Smith theory and the functor T
William G. Dwyer and Clarence W. Wilkerson
University of Notre Dame
Purdue University
x1. Introduction
J. Lannes has introduced and studied a remarkable functor T [L1]
which takes an unstable module (or algebra) over the Steenrod algebra
to another object of the same type. This functor has played an impor-
tant role in several proofs of the generalized Sullivan Conjecture [L1]
[L2] [DMN] and has led to homotopical rigidity theorems for classify-
ing spaces [DMW1] [DMW2]. In this paper we will use techniques
of Smith theory [DW] to calculate the functor T explicitly in certain
key special situations (see 1.1 and 1.3). On the one hand, our calcula-
tion gives general structural information (1.4) about T itself. On the
other hand, up to a convergence question which we will not discuss here
our calculation produces a direct analogue of Smith theory (1.2) for
actions of elementary abelian p-groups on certain infinite-dimensional
complexes; this analogue differs from Smith theory only in that "homo-
topy fixed point set" is substituted for "fixed point set".
We will now state the main results, which are completely algebraic in
nature although they have a geometric motivation. Fix a prime p; the
field Fp with p elements will be the coefficient ring for all cohomology.
Let Ap denote the mod p Steenrod algebra, and U (resp. K) the category
of unstable modules (resp. unstable algebras) over Ap (see [L1]). If R is
an object of K, an unstable Ap R module M is by definition an object
of U which is also an R module in such a way that the multiplication
map RM ! M obeys the Cartan formula; we will denote the category
of Ap R modules by U (R). An object of U (R) typically arises from a
map q : E ! B of spaces; in this case the induced cohomology map q*
makes H* E an object of U (R) for R = H* B .
Let V be an elementary abelian p-group, ie., a finite-dimensional vec-
tor space over Fp , and HV the classifying space cohomology H* BV .
Lannes [L1] has constructed a functor T V : U ! U which is left adjoint
to the functor given by tensor product (over Fp ) with HV and has shown
that T V lifts to a functor K ! K which is also left adjoint to tensoring
______________________________________
Both authors were supported in part by the National Science Foundation.
Typeset by AM S-TEX
2 W. Dwyer and C. Wilkerson
with HV . The adjointness property of T V produces for any space X a
natural map
X : T V(H* X) ! H* Hom (BV; X)
which is often an isomorphism [L1] [L2] [DS]. Given an object R of K
there is a simple way (see x2) of using a particular K-map f : R ! HV
to single out a quotient TfV(R) of T V(R) or for M 2 U (R) a quotient
TfV(M ) of T V(M ). These quotients correspond via to subspaces of
function spaces; more precisely, if q : E ! B is a map of spaces and
f : H* B ! HV is a map in K, then E induces a quotient map E;f from
TfV(H* E) to the cohomology of the subspace of Hom (BV; E) consisting
of maps r : BV ! E with r* q* = f .
For any K-map R ! HV , we will let Sf R denote the multiplicative
subset of R generated by the Bockstein images in R2 of the elements of
R1 which map non-trivially under f . Recall from [W1] [Si] that if M
is an object of U (R) any localization of the form S-1fM inherits an
action of Ap , although this action is not necessarily unstable. Denote
the largest unstable Ap -submodule [DW, 2.2] of such a localization by
Un (S-1fM ).
Theorem 1.1. Let W be an elementary abelian p-group, V a subgroup
of W , and f : HW ! HV the map induced by subgroup inclusion.
Suppose that M is an object of U (HW ) which is finitely-generated as a
module over HW . Then there is a natural isomorphism
TfV(M ) ~= Un S-1f(M ) :
This theorem has a geometric background. Let V and W be as in 1.1.
Suppose that X is a finite CW-complex with a cellular action of W and
let M be the cohomology of the Borel construction EW X = EW xW X
on this action. According to [L2] there is an isomorphism
TfV(M ) ~= H* EW (XV )
where XV is the fixed point set of the action of V on X. Similarly, by
Smith theory [DW, 2.3] there is an isomorphism
H* EW (XV ) ~= Un S-1f(M ) :
The composite of these two maps is the isomorphism of 1.1; the present
paper sprang in part from a desire to produce this isomorphism in a
purely algebraic way without assuming that M ~= EW X for a finite
complex X.
Smith theory and the functor T 3
With Theorem 1.1 in hand, though, it is possible to work backwards
through the above geometric example. Suppose that X is a space with
an action of W but instead of assuming that X is finite assume only
that H* X is finite. Again let M = H* (EW X); a spectral sequence
argument shows that M is finitely generated as a module over HW . Let
XhV denote the homotopy fixed point set [M, p. 581] of the action of
V on X. It is not hard to see that the Borel construction EW (XhV )
is homotopy equivalent to the space of maps BV ! EW X which up
to cohomology cover the map BV ! BW induced by V W , so that
EW X;f produces a map TfV(H* EW X) ! H* EW (XhV ). Theorem 1.1
now computes the domain of EW X;f ; this gives the following corollary.
Corollary 1.2. Let W be an elementary abelian p-group, V a sub-
group of W and f : HW ! HV the map induced by subgroup inclusion.
Let X a space on which W acts and assume that H* X is finite and that
EW X;f is an isomorphism. Then there is a natural isomorphism
H* EW (XhV ) ~= Un S-1fH* (EW X) :
Remark: The conclusion of Corollary 1.2 implies that the localized
restriction map
S-1fH* (EW X) ! S-1fH* EW (XhV )
is an isomorphism. This is the promised extension of Smith theory [H,
Chap. III] to infinite dimensional complexes (the only finiteness condi-
tion on X is a cohomological one). The question of whether EW X;f is
an isomorphism is essentially a spectral sequence convergence problem
[DS] [B] and we intend to consider it in a future note.
Remark: As in [DW, 2.5], for V = W the conclusion of Corollary 1.2
gives an isomorphism H* (XhW ) ~= Fp HW Un S-1fH* (EW X) .
An unstable Ap HV algebra is an object R of K together with a
K-map HV ! R. Denote the category of Ap HV algebras by K(HV ).
(We will also occasionally consider the category K(R) for other objects
R of K.) If M is an R module and I R is an ideal, let MI^ stand
for the completion of M with respect to powers of I, i.e. for the inverse
limit lim sM=Is M .
Theorem 1.3. Let V be an elementary abelian p-group and f : R !
HV a map in K(HV ) with kernel I. Assume that R is finitely generated
as a ring and that M is an object of U (R) which is finitely-generated as
a module over R. Then there is a natural isomorphism
TfV(M ) ~= Un ((S-1fM ) ^I) :
4 W. Dwyer and C. Wilkerson
Remark: Example 4.6 shows that the appearance of something like a
completion is necessary in Theorem 1.3. It is easy to see (4.4) that in
the statement of Theorem 1.3 the localization S-1fM can be replaced
by the localization S-1 M of M with respect to the multiplicative set S
generated by the Bockstein images in H2 BV of the non-zero elements
of H1 BV .
The restriction in Theorem 1.3 that R be an algebra over HV is not
too serious, since it is possible to make any object of K into an object
of K(HV ) by tensoring with HV . If g : R ! HV is a map of K and
f : HV Fp R ! HV is the map of K(HV ) which extends g, then (see 2.2
and the proof of 1.4) there is a natural isomorphism TfV(HV Fp R) ~=
HV Fp TgV(R). In conjunction with Theorem 1.3 this calculation leads
to the following result.
Theorem 1.4. Let V be an elementary abelian p-group, R an object of
K which is finitely generated as a ring and M an object of U (R) which is
finitely generated as a module over R. Then T V(R) is finitely generated
as a ring and T V(M ) is finitely generated as a module over T V(R).
Organization of the paper. In sections 2 through 4 we will prove the
above theorems in the special case in which V is the rank one elementary
abelian p-group Z=p ; for this particular V , T V is simply denoted by T
and HV by H. Section 5 describes the argument that extends the results
to general V . Section 6 contains some auxiliary algebraic material on
K.
x2. Some properties of the functor T
The purpose of this section is to set up some machinery involving the
functor T .
Let f : R ! H be a K-map. The adjoint of f is a K-map T (R) ! Fp ,
which amounts to a ring homomorphism f^ : T (R)0 ! Fp . For M 2
U (R), let Tf (M ) denote the tensor product T (M ) T (R)0Fp , where the
action of T (R)0 on Fp is given by f^. Note that Tf (R) 2 K.
Proposition 2.1. For any K-map f : R ! H the construction Tf (-)
induces functors U (R) ! U (Tf (R)) and K(R) ! K(Tf (R) ). Moreover,
Tf is exact, and preserves tensor products in the sense that if M and N
are objects of U (R) there is a natural isomorphism
Tf (M R N ) ~= Tf (M ) Tf(R) Tf (N ) :
Proof: Most of what is asserted follows from the fact that T is exact
and preserves tensor products [L1]. To see that Tf is exact, note in
Smith theory and the functor T 5
addition that f^makes Fp into a flat module over T (R)0; this flatness is
an algebraic consequence of the fact that T (R)0 is a p-boolean ring [L1,
3.5] (that is, each element x in T (R)0 satisfies the equation xp = x). In
fact, if T (R)0 is finite, then Tf (R) is a summand of T (R).
Proposition 2.2. Let fi : Ri ! H, i = 1, 2 be K-maps with Mi 2
U (Ri) and let f be the product map f1 . f2 : R1 Fp R2 ! H. Then
there is a natural isomorphism
Tf (M1 Fp M2) ! Tf1(M1) Fp Tf2(M2) :
Proof: This is again a consequence of the fact that T preserves tensor
products.
Proposition 2.3. Suppose that R 2 K, M 2 U (R), and that x 2 R is
an element such that x . M = 0. Let f : R ! H be a K-map such that
f (x) 6= 0. Then Tf (M ) = 0.
Proof: By the Cartan formula the annihilator ideal I of M in R is
invariant under Ap and hence the quotient map R ! R=I is a morphism
of K. By Proposition 2.1 Tf (M ) ~= Tf (R=I) Tf(R) Tf (M ) so we will be
done if we can show that Tf (R=I) vanishes in dimension 0. Now Tf (R) ~=
Fp and the map Tf (R)0 ! Tf (R=I)0 is surjective, so non-vanishing
of Tf (R=I)0 implies that Tf (R)0 ! Tf (R=I)0 is an isomorphism and
therefore that f^: T (R)0 ! Tf (R)0 ~= Fp extends to a map T (R=I)0 !
Fp . This is impossible, because by assumption the adjoint map f : R !
H does not extend to a map R=I ! H.
For any object M of U the adjunction map M ! H Fp T (M )
can be combined with the unique algebra map H ! Fp to give a
map M ! T (M ); call this map ffl. (If M = H* X for some space X,
then ffl corresponds via X to the cohomology homomorphism induced
by the basepoint evaluation map Hom (BZ=p ; X) ! X.) If R 2 K,
M 2 U (R) and f : R ! H is a K-map, we will denote the composite
M !fflT (M ) ! Tf (M ) by fflf .
If f : H ! H is the identity map then fflf : H ! Tf (H) is *
*an
isomorphism [L1, 4.2], so that by (2.1) Tf lifts to a functor U (H) !
U (H) (or K(H) ! K(H).
Proposition 2.4 (Splitting Property). Let ' : H ! H be the
identity map and I H the kernel of the unique K-map H ! Fp . Then
for any object R of K(H) there is a natural K(H)-isomorphism
T' (R) ~= H Fp (T' (R)=I . T' (R))
6 W. Dwyer and C. Wilkerson
where the action of H on the tensor product is by multiplication on the
left-hand factor.
Remark: The proof of Proposition 2.4 also shows that if R 2 K(H)
and M 2 U (R) there is an isomorphism T' (M ) ~= H Fp (T' (M )=I .
T' (M )). The action of T' (R) on T' (M ) then respects the tensor product
splittings of both objects.
Proof of 2.4: The adjunction map R ! H Fp T (R) can be combined
with the Hopf algebra coproduct map H ! H Fp H to give a map
R ! H Fp H Fp T (R); the adjoint to this is a map T (R) ! H Fp T (R)
which has as quotient a K(H)-map T' (R) ! H Fp T' (R). The desired
isomorphism oe is the composite of this map with the projection
H Fp T' (R) ! H Fp (T' (R)=I . T' (R)) :
The fact that ' is the identity map insures that the action of H on the
target tensor product is the desired one. Note that oe induces an iso-
morphism on Tor H0(H=I; -) and an epimorphism on Tor H1(H=I; -) (the
latter because Tor H1(H=I; -) vanishes on the free H-module which is the
target of oe). Let C be the cokernel of oe and K the kernel. The fact
that Tor H0(H=I; C) = C=I . C = 0 implies that C = 0, since I is a
connected ideal. The long exact sequence for Tor H*(H=I; -) then shows
that Tor H0(H=I; K) vanishes, which similarly implies K = 0.
x3. Spherical elements
The purpose of this section is to prove Proposition 3.1, which is the
algebraic basis of all of the results in this paper.
Let S H be the multiplicative subset generated by the non-zero
elements of degree 2. The closure of an object M of U (H) is defined to be
Un (S-1 M ); M is closed if the natural map from M to the closure of M
is an isomorphism. An element x of an unstable Ap module is spherical
if ffx = 0 for each strictly positive-dimensional element ff 2 Ap .
Proposition 3.1. If M 2 U (H) is non-zero, closed and finitely gener-
ated as a module over H then M contains a non-zero spherical element.
Let H+ be the subalgebra of H generated by the elements of S; H+ is
an unstable Ap algebra which is isomorphic to the cohomology ring of
CP 1 . The multiplicative set S is contained in H+ and so it is possible
to speak of the closure of an object M of U (H+ ); as above, M is closed
if the natural map from M to the closure of M is an isomorphism. An
object of U is even-dimensional if it vanishes in odd degrees. Proposition
3.1 is a consequence of the following result.
Smith theory and the functor T 7
Proposition 3.2. If M 2 U (H+ ) is non-trivial, closed, even-dimension-
al, and finitely generated as a module over H+ , then M contains a non-
zero spherical element.
Proof of 3.1 (given 3.2): We will assume that p is an odd prime
since the case p = 2 is a little simpler. Consider the map V : M ! M
given by (
Pk (x) |x| = 2k
V(x) = :
fiPk (x) |x| = 2k + 1
The map V is not an Ap -map, but the image of V is an Ap -submodule
of M [Li]. Let N be the closure of the H+ -submodule of M generated
by the image of V. It is clear that N is an Ap H+ submodule of M
which is closed, even-dimensional and finitely generated over H+ . If N
is non-trivial we are done, since by Proposition 3.2 N contains a non-
zero spherical element. If N is trivial, then V is the zero map and M
is the suspension [Li] of a closed Ap H module M 0. We can argue by
induction on the largest degree in which M H+ Fp fails to vanish that
M 0contains a non-zero spherical element; the suspension of this element
is then a non-zero spherical element of M . The induction begins with
the fact that any zero-dimensional class is spherical.
In order to prove Proposition 3.2 we will need some additional no-
tation. Let R be an even-dimensional object of K and M an even-
dimensional object of U (R). Given a degree 2 element a of R and a
degree 2k element x of M , define a(x) by the formula
Xk
a(x) = (-1)iai(p-1)Pk-i (x)
i=0
(cf. [DW, 3.2]). The Cartan formula shows that a(x) is multiplicative
in x whenever this makes sense; in particular, a(a) = 0 implies that
a(x) = 0 if x is a-decomposable in M . We will be particularly inter-
ested in the case R = H+ and will let c denote a chosen fixed degree-two
generator of H+ .
The following lemma is essentially a reformulation of the Adem rela-
tions for the reduced p-th powers.
Lemma 3.3. Let R be an even-dimensional object of K, M an even-
dimensional object of U (R), and a, b degree 2 elements of R. Then, for
any x 2 M , ab(x) = ba(x).
Proof: Suppose that |x| = 2k. Let F2k be the free object in U on a
single generator 2k of dimension 2k; F2k is isomorphic to the submodule
8 W. Dwyer and C. Wilkerson
of H* K(Z=p ; 2k) generated by the fundamental class. By a universality
argument we can assume that x is the element 1 2k of R Fp F2k . By
[AW, 2.7] we can assume that x is the element 1(ck ) of RFp (H+ )k .
In fact, by multiplicativity of we can even prove the lemma by checking
the desired relation on the element x = 1 c of R H+ . In this case
explicit calculation gives
p-1Y
a(x) = xp - ap-1 x = (x + ia)
i=0
and hence
p-1Yp-1Y
ab(x) = (x + ia + jb) :
i=0 j=0
The lemma follows from the fact that this expression is symmetric in a
and b.
Lemma 3.4. Let M 2 U (H+ ) be closed and even-dimensional, and let
x be an element of M . Then there exists an element y of M such that
x = c . y if and only if c(x) = 0.
Proof: If such a y exists, then c(x) = c(c)c(y) = 0. On the
other hand, suppose that c(x) = 0 and let |x| = 2k. The Cartan
formula shows that Pk+i (x=c) = (-1)ici-1 c(x), so that the vanishing
of c(x) implies that Pj (x=c) vanishes for j > k - 1. Since M is even-
dimensional, this easily ([Li] [AW, x 2]) leads to the conclusion that
y = x=c 2 Un (S-1 M ) = M . Then x = c . (x=c) = c . y.
Lemma 3.5. Let M be an even-dimensional object of U (H+ ) and x 2 M
an element of degree 2k with the property that Pix is c-decomposable
for each i > 0. Then the action of Ap on c(x) is given by the formula
Pic(x) = k(p -i1) ci(p-1)c(x) :
Proof: Identify M with the submodule 1 M of N = H+ Fp M ; it is
clear that N is an object of U (H+ Fp H+ ). Let a denote the element
c 1 of H+ H+ and b the element 1 c. Since Pi(x) is b-decomposable
for i > 0, it follows from the multiplicativity of that ba(x) depends
only on the leading term of a(x), in particular,
ba(x) = b((-1)k ak(p-1) x) = (-1)k (ap - bp-1 a)k(p-1) b(x) :
Smith theory and the functor T 9
By definition, however,
Xpk
ab(x) = (-1)iai(p-1)Ppk-i b(x) :
i=1
The proof is finished by equating these two expressions (Lemma 3.3)
and matching up the coefficients of corresponding powers of a.
Proof of 3.2: Let 2k be the largest dimension in which M=(c . M ) is
not zero, and choose an element x 2 M of dimension 2k which is not
divisible by c. It is clear that Pi(x) is c-decomposable for each i > 0,
so the action of Ap on c(x) is given by the formula of Lemma 3.5.
A check with the Cartan formula shows that y = c-k(p-1) c(x) is a
spherical element of M . The element y is non-zero by Lemma 3.4.
x4 The rank one case
This section contains the proofs of Theorems 1.1, 1.3 and 1.4 in the
special case in which the elementary abelian p-group involved is Z=p .
Recall from x3 that S denotes the multiplicative subset of H generated
by the non-trivial elements of degree 2. Throughout this section ' :
H ! H will denote the identity map.
Proposition 4.1. If M 2 U (H) is finitely generated as a module over
H there is a natural isomorphism
T' (M ) ~= Un (S-1 M ):
Remark: An unstable Ap module F is finite if F jis finite-dimensional
for all j and zero for almost all j. Suppose that M 2 U (H) is finitely
generated as a module over H. Propositions 2.4 and 4.1 combine to give
the surprising fact that Un (S-1 M ) splits as a tensor product H Fp F
for some finite Ap module F .
Recall from x2 that T' (H) ~= H so that T' (M ) 2 U (H) if M 2 U (H).
It is easy to see that the map ffl' : M ! T' (M ) is a map of H-modules.
Lemma 4.2. Let M 2 U (H) be the tensor product H Fp F , where
F 2 U is finite. Then ffl' : M ! T' (M ) is an isomorphism.
Proof: Combine Proposition 2.2 and the fact [L1, 4.1] that ffl : F !
T (F ) is an isomorphism.
Lemma 4.3. If M 2 U (H) is finitely generated as a module over H,
then the map S-1 ffl' : S-1 M ! S-1 T' (M ) is an isomorphism.
Proof: Work by induction on the rank rk(S-1 M ) of S-1 M as a module
over S-1 (H+ ). Let M 0= Un (S-1 M ), so that rk (S-1 M 0) = rk(S-1 M )
10 W. Dwyer and C. Wilkerson
and M 0is closed. If M 0= 0 then T' (M ) vanishes by Proposition 2.3;
this case begins the induction.
Suppose then that M 0is not trivial. By Proposition 3.1 M 0contains
a non-zero spherical element x. The annihilator ideal I of x in H is
closed under the action of the Steenrod algebra, but, since M 0embeds
in S-1 M 0, I contains no element of S. It follows from Proposition
6.4 that I is trivial and thus that the cyclic Ap H submodule < x>
of M 0 generated by x is a free H module of rank one. This shows
both that rk (S-1 M 0) > 0 and, by Lemma 4.2, that the map < x> !
T' < x> is an isomorphism. Let M 00= M=< x> . By induction the
map S-1 M 00! S-1 T' (M 00) is an isomorphism. Exactness of T' and
exactness of localization now together imply that S-1 M 0! S-1 T' (M 0)
is an isomorphism. The inductive step is completed by observing that
the map M ! M 0induces isomorphisms S-1 M ~= S-1 M 0and T' (M ) ~=
T' (M 0) (the last by Proposition 2.3).
Proof of 4.1: By Lemma 4.3 the map S-1 ffl' : S-1 M ! S-1 T' (M ) is
an isomorphism. By Proposition 2.4 T' (M ) is a tensor product H Fp F
for some F 2 U , so [DW, 3.6] guarantees that the map T' (M ) !
Un (S-1 T' (M )) is an isomorphism. The proposition follows.
Let f : R ! H be a map in K(H). Any object M of U (R) is an H
module as well as an R module, so it is possible to form T' (M ) as well
as Tf (M ). There is a natural surjection T' (M ) ! Tf (M ).
Lemma 4.4. Let f : R ! H be a map in K(H) with kernel I, and let
M be an object of U (R). Then for each s 0
(1) the map Tf (M ) ! Tf (M=Is M ) is an isomorphism up through
dimension s - 1 and
(2) the map T' (M=Is M ) ! Tf (M=Is M ) is an isomorphism.
Suppose moreover that R is finitely generated as a ring and M is finitely
generated as an R module. Then for each s 0
(3) there is a natural isomorphism Tf (M=Is M ) ~= Un S-1f(M=Is M ).
Proof: To prove (1), observe that the map
Fp ~= Tf (R)0 ! Tf (R=I)0 ~= Fp
is an isomorphism, so by exactness Tf (I) vanishes in dimension 0. By
Proposition 2.1, Tf (I R I R . .R. I) (s factors) vanishes through
dimension s - 1. The rest follows from exactness of Tf . To prove
(2), argue from the fact that I=Is is nilpotent to conclude [L1, 4.3.2]
that the projection map R=Is ! R=I ~= H induces an isomorphism
Smith theory and the functor T 11
hom K (H; H) ! hom K(R=Is ; H) and thus by adjointness an isomor-
phism T (R=Is )0 ~= T (H)0. It follows that there are isomorphisms
Tf (M=Is M ) = T (M=Is M ) T (R)0 Fp
~= T (M=Is M ) T (R=Is)0 Fp
~= T (M=Is M ) T (H)0 Fp = T' (M=Is M ) :
Statement (3) now follows easily from (2) and Proposition 4.1. The
hypotheses imply that M=Is M is finitely generated as a module over
H. It is clear that given an element x of Sf there is an element y of
S such that the image of x in R=Is differs from the image of y by a
nilpotent element; this implies that the natural map S-1 (M=Is M ) !
S-1f(M=Is M ) is an isomorphism.
Proof of 1.3 (for V = Z=p ): Lemma 4.4(1) implies that there is
an isomorphism Tf (M ) ~= limsTf (M=Is M ) and it is clear by inspection
that there is an isomorphism Un ((S-1fM ) ^I) ~= lims Un S-1f(M=Is M ).
The theorem follows from Lemma 4.4(3).
Proof of 1.4 (for V = Z=p ): We will prove only the statement about
T (R). Suppose first that R 2 K(H) and that f : R ! H is a K(H) map.
By Lemma 4.4(1) and exactness there are isomorphisms
Tf (R) ~= limsTf (R=Is ) ~= limsTf (R)=Tf (Is )
andPthe proof of 4.4(1) shows that the associated graded ring gr Tf (R) =
sTf (Is )=Tf (Is+1 ) is generated as an H algebra by Tf (I)=Tf (I2 ) =
Tf (I=I2 ). Lemma 4.3(3) and exactness of Tf imply that Tf (I=I2 ) is
finitely generated as an H module. It follows that Tf (R) is finitely
generated as a ring.
Now choose R 2 K, pick a K-map g : R ! H, and let f : H Fp
R ! H be the product of g with the identity map ' of H. The above
considerations show that Tf (H Fp R) is finitely generated as a ring, and
Proposition 2.2 shows that Tf (H Fp R) is isomorphic to T' (H) Fp
Tg(R) ~= H Fp Tg(R). This proves that Tg(R) is finitely generated.
There are only a finite number of choices for the map g (sinceQR is finitely
generated as a ring) and T (R) is isomorphic to a product g Tg(R)
indexed by these choices (cf. [L1, 3.5]). This implies that Tg(R) is
finitely generated.
The next lemma requires an additional bit of notation. For any K-
map f : R ! HV let f R denote the multiplicative subset of R
consisting of all elements x such that f (x) is not a zero-divisor in HV .
12 W. Dwyer and C. Wilkerson
Lemma 4.5. Let f : R ! H be a map in K(H) and M an object of
U (R). Suppose that R is finitely generated as a ring, that M is finitely
generated as an R module, and that fflf : R ! Tf (R) is an isomorphism.
Then there is a natural isomorphism Tf (M ) ~= Un (-1fM ).
Proof: Calculating with Lemma 4.4 shows that the map fflf : M !
Tf (M ) induces an isomorphism Tf (M ) ! Tf Tf (M ). It follows that
Tf (N ) = 0 if N is either the kernel or cokernel of M ! Tf (M ). The-
orem 1.4 guarantees that N is finitely-generated as an R module, so
by Lemma 4.4(3) the localization -1f(N=IN ) vanishes, and hence by
Nakayama's lemma [AM, p. 21] the localization -1fN itself vanishes. In
other words, the map fflf induces an isomorphism -1fM ! -1fTf (M ).
Now Proposition 2.3 (together with a finite generation argument) shows
that there is an isomorphism Tf (M ) ! Tf Un -1f(M ) and hence that
there is a map fflf from Un -1fM to Tf (Un -1fM ) ~= Tf (M ). The
above considerations produce a map in the other direction from Tf M to
Un -1fTf (M ) ~= Un -1fM . The lemma follows easily.
Proof of 1.1 (for V = Z=p ): Any subgroup of W is a summand, so
the map f : HW ! H is split and can be treated as a map in K(H).
Since fflf : HW ! Tf (HW ) is an isomorphism [L1, 4.2], Lemma 4.5
applies and reduces the proof of the theorem to showing that the natural
map Un S-1f(M ) ! Un -1f(M ) is an isomorphism. Let N denote either
the kernel or the cokernel of the map fflf : M ! Tf (M ) ~= Un -1f(M ).
It is clear that N 2 U (HW ) is a finitely generated module over HW
with the property that -1f(N ) = 0; to finish the proof it is enough to
show that any such module N has S-1f(N ) = 0, or in other words, it is
enough to find an element w of Sf which annihilates N . By the finite
generation of N there exists an element x of f which annihilates N .
Write x = u + v where u belongs to the polynomial subalgebra HW+ of
HW generated by the image of the BocksteinLfi : H1 BW ! H2 BW
and v is nilpotent. By replacing x with xp for large L we can in fact
assume that x = u 2 HW+ ; note that x 6= 0 because the image of x in H
is non-nilpotent. Let I HW+ be the annihilatorTideal of N , and write
the radical of I in HW+ as an intersection i aei of prime ideals aei which
are closed under the action of the Steenrod algebra (Proposition 6.1).
By Proposition 6.3 each aei is generated as an ideal by two-dimensional
classes. If any aei has all of its two-dimensional generators contained in
the kernel of f then I ker (f ), which is impossible because x 2 I.
Consequently, it is possible to choose from each aei a two-dimensional
generator wi such that the image of wi in H is non-zero. It is clear that
Smith theory and the functor T 13
wiQ2 Sf . If w is set equal to a sufficiently high power of the product
iwi, then w is the desired element in Sf \ I.
Example 4.6: The following example from Smith theory illustrates
that it is necessary to include a completion of some type in the state-
ment of Theorem 1.3 . Let p = 2, let R = H* BO(2) and let f : R ! H
be the map induced by an homomorphism Z=2 ! O(2) sending the
non-trivial element of Z=2 to a matrix of determinant -1. Note that
the determinant map O(2) ! Z=2 lifts R to an H algebra and f to a
morphism in K(H). It follows from [L2], say, that Tf (R) is isomorphic
to H* (RP 1 x RP 1 ), but there is only one unstable one-dimensional
generator in S-1fR. Let w1 and w2 be the Stiefel-Whitney classes which
generate H* BO(2) as a polynomial algebra. The additional necessary
one-dimensional generator appears in the completion of the localization
of R as the infinite sum
X1 i
w1(w2=w21)2
i=0
(a formula which can be derived by the a posteriori knowledge that this
generator ff satisfies the equation ff2 + w1ff + w2 = 0).
x5 The general case
In this section we will describe the argument which is used to prove
Theorems 1.1, 1.3 and 1.4 for a general elementary abelian p-group V .
We will use the fact that for V = Z=p these theorems have already been
proven in x4.
Let SV HV be the multiplicative subset generated by the non-
zero elements in the image of the Bockstein map fi : H1 BV ! H2 BV
and HV+ the subalgebra of HV generated by the elements of SV . It
is possible to check that the proofs in x2 and x4 hold almost verbatim
with H replaced by HV , H+ by HV+ , T by T V and S by SV ; the only
exception is the proof of Lemma 4.3 to the extent to which it relies on
Lemma 3.1. To complete the proof of the desired results for V , then, it
is enough to give an appropriate generalization of Lemma 3.1.
The closure of an object M 2 U (HV ) is defined to be Un (SV )-1 M ;
M is closed if the natural map from M to the closure of M is an iso-
morphism.
Lemma 5.1. If M 2 U (HV ) is non-zero, closed and finitely generated
as a module over HV then M contains a non-zero spherical element.
Proof: The argument is by induction on the dimension of V as an
Fp vector space. By Lemma 3.1, we can assume that this dimension
14 W. Dwyer and C. Wilkerson
is greater than one. Write V as a direct sum Z=p W for some W ;
this gives a map g : H ! HV which lifts HV to K(H), as well as
a map f : HV ! H such that f . g is the identity map of H. By
the special case of Theorem 1.3 proven in x4 there is an isomorphism
Tf (M ) ~= Un S-1f(M ). It follows from Proposition 2.4 that there is a
tensor product splitting Tf (M ) ~= H Fp N which is compatible with
the evident [L1, 4.2] splitting Tf (HV ) ~= HV ~= H Fp HW . Since M
is closed, Un S-1f(M ) is isomorphic to M . The conclusion is that M
splits as a tensor product H Fp N where N is an object of U (HW )
which is necessarily non-zero, closed and finitely generated as a module
over HW . By induction N contains a non-zero spherical element x. The
element 1 x 2 H Fp N ~= M is the desired spherical element of M .
x 6 Some algebraic facts
The purpose of this section is to gather together some standard alge-
braic results. Proposition 6.3 is used in the proof of Theorem 1.1 (see
x4) and Proposition 6.4 in the proof of Lemma 4.3 (particularly in the
inductive setting described in x5).
Proposition 6.1. Suppose that R 2 K is evenly graded and finitely
generated as a ring, and that I R is a homogeneous ideal which is
closed under the actionTof Ap . Then the radical of I in R can be written
as an intersection iaei of homogeneous prime ideals aei closed under the
action of Ap .
Proof: Let J be the radical of I (that is, the ideal of all elements x 2 R
such that some power of x lies in I). It is easy to prove by induction on
j and the Cartan formula that Pj x 2 J if x 2 J, in other words, thatT
J is closed under the action of Ap . Write J as an intersection i oei of
(homogeneous) prime ideals (see [AM, Chap. 7]). For each i let oe(1)i
be the set of elements x 2 oei such that Pj x 2 oei for all j 0. The
argument of [AW, p. 138] shows that oe(1)iis a prime ideal contained in
oei, although it is not evident that oe(1)iis closed under the action of Ap .
Iterate the procedure of passing from oei to oe1i to obtain a descending
chain
oei oe(1)i oe(2)i . . .
of prime ideals in R. Such a chain must eventually become constant
[ZS, p. 241]; let aei denote the limiting constant value. It is clear that
aei isTa prime ideal of R which is closed under the action of Ap and that
J = iaei.
The following propositions use some of the notation of x5.
Smith theory and the functor T 15
Proposition 6.2. Let V be an elementary abelian p-group, and sup-
pose that I HV is a non-trivial homogeneous ideal closed under the
action of Ap . Then I contains a non-zero element of HV+ .
Proof: Let e1; : : :; ek be a collection of generators for H1 BV and
t1; : : :; tk their Bockstein images in H2 BV , so that HV+ is the poly-
nomial algebra Fp [t1; : : :; tk ] and HV is isomorphic as an algebra to the
tensor product of HV+ and an exterior algebra on e1; : : :; ek . Pick a
non-zero element x in I. By multiplying x, if necessary, by a suitable
product of ei's we can assume that x has the form te1e2 . .e.k where
t 2 HV+ is non-zero. Define a sequence d1; : : :; dk of integers inductively
by setting d1 = |t|=2 + 1 and di+1 = pdi + 1 (i 1). Define elements
ff0; : : :; ffk-1 of Ap by setting ff0 = fi and ffi+1 = fiPdi+1 ffi (i 0). A
short calculation then shows that
k-1 X pk-1 pk-2
ffk-1 x = tp sgn (oe)toe(1)toe(2). . .toe(k)
oe
where oe runs through the permuation group on k letters. It is clear that
ffk-1 x is a non-zero element of I \ HV+ .
Proposition 6.3. Suppose that V is an elementary abelian p-group
and that I HV+ is a homogeneous prime ideal which is closed under
the action of Ap . Then I is generated as an ideal by its elements of
dimension 2.
Proof: This result is due to Serre ([Se], cf. [AW, 1.11]).
Proposition 6.4. Suppose that V is an elementary abelian p-group
and that I HV is a non-trivial homogeneous ideal closed under the
action of Ap . Then I contains an element of SV .
Proof: Let J be the intersection of the radical of I with HV+T . By
Propositions 6.1 and 6.3, J can be written as an intersection i aei of
prime ideals in HV+ each of which is generated by elements of dimension
2. Proposition 6.3 guarantees that J is not trivial, so each of the aei is
also a non-trivial ideal. Pick non-zero two-dimensional elements xi 2 aei
and let x be the product of the xi's. It is clear that some power of x lies
in I \ SV .
Remark: The top Dickson invariant c0 [W2] in HV is by definition
the product of the two-dimensional elements of SV . It follows from
Proposition 6.4 that any ideal I of the indicated type actually contains
some power of c0.
16 W. Dwyer and C. Wilkerson
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University of Notre Dame, Notre Dame, Indiana 46556 USA
Purdue University, West Lafayette, Indiana 47907 USA