p cohomology of the space of maps B E ! X. More generally,
associated to any K-map f : R ! H *(B E; Fp) is a "component" TE;f (R)
of TE (R) (see [16, 2.5]); for R = H *(X; Fp) this is an algebraic approx-
imation to the mod p cohomology of an appropriate component of the
space of maps B E ! X.
Our main theorem is the following one.
1.1 Theorem. Suppose that R is an object of K which as an algebra is
a connected finitely generated polynomial algebra over Fp. Then for any
elementary abelian p-group E and K-map f : R ! H *(B E; Fp), TE;f (R)
is also a finitely generated polynomial algebra over Fp.
The proof of 1.1 proceeds by combining a characterization of graded
polynomial algebras in terms of K"ahler differentials with a study of how
________________
The authors were supported in part by the National Science Foundation
Typeset by AM S-TEX
1
2 W. G. DWYER AND C. W. WILKERSON
the functor TE interacts with the process of constructing the K"ahler
differentials.
Theorem 1.1 leads pretty directly to three other results. If K is a
subgroup of the group G, let CG (K) denote the centralizer in G of K.
1.2 Theorem. Suppose that G is a compact Lie group with H *(B G; Fp)
a polynomial algebra over Fp. Then for any elementary abelian p-
subgroup E of G, H *(B CG (E); Fp) is also a polynomial algebra over Fp.
Proof. Suppose that E is an elementary abelian p-subgroup of G and
that f : H *(B G; Fp) ! H *(B E; Fp) is the map induced by the inclusion
E ! G. By [15], H *(B CG (E); Fp) is isomorphic as an object of K to
TE;f (H *(B G; Fp)). The theorem follows from 1.1.
Parallel to this is a theorem about p-compact groups [11]. If h: Y ! X
is a homomorphism of p-compact groups [11, 3.1], let CX (h(Y )) denote
the centralizer of h(Y ) in X [11, 3.5].
1.3 Theorem. Suppose that X is a p-compact group with H *(B X; Fp) a
polynomial algebra over Fp. Then for any elementary abelian p-group E
and homomorphism h: E ! X, H *(B CX (h(E)); Fp) is also a polynomial
algebra over Fp.
Proof. By definition, B CX (h(E)) is the component of the space of maps
from B E to B X containing the map B h. Denote by f : H *(B X; Fp) !
H *(B E; Fp) the map induced by Bh. According to [12, pf. of 8.1],
H *(B CX (h(E)); Fp) is isomorphic to TE;f (H *(B X; Fp)). The theorem
follows from 1.1.
Suppose that k is a field and that V is a finite-dimensional vector
space over k. We let V # denote the dual of V , and Sym (V # ) the
symmetric algebra on V # . If k is infinite, Sym (V # ) is the alg*
*ebra
of polynomial functions on V ; in any case, Sym (V # ) is a polynomial
algebra over k isomorphic to k[x1; : : :; xd], where d = dim k(V ). Any
group W of automorphisms of V acts on Sym (V # ) in a natural way
and has an associated fixed subalgebra Sym (V # )W . The following is a
theorem of Nakijima [18, 2.1.2]. See [14] for a version that applies to the
case in which the action of W on V is irreducible, as well as for many
examples. We give another proof in x5.
1.4 Theorem. Suppose that V is a finite dimensional vector space over
the field Fp and that W is a subgroup of Aut (V ). Let U be a subset of V ,
and WU the subgroup of W consisting of elements which fix U pointwise.
Then if Sym (V # )W is a polynomial algebra over Fp, so is Sym (V # )WU .
KA"HLER DIFFERENTIALS AND T 3
Reflection groups and Steinberg's theorem. Both 1.2 and 1.4 have
characteristic zero versions which are consequences of the following the-
orem of Steinberg. Recall that if V is a finitely generated free module
over a domain k, an element w of finite order in Aut k (V ) is said to be a
reflection if image (w - IdV ) has rank 1. (Sometimes such a w is called
a generalized reflection or a pseudoreflection, and the word reflection
is reserved for the case in which the order of w is 2. We will use the
word reflection in the wider sense.) A subgroup W Aut k(V ) is said
to be generated by reflections (or said to be a reflection group) if it is
generated by the reflections that it contains.
1.5 Theorem. (Steinberg, [22, Thm. 1.5]) Suppose that k is a field of
characteristic zero, and that W GL(n; k) is a finite group generated
by reflections. Let U be a subset of kn and WU the subgroup of W
consisting of elements which fix U pointwise. Then WU is also generated
by reflections.
The tie connecting 1.5 to 1.2 and 1.4 is the following classical result.
1.6 Theorem. [8] [21] [7] [20] [5] [2] Suppose that V is a finite *
* di-
mensional vector space over a field k of characteristic zero, and that
W Aut (V ) is a finite group. Then Sym (V # )W is a polynomial alge-
bra over k if and only if W is generated by reflections.
Remark. If the field k in 1.6 has finite characteristic and Sym (V # )W is
a polynomial algebra, then W is generated by reflections [20], but the
converse does not necessarily hold (see for instance [23]).
Here are the characteristic zero versions of 1.2 and 1.4.
1.7 Theorem. Suppose that G is a compact Lie group with the prop-
erty that H *(B G; Q) is a polynomial algebra over Q. Let T 0be a toral
subgroup of G. Then H *(B CG (T 0); Q) is also a polynomial algebra.
Remark. If G is a connected compact Lie group, then H * (B G; Q) is
always a polynomial algebra.
Proof of 1.7. Suppose that K is a compact Lie group with maximal torus
TK and Weyl group WK = Norm K (TK )=TK . Let VK = Q ss1TK ; the
group WK acts by conjugation on TK and hence on VK . Recall that the
obvious restriction map gives an isomorphism
(1.8) H *(B K; Q) ~= H *(B TK ; Q)WK ~=Sym (VK# )WK :
Now let TG be a maximal torus of G. By adjusting T 0up to conjugacy
we can assume that T 0 TG . Let V = V (G); by 1.8 and 1.6, the image
4 W. G. DWYER AND C. W. WILKERSON
of the map W (G) ! Aut (V ) is generated by reflections. Let U V be
the image of ss1T 0and let WU W (G) denote the subgroup of elements
which fix U pointwise. By 1.5, the image of the map WU ! Aut (V ) is
generated by reflections. It is easy to identify WU with the Weyl group
of CG (T 0), and it follows from another application of 1.8 and 1.6 that
H *(B CG (T 0); Q) = Sym (V # )WU is a polynomial algebra.
1.9 Theorem. Suppose that V is a finite dimensional vector space over
Q, and that W Aut (V ) is a finite group. Let U be a subset of V , and
WU the subgroup of W consisting of elements which fix U pointwise.
Then if Sym (V # )W is a polynomial algebra over Q, so is Sym (V # )WU .
Proof. This is a direct combination of 1.5 and 1.6. The theorem also
holds if Q is replaced by any other field of characteristic zero.
In some sense, then, 1.2 and 1.1 can be viewed as generalizations of
Steinberg's theorem 1.5 to finite characteristic. We give the following
relatively simple example to show that some adjustment in the statement
of the theorem is necessary in order to obtain such a generalization.
1.10 Examples. There is a rank 3 free abelian subgroup of SL(5; Z)
and an element u 2 (Z)5 such that for any prime p the following con-
dition holds: the mod p reduction of acts faithfully on (Fp)5 as
a group generated by reflections, but the stabilizer in of the mod p
reduction of u is a subgroup of order p not generated by reflections.
Organization of the paper. For the rest of the paper, p denotes a fixed
prime number. Section 2 describes a way to recognize polynomial al-
gebras over Fp by looking at modules of K"ahler differentials. Section
3 shows that Lannes's functor T preserves K"ahler differentials, and x4
shows that T preserves the properties of these differentials which char-
acterize polynomial algebras. The proof of 1.1 is immediate. In x5 there
is an explanation of how to obtain 1.4 from 1.1. Finally, x6 gives a
construction of Example 1.10.
Acknowledgments. The authors thank L. Avramov for discussions on the
commutative algebra points in section 2. The formulation of Theorem
2.3 was prompted by a more general conjecture mentioned in a talk by
C. Weibel. We note that Campbell-Huges-Shank [6] have independently
found examples with properties similar to 1.10.
x2. K"ahler Differentials and Polynomial Algebras
We first need a recognition principle for polynomial algebras. If R is
a commutative algebra over a commutative base ring k, there is a short
KA"HLER DIFFERENTIALS AND T 5
exact sequence of R k R-modules
0 ! J ! R k R -! R ! 0
where the map is multiplication on R. The two inclusions R ! R k 1
and R ! 1 k R give possibly different left R-module structures on J.
2.1 Definition. The module of K"ahler differentials of R relative to k,
denoted (R|k) , is J=J2 = J= image (J k J -! J):
2.2 Remark. The action of R k R on (R|k) factors through the multi-
plication map , and so the above two R-module structures on J induce
the same R-module structure on (R|k) . Let S = R k R. The module
(R|k) can be identified with J S R or equivalently, by a long exact
sequence argument, with Tor S1(R; R).
There is a map d: R ! (R|k) which is a universal k-linear deriva-
tion and sends x 2 R to the equivalence class of x 1 - 1 x. If R
has a set {x1; : :;:xn } of k-algebra generators, (R|k) is generated as an
R-module by {dx1; : : :; dxn }; if R is a polynomial algebra k[x1; : :;:xn ],
then (R|k) is the free R-module on the classes dxi. Under some con-
ditions a converse of this last observation holds.
2.3 Theorem. Let k be a perfect field of characteristic p and R a finitely
generated connected graded algebra over k. Suppose that (R|k) is a free
R-module, and that R has no nilpotent elements. Then R is isomorphic
to a polynomial algebra k[x1; : :;:xn ].
Remark. The integer n which appears in 2.3 equals rank R ((R|k) ). The
assumption that R has no nilpotent elements cannot be removed from
2.3. To see this, note that if R is a connected graded bicommutative Hopf
algebra over k, a standard untwisting argument gives an isomorphism
TorRkR* (R; R) ~= R k Tor R*(k; k) :
In particular, (R|k) is free as an R-module. Now take R to be, say, a
primitively generated Hopf algebra with one even dimensional generator
x truncated at height p. Then (R|k) is free as an R-module but R is
not a polynomial algebra.
Proof of 2.3. This follows from the Jacobian Criterion for smoothness;
see for example [17, Th. 30.3] or [13, Cor. 16.22]. The particular case we
have is simpler than the general one and so we include an elementary
proof. Since R is finitely generated and connected, there exists a surjec-
tion OE: S = k[x1; : :;:xn ] ! R, where the xi are polynomial variables of
6 W. G. DWYER AND C. W. WILKERSON
positive grading. Choose OE in a minimal way so that the induced map on
indecomposable quotients is an isomorphism. Let I be the kernel of OE;
the goal is to show that I is zero. Assume not, select a nonzero element
f 2 I of minimal grading m > 0, and renumber the variables xi if neces-
sary so that {xi}n0i=1is the set of variables of grading less than m. Since
OE is an isomorphism on indecomposable quotients, f can be written as
a polynomial f (x1; : : :; xn0 ).
The map OE: S ! R is an isomorphism in gradings below m, as are the
induced maps (S|k) ! (R|k) and k S (S|k) ! k R (R|k). Now
the elements {dxi}n0i=1have linearly independent images in k S (S|k),
and hence the elements {dOE(xi)}n0i=1are linearly independent in k R
(R|k). This implies that {dOE(xi)}n0i=1is part of an R-basis for (R|k).
Since OE(f ) vanishes, so does dOE(f ), and we get an equation
Xn0
0 = dOE(f ) = OE(@f =@xi)dOE(xi) :
i=1
As above, though, the elements {dOE(xi)}n0i=1are linearly independent
over R, so the equation implies that OE(@f =@xi) vanishes for i n0. Since
@f =@xi is of grading less than m we conclude, from the choice of m, that
@f =@xi vanishes for i n0 and hence for all i (because f only depends
on {xi}n0i=1). Since k is perfect and S is a polynomial algebra, there
exists a polynomial g 2 S such that f = (g)p, and, because R contains
no nilpotent elements, g must also be in the kernel of OE. But g has
grading m=p < m, and so the existence of g contradicts the assumption
that OE is an isomorphism in gradings below m.
x3. K"ahler Differentials and T
As in [19, x1], let Ap be the mod p Steenrod algebra, U the category
of unstable modules over Ap, and K the category of unstable algebras
over Ap. An object of U or of K is a nonnegatively graded Fp-vector
space with an action of Ap and, in the case of K, a graded commuta-
tive multiplication which obeys the Cartan formula and a p'th power
condition [19, 1.3, 1.4]. There is a forgetful functor K ! U . If X is a
space, the cohomology algebra H *(X; Fp), with its usual cup product
structure, belongs to K.
Let E be an elementary abelian p-group and B E its classifying space,
Lannes has studied the functor TE : U ! U which is left adjoint to the
functor which sends M 2 U to M Fp H *(B E; Fp) (note that the action
of Ap on such a tensor product is given by the Cartan formula). The
functor TE has some remarkable algebraic properties; in particular, it is
KA"HLER DIFFERENTIALS AND T 7
exact, preserves tensor products over Fp up to natural isomorphism, and
lifts to an identically named functor K ! K [19, x3]. The topological
significance of TE has to do with its usefulness for computing the coho-
mology of function spaces, but we are interested in it from an algebraic
point of view.
There is a slight refinement of TE which has topological applications in
computing the cohomology of individual components of function spaces.
Suppose that R 2 K, and that f : R ! H * (B E; Fp) is a K-map. By
adjointness f corresponds to a K-map f [: TE (R) ! Fp. Since the range
of f [ is concentrated in degree 0, f [ amounts to an ordinary ring ho-
momorphism f [: TE0(R) ! Fp; this homomorphism makes Fp into a
module over TE0(R). Indeed, it follows from the fact that TE0(R) is a
p-Boolean algebra [19, 3.8] that Fp is a flat module over TE0(R). We
define TE;f (R) = Fp TE0R TE (R). More generally, let U (R) be the cat-
egory in which an object is an element M 2 U together with a U -map
RFp M ! M which makes M into a module over R. If M 2 U (R) then
TE (M ) 2 U (TE (R)) (this follows from the fact that TE preserves tensor
products) and we define TE;f M = Fp TE0R TE M ~= TE;f (R)TE R TE M .
The construction TE;f gives an exact functor U (R) ! U (TE;f R). In par-
ticular, TE;f (R) is flat as a module over TE R.
Remark. Suppose that Hom K(R; H *(B E; Fp)) is a finite set; this is al-
ways the case if R is finitely generated as an algebra (since H *(B E; Fp)
is finite in each dimension). In this situation TE0(R) is isomorphic as
a ring to a direct product of copies of Fp, indexed by the K-maps
f : R ! H *(B E; Fp). There is a corresponding product decomposition
[19, 3.8.6] Y
TE R ~= TE;f R :
f
For the rest of this section we assume that E is a fixed elementary
abelian p-group and write T for TE and Tf for TE;f .The goal of this
section is to prove the following two propositions.
3.1 Proposition. Let R be an object of K and f : R ! H *(B E; Fp) a
K-map. Then there is a natural isomorphism of Tf R-modules
Tf (R|Fp) ~=(Tf R|Fp) :
3.2 Remark. An object R of K need not be commutative as a ring (in
general it is graded commutative), but we follow 2.2 in defining (R|Fp)
as Tor S1(R; R), where S = R Fp R. It will become clear below how to
identify this as a object of U (R).
8 W. G. DWYER AND C. W. WILKERSON
Note that if R 2 K is commutative for the transparent reason that
p = 2 or that R is concentrated in even degrees, then T (R) and hence
any Tf (R) are commutative for the same reason [19, 3.6].
An algebra R is said to be nilpotent free if the only nilpotent element
of R is the zero element.
3.3 Proposition. Let R be an object of K which is nilpotent free. Then
for any K-map f : R ! H *(B E; Fp), Tf R is nilpotent free.
The proof of 3.1 depends on two lemmas.
3.4 Lemma. Let R and S be objects of K, g: S ! R and f : R !
H *(B E; Fp) a pair of K-maps with g surjective, and M an object of
U (R). Let h = gf . Then Th M is naturally isomorphic to Tf M as an
object of U (Th S).
Proof. Observe that in the statement of 3.4, M is considered to be an
S-module via the homomorphism g: S ! R, and Tf M is a Th S-module
via the induced homomorphism Th S ! Tf R.
[
The adjoint h[ of h is the composite T (S) -T-(g)-!T (R) -f! Fp. Since
g is surjective and T is exact, T (g) is also surjective and in particular
the map T 0(S) ! T 0(R) induced by T (g) is surjective. It is now clear
that for any T 0(R)-module N (such as T (M )) the map Fp T 0S N !
Fp T 0R N is an isomorphism, and the lemma follows.
Note that if R is an object of K and M , N are objects of U (R), then for
any i 0 the R-module Tor Ri(M; N ) is also in a natural way an object of
U (R); the Ap action on these R-modules can be obtained, for instance,
by letting Ap act via the Cartan formula on the bar construction [19,
6.4].
3.5 Lemma. Suppose that R is an object of K, f : R ! H * (B E; Fp)
is a K-map, and M , N are objects of U (R). Then there are natural
isomorphisms in U (Tf R):
Tf (Tor Ri(M; N )) ~= Tor TfRi(Tf M; Tf N ) i 0 :
Proof. Since T is exact and preserves tensor products, it follows as
in [19, 6.4.2] that there are natural isomorphisms T (Tor Ri(M; N )) ~=
Tor TiR(T M; T N ) in U (T R). Since Tf (R) is flat (in fact projective [19,
pf. of 6.4.3]) as a T (R)-module, there are natural isomorphisms
Tf (R) T R T orTiR(T M; T N ) ~= Tor TiR(Tf M; T N ) :
KA"HLER DIFFERENTIALS AND T 9
Again by flatness there is an isomorphism between Tor TiR(Tf M; T N )
and Tor TfRi(Tf M; Tf N ) (cf. [19, pf. of 6.4.3]). Combining these isomor-
phisms gives the desired result.
Proof of 3.1. Let S = R Fp R and let h: S ! H *(B E; Fp) be the
composite of f with the multiplication map S ! R. By 3.5, there is
a natural isomorphism Th (Tor S1(R; R)) ~= Tor ThS1(Th R; Th R). By 3.4
there are natural isomorphisms Th (Tor S1(R; R)) ~= Tf (Tor S1(R; R) and
Th (R) ~= Tf (R). Finally, by using the fact that T preserves tensor
products over Fp we obtain an isomorphism
Th (S) = T (S) T 0S Fp ~= Tf (R) Fp Tf (R)
To finish up, identify (R|Fp) with Tor S1(R; R) (3.2).
Proof of 3.3. Suppose that R is an object of K which is nilpotent free.
We first show that T R is nilpotent free. Since R is graded commutative,
it is clear that either R is concentrated in even degrees, or p = 2. Let
(R) be the object of U constructed as in [19, 1.7.2] by multiplying the
degrees of elements in R by a factor of p. There is a natural U -map
R : (R) ! R which loosely speaking sends each element of R to its
p'th power. Since R is nilpotent free, the map R is a monomorphism.
By [19, 3.4], the map T R can be identified with T (R ). Since T is exact,
it follows that T R is a monomorphism and hence that T R is nilpotent
free.
The final step is to show that Tf (R) is nilpotent free. The functors
T and Tf commute with colimits (because T is a left adjoint and Tf is
obtained from T by a tensor product). This implies that it is enough
to treat the case in which R is a finitely generated object of K, i.e.,
generated by a finite number of elements under product, sum, and the
operation of Ap. In this case there are only a finite number of K-maps
R ! H *(B E; Fp) and T 0(R) is isomorphic as a ring to a direct product
of copies of Fp, one for each such K-map [19, 3.8.6]. It follows that
Tf (R) is a direct factor, as a ring, of T (R), and so Tf (R) is nilpotent
free if T (R) is.
4. Free R modules and T
We continue using the notation of the previous section in letting E
stand for an elementary abelian p-group and writing T and Tf for TE and
TE;f respectively. Our goal is to complete the proof of 1.1 by showing
that Tf preserves the freeness of modules, in the following sense.
10 W. G. DWYER AND C. W. WILKERSON
4.1 Proposition. Suppose that R is an object of K and that M is
object of U (R) which is free as an R-module. Then for any K-map
f : R ! H *(B E; Fp), Tf (M ) is free as a Tf (R)-module.
Proof of 1.1. This consists of stringing together 3.1, 4.1, 3.3, and 2.3.
Note that Tf (R) is finitely generated as a polynomial algebra because
in general T preserves finite generation of algebras [10, 1.4].
Proof of 4.1. Since Tf (R) is a connected graded algebra, in order to
show that Tf (M ) is a free module over Tf (R) it is enough to show that
the groups Tor TfRi(Fp; Tf M ) vanish for i > 0 (actually, vanishing of
Tor 1 would be enough). Let H = H *(B E; Fp). The map f [: Tf (R) !
Fp that is used in computing Tor is adjoint to f : R ! H. It follows
from naturality that f [is the composite of Tf (f ): Tf (R) -! Tf (H) with
the map [: Tf (H) ! Fp adjoint to the identity map : H ! H. In
particular, for any Tf (R)-module N there is a natural isomorphism
Fp TfR N ~= Fp TfH (Tf H TfR N ) :
This gives rise to a composition of functors spectral sequence
E2i;j= Tor TfHi(Fp; Tor TfRj(Tf H; N )) ) Tor TfRi+j(Fp; N ) :
We will show that for N = Tf (M ) this spectral sequence has E2i;j= 0
for (i; j) 6= (0; 0).
Note first of all that the groups Tor Rj(H; M ) vanish for j > 0 be-
cause M is free as an R-module (here R acts on H via f ). By 3.5,
Tor TfRj(Tf H; Tf M ) = 0 for j > 0. Now consider the object
U = Tor TfR0(Tf H; Tf M ) = Tf (H R M )
as a module over Tf (H).QBy [19, 3.8.6] and [19, 3.9], Tf (H) is isomorphic
to a direct product ffH(ff) of copies of H indexed by the (finite)
collection of K-maps ff: H ! H with ff . f = f . Under this isomorphism,
the map [: Tf (H) ! Fp is obtained by composing projection on the
component H() with the unique ring homomorphism H() = H ! Fp.
Since H() = H is flat (in fact projective) over Tf (H), there are natural
isomorphisms
TorTfHi (Fp; U ) ~= Tor Hi(Fp; H TfH U ) i 0 :
KA"HLER DIFFERENTIALS AND T 11
The projection Tf (H) ! H() can alternatively be interpreted as the
natural map Tf (H) ! T (H). This implies that there are isomorphisms
H TfH U ~= T H TfH Tf H TfR Tf R T R T M
~= T H T H (T H T R T M )
~= TH T H T (H R M )
~= T (H R M )
where we have used the fact that T is exact and preserves tensor products
to give the isomorphism T H T R T M ~= T (H R M ) (cf. 3.5). The
desired vanishing now follows from the fact that if N is any object of
U (H), in particular HR M , then T (N ) is free as a module over T (H) ~=
H [10, 2.4] [16, 4.5].
x5. Rings of invariants and T
In order to deduce Theorem 1.4 from Theorem 1.1 we have to find a
connection between rings of invariants, algebras over Ap, and the func-
tor T . This is provided by the following construction.
5.1 Definition. Suppose that V is a finite dimensional vector space
over Fp. The enhanced symmetric algebra S(V # ) is the unstable al-
gebra over Ap freely generated as a commutative algebra by the ele-
ments of V # , which are considered to have grading two. The action of
Ap on S(V # ) is the unique one allowed by the usual unstable algebra
conditions.
5.2 Remark. The action of Ap on S(V # ) can be described explicitly
as follows. If p is odd and x 2 V # = S(V # )2, then P1 (x) = xp 2
S(V # )2p, fi(x) = 0, and Pi(x) = 0 for i > 1; Ap acts on products
of two-dimensional classes by the Cartan formula. The same formulas
work for p = 2 if fi is interpreted as Sq 1 and Pi as Sq 2i. The algebra
S(V # ) is isomorphic as an element of K to the cohomology ring of a
product of copies of CP 1 , where the number of factors in the product
is the dimension of V ; S(V # ) is also isomorphic to the subalgebra of
H *(B V ; Fp) generated by the Bocksteins of one-dimensional cohomology
classes. In particular S(V # ) is functorial in V , and is isomorphic as an
algebra to Sym (V # ); it differs from Sym (V # ) only in that it is explicit*
*ly
graded and has a specified action of Ap.
5.3 Lemma. Suppose that E and V are finite-dimensional vector spaces
over Fp. Then there is a natural bijection
Hom K (S(V # ); H *(B E; Fp)) ~= Hom (E; V )
12 W. G. DWYER AND C. W. WILKERSON
and a natural K-isomorphism between TE S(V # ) and S(V # )Hom (E;V ).
Remark. The notation S(V # )Hom (E;V ) from 5.3 denotes the collection
of set-maps from Hom (E; V ) to S(V # ), with pointwise ring operations;
equivalently, this is a product of copies of S(V # ) indexed by the el-
ements of Hom (E; V ). Under the isomorphism of 5.3 the action of
Aut (V ) on TE S(V # ) corresponds to the diagonal action of Aut (V ) on
S(V # )Hom (E;V ), in other words, the action which given g 2 Aut (V ),
OE 2 Hom (E; V ), and ff: Hom (E; V ) ! S(V # ) has (g . ff)(OE) = gff(g-1 OE*
*).
The first statement in 5.3 is related to the calcualtion of Adams,
Gunawardena and Miller in [1, p. 438] but simpler, because it involves
morphisms of unstable algebras (not modules) over the Steenrod algebra.
Proof of 5.3. Let H = H* denote H *(B E; Fp). The calculation of
Hom K (S(V # ); H) is as follows. Any f 2 Hom K (S(V # ); H) is an al-
gebra map and so is determined by its effect on the copy of V # in
dimension 2. Clearly f (V # ) lies in the kernel of the Bockstein map
fi: H2 ! H3 . Since H"*(B E; Z) is of exponent p, this kernel is isomor-
phic to H1 = E# via fi: H1 ! H2 (cf. [19, 1.5]). Thus any such f gives
a map V # ! E# or equivalently a map E ! V . It is now easy to check
that any homomorphism V # ! E# arises from a unique f .
For any object R of K there is a natural map R ! TE (R) adjoint to
the morphism RFp H* ! RFp Fp ~= R induced by the unique K-map
H ! Fp. For any f : R ! H there is a composite map fflf : R ! TE R !
TE;f R. The last statement in the lemma follows from the fact that for
any f : S(V # ) ! H the map fflf : S(V # ) ! TE;f S(V # ) is an isomorphism.
The most economical way to obtain this is from a geometric theorem
of Lannes [19, 9.6] [15]. Let G denote the circle group S1 = SO(2).
The algebra S(V # ) is isomorphic to H * (B Gd; Fp), where d = dim V
(cf. 5.2). By Lannes' theorem, a K-map f : S(V # ) ! H corresponds
to a homomorphism h(f ): E ! SGd, and Tf S(V # ) is then naturally
isomorphic to the cohomology of the classifying space of the centralizer
in Gd of the image of h(f ). Since Gd is abelian, this centralizer is Gd
itself, and the cohomology of its classifying space is again S(V # ).
Proof of 1.4. Let S = S(V # ) and R = SW . Suppose that E is an
elementary abelian p-group. It follows directly from the exactness of T
that there is a K-isomorphism TE R = TE (SW ) ~= (TE S)W [19, 3.9.5].
By 5.3, this gives isomorphisms
TE (R) = TE (SW ) ~= Map (Hom (E; V ); S)W
(5.4) ~= Map W (Hom (E; V )S) :
~= Q {OE}SWOE
KA"HLER DIFFERENTIALS AND T 13
Here the product in the third line is taken over the set of orbits {OE} of
the action of W on Hom (E; V ), and WOEdenotes the isotropy subgroup
in W of an orbit representative OE. To pass from the second to the third
line we have used the fact that if A and B are W -sets and W acts
transitively on A, there is an isomorphism Map W (A; B) ~= BWa , where
a is any element of the orbit a.
It follows from the dimension 0 part of 5.4 that K-maps f : R !
H *(B E; Fp) (equivalently ring homomorphisms T 0R ! Fp) are in bijec-
tive correspondence with orbits {OE} of the action of W on Hom (E; V ).
Moreover, for any such f , Tf (R) is isomorphic to SWOE . Let U be the cho-
sen subset of V , E the linear span of U , and OE: E ! V the inclusion. The
WU (the subgroup of W consisting of elements which fix U pointwise)
is equal to WOE (the subgroup of W consisting of elements which under
composition leave the map OE unchanged). Let f : R ! H *(B E; Fp) cor-
respond to {OE}. Since R is a polynomial algebra by assumption (finitely
generated for elementary transcendence degree reasons), it follows from
1.1 that Tf (R) ~= SWOE ~= SWU ~= Sym (V # )WU is also a polynomial
algebra.
x6 A universal example with non-reflection stabilizers
We provide a "universal" example to satisfy the promise of Example
1.10.
6.1 Example. Let be the rank 3 free abelian subgroup of GL(5; Z)
generated by the matrices
0 1 0 1 0 0 1 0 1 0 0 1 01 0 1 0 0 0 01
BB 0 1 1 0 0 C B 0 1 0 0 0C B 0 1 0 0 1C
B@ 0 0 1 0 0 CC ; BB 0 0 1 0 0CC ; BB 0 0 1 0 0CC ;
0 0 0 1 0 A @ 0 0 0 1 0A @ 0 0 0 1 0A
0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
denoted r, s and t respectively. The stabilizer of (0; 0; 1; -1; -1)T in
is the subgroup generated by the product z = rst. Then
0 1 0 1 1 0 1
B 0 1 1 0 1 C
z = BB 0 0 1 0 0 CC
@ 0 0 0 1 0 A
0 0 0 0 1
and so z - Id has rank 2. Let ae: GL(5; Z) ! GL(5; Fp) be the mod p
reduction map. Then the images of {r; s; t} are reflections of order p,
14 W. G. DWYER AND C. W. WILKERSON
and ae() is an elementary abelian p-group of rank 3. In this image, the
stabilizer of the reduction mod p of (0; 0; 1; -1; -1)T is the subgroup of
order p generated by ae(z) and thus contains no reflections.
Proof. The verification is a straightforward calculation. The matrices
have been chosen so that multiplication of the matrices corresponds
to addition of the upper right 2 x 3 blocks - if A and B are matrices
which are zero off this block, then AB = 0 and (I + A)(I + B) =
(I + B)(I + A). Hence the group is commutative and isomorphic to
the free abelian group of rank 3, and ae() is a rank 3 elementary abelian
p-group. Clearly, ra (sb(tc((0; 0; 1; -1; -1)T ))) = (a-b; a-c; 1; -1; -1)T ,
so the stabilizer of (0; 0; 1; -1; -1)T consists of the elements ra sbtc with
a = b and a = c. This is the group generated by za , where z = rst. An
identical calculation works mod p.
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University of Notre Dame, Notre Dame, Indiana 46556
E-mail address: dwyer.1@nd.edu
Purdue University, West Lafayette, Indiana 47907
E-mail address: wilker@math.purdue.edu
Processed September 16, 1997