Modular coinvariants and the mod p homology of QSk

Submitted exclusively to the London Mathematical Society
doi:10.1112/0000/000000

Modular coinvariants and the mod p homology of QS k
Phan Hoàng Chơn
Abstract
In this paper, we use the modular coinvariants theory to establish a complete set of relations
of the mod p homology of {QS k }k≥0 , for p odd, as a ring object in the category of coalgebras,
so called a coalgebraic ring or a Hopf ring. Beside, we also describe the action of the mod p
Dyer-Lashof algebra as well as one of the mod p Steenrod algebra on the coalgebraic ring.

1. Introduction
Let G∗ (−) be an unreduced multiplicative cohomology theory. Then, G∗ (−) can be
represented unstably by the infinite loop spaces Gn of its associated Ω-spectrum (i.e. Gk (X) ∼
=
[X, Gk ] naturally and ΩGk+1 Gk , where we denote by [X, Y ] the homotopy classes of unbased
maps from X to Y ). The collection of these spaces G∗ = {Gk }k∈Z is considered as a graded
ring space with the loop sum
m : Gk × Gk → Gk
and the composition product
µ : Gk × G → Gk+ .
Therefore, the homology of {Gk }k∈Z (beside the usual addition and coproduct) has two
operations, which are denoted by and ◦, respectively, induced by m and µ. These operations
make the homology of {Gk }k∈Z a ring object in the category of coalgebras, which is called a
Hopf ring or coalgebraic ring (see Ravenel-Wilson [26], and Hunton-Turner [9]). The Hopf ring
structure actually becomes an important tool to study the homology of Ω-spectrum as well
as the unreduced generalized multiplicative cohomology theory, and it is of interest in study
of algebraic topologists. For example, the Hopf ring for complex cobordism M U is studied
by Ravenel-Wilson [26], the Hopf ring for Morava K-theory is studied by Wilson [28] and
for connective Morava K-theory by Kramer[19], Boardman-Kramer-Wilson [2]. Recently, the
Hopf ring structure for BP and KO, KU are respectively investigated by Kashiwabara [11],

Kashiwabara-Strickland-Turner [16] and Mortion-Strickland [23].
Let QS k = limΩn Σn S k be the infinite loop space of the sphere S k . Then {QS k }k≥0 is an
−→

Ω-spectrum, called the sphere spectrum, therefore, the mod p homology of {QS k }k≥0 also has
a Hopf ring structure. Moreover, it is well known that all spectra are module spectra over the
sphere spectrum, so the mod p homology of any infinite loop space becomes an H∗ QS 0 -module
or {H∗ QS k }k≥0 -module object in the category of coalgebras, which is called a coalgebraic
module. As is well-known, (see Kashiwabara [14]) the mod p homology of an infinite loop
space has an A-H∗ QS 0 -coalgebraic module structure. Beside, from the result of May [3], the
mod p homology of an infinite loop space also has a so-called A-R-allowable Hopf algebra, i.e.,
it is a Hopf algebra on which both the Steenrod and the Dyer-Lashof algebra act satisfying
some compatibility conditions. Thus, understanding the coalgebraic ring structure of H∗ QS 0

2000 Mathematics Subject Classification 55P47, 55S12 (Primary), 55S10, 20C20 (Secondary).
This work is partial supported by a NAFOSTED gant.


Page 2 of 22

PHAN HOÀNG CHƠN

plays important role in the study of homology of infinite loop spaces as well as in the study
of the category of A-H∗ QS 0 -coalgebraic modules and one of A-R-allowable Hopf algebras, and
relationship between them.
By the results of Araki-Kudo [1], Dyer-Lashof [5] and May [3], the mod p homology of
{QS k }k≥0 is generated as a Hopf ring by Qi [1], i ≥ 0, σ (for p = 2) and by Qi [1], i ≥ 0,
βQi [1], i ≥ 1, σ (for p odd), where Qi is the ith homology operation (which is called the DyerLashof operation), [1] ∈ H∗ QS 0 is the image of the non-base point generator of H0 S 0 under
the homomorphism H0 S 0 → H0 QS 0 induced by the inclusion S 0 → QS 0 and σ is the image of
the basis element of H1 S 1 under the homomorphism H1 S 1 → H1 QS 1 induced by the inclusion

S 1 → QS 1 . This actually corresponds the fact that the Quillen’s approximation map of finite
groups by elementary abelian subgroups is a monomorphism [25]. However, a long time, no
one undertook to sudy the relations until the importance of the coalgebraic ring structure of
H∗ QS k is clearly made again from works of Hunton-Turner [9] and Kashiwabara [13] (which
develop the homological algebra for the category of modules over a Hopf ring). These works are
maybe the main motivation for study in [27] and [6], which give a description of a complete set
of relations as a Hopf ring of H∗ QS k for p = 2. Later, it was discovered in [12] that the nice
description of the complete set of relations comes from the fact that the Quillen’s map for the
symmetric groups is actually an isomorphism at the prime 2 (see [7]). Also according to [7],
the map is no longer an isomorphism for odd primes, therefore, it is difficult to generalize the
results in [27] and [6] for odd primes. However, in the Brown-Peterson cohomology theory, the
Quillen’s map of the symmetric groups is also an isomorphism [8]. This fact allows to generalize
the results in [27] and [6] for the Bockstein-nil homology of H∗ QS k [15]. Thus, the describing
of a complete set of relations as a Hopf ring for {H∗ QS k }k≥0 is not only important but also
difficult.
In this work, we discover that the isomorphism between the dual of R[n] and the image of the
restriction map from the cohomology of the symmetric group Σpn to the elementary abelian
p-group of rank n, Vn , is the main key to establish the nice description of the complete set
of relations as above discussion, where R[n] denote the subspace of the Dyer-Lashof algebra
spanned by all monomials of length n. Using this idea and modifying the framework in [27]
allows us to obtain a nice description of the complete set of relations as a Hopf ring of
{H∗ QS k }k≥0 for p odd. In more detail, we construct a new basis for B[n]∗ , which is the
dual of the image of the restriction map from the cohomology of the symmetric group Σpn to
the cohomology of Vn [24]. Using the basis and combining with the fact that the induced in
(n)
homology of the Kahn-Priddy transfer, tr∗ , is multiplicative and GLn -invariant to investigate,
we obtain an analogous description of a complete set of relation of {H∗ QS k }k≥0 as a coalgebraic
ring for odd primes. This fact again confirms the closely correspondence between the Hopf ring
structure of {H∗ QS k }k≥0 and the Quillen’s map of the symmetric groups. The results in [27],
[6] as well as in [15] can be deduce from our results by letting p = 2 or killing the action of

the Bockstein operation for p odd. It should be noted that much of our work rests on previous
results with a suitable modifying. For example, relations (4.1)-(4.3) (see Proposition 4.7) can be
(2)
followed from the multiplicativity and the GL2 -invariant of tr∗ as the case of p = 2. However,
the relation (4.4) is here difference. In deed, for p = 2 or for the Bocktein-nil homology, the
general case of the relation can be simple implied from the case of the length 1 and other
relations, but here it is impossible because of the action of Bocktein operation.
In this paper, additive base of (H ∗ BVn )GLn , (H∗ BVn )GLn as well as the cokernel of
/ (H ∗ BVn )GLn are also established. Beside, using the similar
the restriction map H ∗ BΣpn
method of Turner [27], we give descriptions of the action of the mod p Steenrod algebra A and
the action of the mod p Dyer-Lashof algebra R on the Hopf ring as relative results.
The paper is divided into five sections. The first two sections are preliminaries. In Section
2, we review some main points of the Dickson-Mùi algebra, the image of the restriction from


MODULAR COINVARIANTS AND THE MOD p HOMOLOGY OF QS k

Page 3 of 22

the cohomology of the symmetric group Σpn to the cohomology of the elementary abelian pgroup of rank n, Vn , as well as the mod p Dyer-Lashof algebra. In Section 3, we construct new
additive base for (H ∗ BVn )GLn , (H∗ BVn )GLn , the cokenel as well as the dual of the image of
/ (H ∗ BVn )GLn . By the results of May [3], the new basis of the
the restriction map H ∗ BΣpn
dual of the image of the restriction map is considered as an additive basis of the subspace R[n]
of the mod p Dyer-Lashof algebra. The Hopf ring for {H∗ QS k }k≥0 as well as the actions of
the Steenrod algebra and the Dyer-Lashof algebra on {H∗ QS k }k≥0 are respectively presented
in two final sections.

2. Preliminaries

In this section, we review some main points of the Dickson-Mùi algebra and the image of
the restriction from the cohomology of the symmetric group Σpn to the cohomology of the
elementary abelian p-group of rank n, Vn . We also review some basic properties of the mod p
Dyer-Lashof algebra.
2.1. Modular invariant
Let Vn be an n-dimensional Fp -vector space, where p is an odd prime number. It is well-known
that the mod p cohomology of the classifying space BVn is given by
H ∗ BVn = E(e1 , · · · , en ) ⊗ Fp [x1 , · · · , xn ],
where (e1 , · · · , en ) is a basis of H 1 BVn = Hom(Vn , Fp ), xi = β(ei ) for 1 ≤ i ≤ n with β
the Bockstein homomorphism, E(e1 , · · · , en ) is the exterior algebra generated by ei ’s and
Fp [x1 , · · · , xn ] is the polynomial algebra generated by xi ’s.
Let GLn denote the general linear group GLn = GL(Vn ). The group GLn acts on Vn and
then on H ∗ BVn according to the following standard action
ais xi ,

(aij )xs =

(aij ) ∈ GLn .

ais ei ,

(aij )es =

i

i

The algebra of all invariants of H ∗ BVn under the actions of GLn is computed by Dickson [4]
and Mùi [24]. We briefly summarize rtheir results. For any n-tuple of non-negative integers
j

(r1 , . . . , rn ), put [r1 , · · · , rn ] := det(xpi ), and define
Ln,i := [0, · · · , ˆi, . . . , n];

Ln := Ln,n ;

qn,i := Ln,i /Ln ,

for any 1 ≤ i ≤ n. In particular, qn,n = 1 and by convention, set qn,i = 0 for i < 0. The degree
of qn,i is 2(pn − pi ). Define
Vn := Vn (x1 , · · · , xn ) :=

(λ1 x1 + · · · + λn−1 xn−1 + xn ).
λj ∈Fp

Another way to define Vn is that Vn = Ln /Ln−1 . Then qn,i can be inductively expressed by
the formula
p
qn,i = qn−1,i−1
+ qn−1,i Vnp−1 .

For non-negative integers k, rk+1 , . . . , rn , set

[k; rk+1 , · · · , rn ] :=

1
k!

e1
·
e1

xp1

rk+1

·
rn
xp1

···
···
···
···
···
···

en
·
en
rk+1

xpn

·
rn
xpn

.


Page 4 of 22

PHAN HOÀNG CHƠN

For 0 ≤ i1 < · · · < ik ≤ n − 1, we define
Mn;i1 ,...,ik := [k; 0, · · · , ˆi1 , · · · , ˆik , · · · , n − 1],
Rn;i1 ,··· ,ik := Mn;i1 ,...,ik Lnp−2 .
The degree of Mn;i1 ,··· ,ik is k + 2((1 + · · · + pn−1 ) − (pi1 + · · · + pik )) and then the degree of
Rn;i1 ,··· ,ik is k + 2(p − 1)(1 + · · · + pn−1 ) − 2(pi1 + · · · + pik ).
We put Pn := Fp [x1 , · · · , xn ]. The subspace of all invariants of H ∗ BVn under the action of
GLn is given by the following theorem.
Theorem 2.1 (Dickson [4], Mùi [24]).
(i) The subspace of all invariants of Pn under the action of GLn is given by
D[n] := PnGLn = Fp [qn,0 , · · · , qn,n−1 ].
(ii) As a D[n]-module, (H ∗ BVn )GLn is free and has a basis consisting of 1 and all elements
of {Rn;i1 ,··· ,ik : 1 ≤ k ≤ n, 0 ≤ i1 < · · · < ik ≤ n − 1}. In other words,
n
∗

(H BVn )

GLn

=

PnGLn

Rn;i1 ,··· ,ik PnGLn .

⊕
k=1 0≤i1 <···

(iii) The algebraic relations are given by
2
Rn;i
= 0,
k−1
Rn;i1 · · · Rn;ik = (−1)k(k−1)/2 Rn;i1 ,··· ,ik qn,0

for 0 ≤ i1 < · · · < ik < n.
Let B[n] be the subalgebra of (H ∗ BVn )GLn generated by
(1) qn,i for 0 ≤ i ≤ n − 1,
(2) Rn;s for 0 ≤ s ≤ n − 1,
(3) Rn;s,t for 0 ≤ s < t ≤ n − 1.
Mùi shows that, [24], the algebra B[n] is the image of the restriction from the cohomology
of the symmetric group Σpn to the cohomology of the elementary abelian p-group of rank n,
Vn .
In [3], May shows that n≥1 B[n] is isomorphic to the dual of the Dyer-Lashof algebra.
2.2. The Dyer-Lashof algebra
Let us recall the construction of the Dyer-Lashof algebra. Let F be the free algebra generated
by {f i |i ≥ 0} and {βf i |i > 0} over Fp , with |f i | = 2i(p − 1) and |βf i | = 2i(p − 1) − 1. Then
F becomes a coalgebra equipped with coproduct ψ : F → F ⊗ F given by
ψf i =

f i−j ⊗ f j ;

ψβf i =

βf i−j ⊗ f j +

f i−j ⊗ βf j .

Elements of F are of the form
f I,ε = β 1 f i1 · · · β n f in ,
where (I, ε) = ( 1 , i1 , · · · , n , in ) with j ∈ {0, 1} and ij ≥ j for 1 ≤ j ≤ n. The degree of f I,ε
is equal to 2(p − 1)(i1 + · · · + in ) − ( 1 + · · · + n ). Let l(f I,ε ) = n denote the length of (I, ε) or
f I,ε and let the excess of (I, ε) or f I,ε be denoted and defined by exc(f I,ε ) = 2i1 − 1 − |f I ,ε |,
where (I , ε ) = ( 2 , i2 , · · · , n , in ). In other words,
n

exc(f I,ε ) = 2i1 −

1

− 2(p − 1)

n

ij +
j=2

j.
j=2


MODULAR COINVARIANTS AND THE MOD p HOMOLOGY OF QS k

Page 5 of 22

The excess is defined ∞ if (I, ε) = ∅ and we omit j if it is 0. The element f I,ε is as
having non-negative excess if f It ,εt is non-negative excess for all 1 ≤ t ≤ n, where (It , εt ) =

( t , it , · · · , n , in ).
The algebra F is a Hopf algebra with unit η : Fp → F and augmentation : F → Fp sending
f 0 to 1 and others to zero.
Let T = F/Iexc , where Iexc is the two-sided ideal of F generated by all elements of negative
excess. Then T inherits the structure of a Hopf algebra. Denote the image of f I,ε by eI,ε . The
degree, length, excess described above passes to T .
Let IAdem be the two-sided ideal of T generated by elements
er es −

(p − 1)(i − s) − 1 r+s−i i
e
e , r > ps;
pi − r

(−1)r+i
i

er βes −

(p − 1)(i − s)
βer+s−i ei
pi − r

(−1)r+i
i

(−1)r+i

+
i

(p − 1)(i − s) − 1 r+s−i i
e
βe , r ≥ ps.
pi − r − 1

These elements are called Adem relations. The quotient algebra R = T /IAdem is called the
Dyer-Lashof algebra. We denote the image of eI,ε by QI,ε , then Qi and βQi satisfy the Adem
relations:
(p − 1)(i − s) − 1
Qr Qs =
(−1)r+i
Qr+s−i Qi , r > ps;
(2.1)
pi
−
r
i
Qr βQs =

(p − 1)(i − s)
βQr+s−i Qi
pi − r

(−1)r+i
i

−

(−1)

r+i

i

(p − 1)(i − s) − 1
Qr+s−i βQi , r ≥ ps.
pi − r − 1

(2.2)

Let P∗r be the dual to the Steenrod cohomology operation P r , then the Nishida relations
hold:
(p − 1)(s − r)
P∗r Qs =
(−1)r+i
Qs−r+i P∗i ;
r
−
pi
i
P∗r βQs =

(−1)r+i
i

(p − 1)(s − r) − 1
βQs−r+i P∗i
r − pi
(−1)r+i

+
i
I,ε

(p − 1)(s − r) − 1
Qs−r+i P∗i β.
r − pi − 1

A monomials Q
is called admissible if (I, ε) is admissible (i.e. a string (I, ε) =
( 1 , i1 , · · · , n , in ) is admissible if pik − k ≥ ik−1 for 2 ≤ k ≤ n).
Let R[n] be the subspace of R spanned by all monomials of length n. Due to the form of the
Adem relations, R[n] has an additive basis consisting of all admissible monomials of length n
and non-negative excess, which is called the admissible basis.
Next, we recall the structure of the dual of the Dyer-Lashof algebra. For p = 2, the structure
is studied by Madsen [20]. He shows that R[n]∗ is isomorphic to the Dickson algebra. For p
odd, May [3] shows that R[n]∗ is isomorphic to a proper subalgebra of the Dickson-Mùi algebra
(see also Kechagias [18]).
For convenience we shall write I instead of (I, ε).


Page 6 of 22

PHAN HOÀNG CHƠN

Let In,i , Jn;i , Kn;s,i be admissible sequences of non-negative excess and length n as follows
In,i = (pi−1 (pn−i − 1), · · · , pn−i − 1, pn−i−1 , · · · , 1);
Jn;i = (pi−1 (pn−i − 1), · · · , pn−i − 1, (1, pn−i−1 ), · · · , 1);
Kn;s,i = (pi−1 (pn−i − 1) − ps−1 , · · · , pi−s (pn−i − 1) − 1),

(1, pi−s−1 (pn−i − 1)), pi−s−2 (pn−i − 1), · · · , p(pn−i − 1),
(1, pn−i − 1), pn−i−1 , · · · , 1).
Then the excess of QIn,i is 0 if 0 < i ≤ n − 1 and 2 if i = 0; and
exc(QJn;i ) = 1, 0 ≤ i ≤ n − 1;
exc(QKn;s,i ) = 0, 0 ≤ s < i ≤ n − 1.
Let ξn,i = (QIn,i )∗ , 0 ≤ i ≤ n − 1, τn;i = (QJn;i )∗ , 0 ≤ i ≤ n − 1, and σn;s,i = (QKn;s,i )∗ , 0 ≤
s < i ≤ n − 1, with respect to the admissible basis of R[n].
The following theorem gives the structure of the dual of the Dyer-Lashof algebra.
Theorem 2.2 (May [3], see also Kechagias [18]). As an algebra, R[n]∗ is isomorphic to
the free associative commutative algebra over Fp generated by the set {ξn,i , τn;i , σn;s,i : 0 ≤ i ≤
n − 1, 0 ≤ s < i}, subject to relations:
2
(i) τn,i
= 0, 0 ≤ i ≤ n − 1;
(ii) τn;s τn;i = σn;s,i ξn,0 , 0 ≤ s < i ≤ n − 1;
(iii) τn;s τn;i τn;j = τn;s σn;i,j ξn,0 , 0 ≤ s < i < j ≤ n − 1;
2
(iv) τn;s τn;i τn;j τn;k = σn;s,i σn;j,k ξn,0
, 0 ≤ s < i < j < k ≤ n − 1.
The relationship between the dual of the Dyer-Lashof algebra and the modular invariants is
given by the following theorem.
Theorem 2.3 (Kechagias [17], [18]). As algebras over the Steenrod algebra, R[n]∗ is
isomorphic to B[n] via the isomorphism Φ given by Φ(ξn,i ) = −qn,i , Φ(τn;i ) = Rn;i , 0 ≤ i ≤
n − 1 and Φ(σn;s,i ) = Rn;s,i , 0 ≤ s < i ≤ n − 1.

3. Additive base of modular (co)invariants
In this section, we construct a new basis for B[n]∗ , which is a useful tool for the Section 4.
Since R[n] ∼
= B[n]∗ , the basis can be considered is a basis of R[n]. Beside, some additive base of
the Dickson-Mùi invariants (H ∗ BVn )GLn , the Dickson-Mùi coinvariants (H∗ BVn )GLn as well

/ (H ∗ BVn )GLn are
as the cokernel of the restriction map of the symmetric group H ∗ BΣpn
established.
We order the set of tuples I = ( 1 , i1 , · · · , n , in ) by the ordering defined inductively as follows
(1) ( 1 , i1 ) < (ω1 , j1 ) if 1 + i1 < ω1 + j1 or i1 + 1 = j1 + ω1 , 1 < ω1 ;
(2) ( 1 , i1 , · · · , k , ik ) < (ω1 , j1 , · · · , ωk , jk ) if:
(a) I = ( 1 , i1 , · · · , k−1 , ik−1 ) < (ω1 , j1 , · · · , ωk−1 , jk−1 ) = J or
(b) I = J, ik + pk−1 k < jk + pk−1 ωk or
(c) I = J, ik + pk−1 k = jk + pk−1 ωk and k < ωk .
It should be noted that, when k = ωk = 0 for all k, the above ordering coincides with the
lexicographic ordering from the left.
in
i1
1
n
(respect, V I , xI ) is called less than q J (respect,
A monomial q I = Rn;0
qn,0
· · · Rn;n−1
qn,n−1
J
J
V , x ) if I < J.


MODULAR COINVARIANTS AND THE MOD p HOMOLOGY OF QS k

Page 7 of 22

Then we obtain the following lemmas.

Lemma 3.1. For is ≥ 0, we have
(p−1)i1 p(p−1)(i1 +i2 )
x2

in
i1
= x1
· · · qn,n−1
qn,0

n−1

· · · xnp

(p−1)(i1 +···+in )

+ greater.

Proof. For 0 ≤ s ≤ n − 1, using the inductive formula
p
qn,s = qn−1,s−1
+ qn−1,s Vnp−1 ,

we can express qn,s in Vi ’s as follows
qn,s = (Vs · · · Vn )p−1 + greater.
It implies
(p−1)i1

i1
in

qn,0
· · · qn,n−1
= V1

· · · Vn(p−1)(i1 +···+in ) + greater

Moreover, by the definition
n−1

(λ1 x1 + · · · + λn−1 xn−1 + xn ) = xpn

Vs =

+ greater.

λi ∈Fp

So that, we have
(p−1)i1 p(p−1)(i1 +i2 )
x2

i1
in
qn,0
· · · qn,n−1
= x1

n−1

· · · xnp

(p−1)(i1 +···+in )

+ greater.

The proof is complete.
For any string of integers I = ( 1 , i1 , . . . , n , in ), with i1 ∈ Z, is ≥ 0, 2 ≤ s ≤ n, and
{0, 1}, we put b(I) = s s and m(I) = max{ s : 1 ≤ s ≤ n}.
Lemma 3.2. For I = ( 1 , i1 , . . . ,
m(I) + b(I) ≥ 0, we have

n , in ),

with i1 ∈ Z, is ≥ 0, 2 ≤ s ≤ n,

s

(p−1)(i1 +b(I))−

1

n−1

· · · enn xnp

(p−1)(i1 +···+in +b(I))−pn−1

n

Proof. From the proof of Lemma 3.1, we have

(p−1)i1

i1
in
qn,0
· · · qn,n−1
= V1

(p−1)i1

= L1

· · · Vn(p−1)(i1 +···+in ) + greater
(p−1)(i1 +···+in )

(p−1)(i1 +i2 )

L2

(p−1)(i1 +i2 )

···

L1

Ln

(p−1)(i1 +···+in )

+ greater

Ln−1

(p−1)(i1 +···+in )

Ln

=

(p−1)i2

L1

(p−1)in

· · · Ln−1

+ greater.

Since Rn;s = Mn;s Lp−2
n , for 0 ≤ s ≤ n − 1, we obtain
i1
in
1
n
Rn;0
qn,0
· · · Rn;n−1
qn,n−1
(p−1)(i1 +···+in +b(I))−b(I)

1
n
= Mn;0
· · · Mn;n−1

Ln

(p−1)i2

L1

(p−1)in

· · · Ln−1

(p−1)(i1 +b(I))−b(I)

1
n
= Mn;0
· · · Mn;n−1
V1

∈

∈ {0, 1}, and i1 −

i1
in

1
n
Rn;0
qn,0
· · · Rn;n−1
qn,n−1

=(−1) 2 +···+(n−1) n e11 x1
+ greater.

s

+ greater

· · · Vn(p−1)(i1 +···+in +b(I))−b(I) + greater.


Page 8 of 22

PHAN HOÀNG CHƠN

Since is ≥ 0, 2 ≤ s ≤ n and i1 − m(I) + b(I) ≥ 0, applying the proof of Lemma 3.1, we get
i1
in
1
n
Rn;0
qn,0
· · · Rn;n−1
qn,n−1

(p−1)(i1 +b(I))−b(I)

1
n
= Mn;0
· · · Mn;n−1
x1

n−1

· · · xnp

(p−1)(i1 +···+in +b(I))−pn−1 b(I)

+ greater.
Moreover, for 0 ≤ s ≤ n − 1,
s−1

Mn;s = (−1)s x1 xp2 · · · xsp
s−1

s+1

s+1

n−1

es+1 xps+2 · · · xpn

+ greater,

n−1

in other words, x1 xp2 · · · xsp es+1 xps+2 · · · xpn
is the least monomial occurring non-trivially
in Mn;s . Indeed, it is sufficient to compare the order of n following monomials.
e1 x2 xp3 · · · xsp

s−2

s−1

s+1

n−1

xps+1 xps+2 · · · xpn

,

····································
2

x1 xp2 xp3 · · · xsp

s−1

s+1

n−1

es+1 xps+2 · · · xpn

,

····································
2

x1 xp2 xp3 · · · xps

s−1

s+1

n−1

xps+1 · · · xpn−1 en .

By directly checking, we have the assertion.
Combining these facts, we have the assertion of the lemma.
Proposition 3.3. For any n ≥ 1, as an Fp -vector space, (H ∗ BVn )GLn has a basis
i1
in
1
n
consisting of all elements q I = Rn;0
qn,0
· · · Rn;n−1
qn,n−1
for i1 ∈ Z, is ≥ 0, 2 ≤ s ≤ n, s ∈ {0, 1}

and i1 − m(I) + b(I) ≥ 0.
i1
in
1
n
Proof. From Theorem 2.1, {q I = Rn;0
qn,0
· · · Rn;n−1
qn,n−1
: i1 − m(I) + b(I) ≥ 0} is a set
of generators of (H ∗ BVn )GLn .
Moreover, from Lemma 3.2, this set is linear independent.

i1
in
1
n
Proposition 3.4. For any n ≥ 1, the set of elements q I = Rn;0
qn,0
· · · Rn;n−1
qn,n−1
for
i1 ∈ Z, is ≥ 0, 2 ≤ s ≤ n, s ∈ {0, 1} and 2i1 + b(I) ≥ 0, provides an additive basis for B[n].

Proof. From Proposition 3.3, we see that the set in the proposition is the subset of a basis
of (H ∗ BVn )GLn , therefore, it is linear independent.
Moreover, since, for 0 ≤ s < t ≤ n − 1,
−1
Rn;s,t = Rn;s Rn;t qn,0
,

every elements in B[n] can be written as a linear combination of elements of the set.
Corollary 3.5. For any n ≥ 1, as an Fp -vector space, the cokernal of the restriction map
/ H ∗ (BVn )GLn has a basis consisting of all elements that are the images under
H ∗ (BΣpn )
i1
in
1
n
the quotient map of all elements of the form Rn;0
qn,0
· · · Rn;n−1
qn,n−1
for i1 ∈ Z, is ≥ 0, 2 ≤
s ≤ n, s ∈ {0, 1} and m(I) − b(I) ≤ i1 < −b(I)/2.
in
i1
n
1
qn,0
· · · Rn;n−1
qn,n−1
: 2i1 + b(I) ≥ k} is
For k ≥ 0, the subspace of B[n] generated by {Rn;0
a subalgebra of B[n], which is denoted by Bk [n]. It is immediate that B0 [n] = B[n].


MODULAR COINVARIANTS AND THE MOD p HOMOLOGY OF QS k

Page 9 of 22

Let ui ∈ H1 BVn be the dual of ei and let vi ∈ H2 BVn be the dual of xi . Then the homology
of Vn , H∗ BVn , is the tensor product of the exterior algebra generated by ui ’s and the divided
[t]
power algebra generated by vi ’s. We denote by vi the t-th divided power of vi . Since R[n] is
isomorphic to B[n]∗ , R[n] is considered the quotient algebra of (H∗ BVn )GLn . The following
theorem provides an additive basis for B[n]∗ and then for R[n].
Theorem 3.6. For k ≥ 0, the set of all elements
[(p−1)(i1 +b(I))−

[u11 v1

for i1 ∈ Z, is ≥ 0, 2 ≤ s ≤ n,
Bk [n]∗ , the dual of Bk [n].

s

1]

· · · unn vn[p

n−1

(p−1)(i1 +···+in +b(I))−pn−1

n]

],

∈ {0, 1}, and 2i1 + b(I) ≥ k provides an additive basis for

Proof. Denote
q( 1 , i1 , · · · ,
(−1)

n , in )

2 +···+(n−1)

n

=
[(p−1)(i1 +b(I))−

u11 v1

1]

· · · unn vn[p

n−1

(p−1)(i1 +···+in +b(I))−pn−1

n]

.

From Lemma 3.2, we see that
i1

in
1
n
Rn;0
qn,0
· · · Rn;n−1
qn,n−1
, q(ω1 , s1 , · · · , ωn , sn )

=

0, (ω1 , s1 , · · · , ωn , sn ) < ( 1 , i1 , · · · ,
1, (ω1 , s1 , · · · , ωn , sn ) = ( 1 , i1 , · · · ,

n , in );
n , in ).

Therefore, the set of all [q( 1 , i1 , · · · , n , in )] satisfying the condition in the theorem provides
a basis of Bk [n]∗ .
Moreover, since Bk [n]∗ is a quotient algebra of (H∗ BVn )GLn ,
[q( 1 , i1 , · · · ,
=

n , in )]
[(p−1)(i1 +b(I))−
[u1 v1
1

1]

· · · unn vn[p

n−1

(p−1)(i1 +···+in +b(I))−pn−1

n]

].

Hence, we have the assertion of the theorem.
It should be noted that, when k = 0, the basis mentioned in Theorem 3.6 is not the dual
basis of the one in Proposition 3.4.
Using the proof is similar to the proof of Theorem 3.6, we have the following proposition.
Proposition 3.7. For n ≥ 1, the set of all elements
[(p−1)(i1 +b(I))−

[u11 v1

for i1 ∈ Z, is ≥ 0, 2 ≤ s ≤ n,
(H∗ BVn )GLn .

s

1]

· · · unn vn[p

n−1

(p−1)(i1 +···+in +b(I))−pn−1

n]

],

∈ {0, 1}, and i1 + b(I) − m(I) ≥ 0 provides an additive basis for

Let Ik be the ideal of R generated by all monomials of excess less than k. The quotient
algebra R/Ik is denoted by Rk . And we also denote by Rk [n] the subspace of Rk spanned by
all monomial of length n. Then, we have the following proposition.
Proposition 3.8. As algebras over the Steenrod algebra, Rk [n]∗ ∼
= Bk [n] via the
isomorphism give in Theorem 2.3.


Page 10 of 22

PHAN HOÀNG CHƠN

Proof. For a string of integers e = (e1 , · · · , ej ) such that 1 ≤ e1 < · · · < ej ≤ n, we put
Kn;e1 ,e2 + · · · + Kn;ej−1 ,ej ,
if j is even,
Kn;e1 ,e2 + · · · + Kn;ej−2 ,ej−1 + Jn;ej , if j is odd,

Ln;e =

and Ln;e is the string of all zeros if e is empty. Here we mean ( 1 , i1 , · · · , n , in ) +
( 1 , j1 , · · · , n , jn ) to be the string (ω1 , t1 , · · · , ωn , tn ) with ts = is + js and ωs = s +
s (mod 2).

In [3, p.38], May shows that for any string I of non-negative excess, it can be uniquely
expressed in the form
n−1

I=

ti In,i + Ln;e ,
i=0

for some string e, and exc(I) = 2t0 + exc(Ln;e ). By the same argument of the proof of Theorem
3.7 in [3, p.29], we obtain that the set of all monomials
i1
in
2
ξn,0
· · · ξn,n−1
,
(σn;e1 ,e2 · · · σn;ej−2 ,ej−1 ) 1 τn;e
j

2i1 +

2

≥k

∗

provides an additive basis of Rk [n] .
Using relation (ii) in Theorem 2.2, above monomials can be written in the form (up to a

sign)
i1
in
1
n
τn;0
ξn,0
· · · τn;n−1
ξn,n−1
,

2i1 + b(I) ≥ k.

i1
1
τn;0
ξn,0

in
n
It implies that the set of all monomials
· · · τn;n−1
ξn,n−1
, 2i1 + b(I) ≥ k is a basis of
Rk [n]∗ .
By the definition of Bk [n] and Theorem 2.3 we have the assertion of the proposition.

4. The Hopf ring structure of H∗ QS k
In this section, we use results of the modular (co)invariants in above sections to describe a
complete set of relations for {H∗ QS k }k≥0 as a Hopf ring.

Let [1] ∈ H∗ QS 0 be the image of non-base point generator of H0 S 0 under the map induced
by the canonical inclusion S 0 → QS 0 and let σ ∈ H∗ QS 1 be the image of the generator of
H1 S 1 under the homomorphism induced by the inclusion S 1 → QS 1 . Note that the element σ
is usually known as the homology suspension element because σ ◦ x is the homology suspension
of x. From the results of Dyer-Lashof [5] and May [3], we have
Theorem 4.1 (Dyer-Lashof [5], May [3]). The mod p homology of {QS k }k≥0 is given by
H∗ QS 0 = P [QI [1] : I admissible, exc(I) +
k

I

◦k

1

H∗ QS = P [Q (σ ) : I admissible, exc(I) +

> 0] ⊗ Fp [Z],
1

> k], k > 0.

Some basic properties are given in the following theorem.
Theorem 4.2 (May [3], [22]). For b, f ∈ H∗ QS k ,
(i) P∗k (b ◦ f ) = i P∗i (b) ◦ P∗k−i (f ) and β(b ◦ f ) = β(b) ◦ f + (−1)degb b ◦ β(f ).
(ii) Qk (b) ◦ f = i Qk+i (b ◦ P∗i (f )).
(iii) βQk (b) ◦ f = i βQk+i (b ◦ P∗i (f )) − i (−1)degb Qk+i (b ◦ P∗i β(f )).
In [10], Kahn and Priddy constructed the transfer
tr(1) : (BV1 )+ → QS 0 .

MODULAR COINVARIANTS AND THE MOD p HOMOLOGY OF QS k

Page 11 of 22

(1)

The induced transfer tr∗ : H∗ (BV1 )+ → H∗ QS 0 sends u v [i(p−1)− ] to β Qi [1] and others to
zero.
Let ψ : Σm × Σn → Σmn be the permutation product of symmetric groups; and let In : Vn →
Σpn be the composition
ψ×···×ψ

Vn = V1 × · · · × V1 → Σp × · · · × Σp −−−−−→ Σpn .
By the results of Madsen and Milgram [21, Theorem 3.10], we have the following
commutative diagram
/ BΣpn

BIn

BVn
tr (1) ×···×tr (1)

i


QS 0 × · · · × QS 0


/ QS 0

µ

where µ is the composition product in QS 0 . Therefore, we get the Kahn-Priddy’s transfer
tr(n) = µ ◦ (tr(1) × · · · × tr(1) ) : BVn → QS 0 .
(n)

The induced transfer in homology tr∗ : H∗ BVn → H∗ QS 0 sends the “external product”
in H∗ BVn (with respect to the decomposition BVn BVr × BVn−r ) to the circle product in
H∗ QS 0 . In other words, we have
(n)

[i (p−1)−

tr∗ (u11 v1 1
=

1]

· · · unn vn[in (p−1)−

(1)
[i (p−1)−
tr∗ (u11 v1 1

1]

n]

)

(1)

) ◦ · · · ◦ tr∗ (unn vn[in (p−1)−

n]

).

(n)

Since tr = i ◦ BIn and GLn is the “Weyl group” of the inclusion Vn ⊂ Σpn , we have an
(n)
important feature of the map tr∗ is that they factor through the coinvariant of the general
linear group. In other words, the diagram
(n)

tr∗

H∗ BVn

/ H∗ QS 0
?

p


(H∗ BVn )GLn
is commutative.
Moreover, we have the following proposition.
(n)

Proposition 4.3. The transfer tr∗

factors through (B[n])∗ . In other words, the diagram
(n)

tr∗

H∗ BVn
p


∗
B[n]

/ H∗ QS 0
?
ϕn

commutes.
∗
∗
= (BIn )∗ ◦ i∗ , the image of tr(n)
is
Proof. Since the induced transfer on cohomology tr(n)
∗
∗
∗
contained in the image of the restriction (BIn ) : H BΣpn → H BVn . Moreover, from Mùi

Page 12 of 22

PHAN HOÀNG CHƠN

[24, Chapter 2, Theorem 6.1], the image of the restriction (BIn )∗ is B[n] ⊂ (H ∗ BVn )GLn .
Therefore, the assertion of the proposition is immediate from the dual.
For any I = ( 1 , i1 , · · · , n , in ), with i1 ∈ Z, is ≥ 0, 2 ≤ s ≤ n, s ∈ {0, 1}, and i1 + b(I) −
in
i1
n
1
with respect to the
qn,n−1
, · · · , Rn;n−1
qn,0
m(I) ≥ 0, let E( 1 ,i1 ,··· , n ,in ) is the dual of Rn;0
monomials basis given in Proposition 3.3; and we use the same notation E( 1 ,i1 ,··· , n ,in ) to
(n)
denote its image under the transfer tr∗ . In particular, E( ,k) = β Qk [1].
We have another description of the homology of {QS k }k≥0 as follows.
Theorem 4.4. The homology of QS k is given by
H∗ QS 0 = P [E(

1 ,i1 +b(I))

◦ · · · ◦E(

n−1 (i +···+i +b(I))−∆
n ,p

1
n
n n)

n ≥ 1, 2i1 + b(I) +

:

1

> 0] ⊗ Fp [Z],

and for k > 0,
H∗ QS k = P [σ ◦k ◦ E(

1 ,i1 +b(I))

◦ · · · ◦E(

n−1 (i +···+i +b(I))−∆
n ,p
1
n
n n)

n ≥ 1, 2i1 + b(I) +
where ∆s =

ps−1 −1
p−1

:

1

> k],

= 1 + · · · + ps−2 , s ≥ 2, and ∆1 = 0.

In order to prove the theorem, we need two following lemmas.
Lemma 4.5. For n ≥ 1,
σ ◦k ◦ E(

1 ,i1 +b(I))

◦ · · · ◦E(

2 +···+(n−1) n

= (−1)

n−1 (i +···+i +b(I))−∆
n ,p
1
n
n n)

β Qj1 · · · β n Qjn (σ ◦k ) +
1

QK (σ ◦k ),

where
n−s−1

js = pn−s (i1 + · · · + is + b(I)) +

p (pn−s− − 1)is+

+1

− δn (s),

=0

δn (s) = pn−s−1

n

+ ··· +

s+1 ,

exc(K) < exc(J) = 2i1 + b(I).

Proof. The n = 1 case is immediate.
Using Theorem 4.2 and Nishida’s relations, we obtain the assertion of the lemma for n = 2.
We shall prove the case n ≥ 3 by induction. It is sufficient to prove in the case n = 3. By
the inductive hypothesis, the element
E(

2 ,p(i1 +i2 +b(I))− 2 )

◦ E(

p2 −1
2
3 ,p (i1 +i2 +i3 +b(I))− p−1 3 )

= β 1 Qp(i1 +t2 +b(I))− 2 [1] ◦ β 3 Qp

2

2

−1
(i1 +i2 +t3 +b(I))− pp−1

can be written as follows
(−1) 3 β 2 Qp

2

(i1 +i2 +b(I))+p(p−1)i3 −(

2 +p 3 )

β 3 Qp(i1 +i2 +i3 +b(I))− 3 [1]

+ other terms of smaller excess.

3

[1]


MODULAR COINVARIANTS AND THE MOD p HOMOLOGY OF QS k

Therefore, y = E(
ten as
(−1)

1 ,i1 +b(I))

◦ E(

2 ,p(i1 +i2 +b(i))− 2 )

◦ E(

Page 13 of 22

p2 −1
2
3 ,p (i1 +i2 +i3 +b(I))− p−1 3 )

can be writ-

β 1 Qi1 +b(I)+k

3

k
(P∗k (β

2

Qp

2

(i1 +i2 +b(I))+p(p−1)i3 −(

2 +p 3 )

β 3 Qp(i1 +i2 +i3 +b(I))− 3 [1]))

+ other terms of smaller excess.
We observe that, for k ≥ pi,
P∗k (β 2 Qp

2

(i1 +i2 +b(I))+p(p−1)i3 −(

β 3 Qp(i1 +i2 +i3 +b(I))− 3 [1]) =
(p − 1)[p (i1 + i2 + b(I)) + p(p − 1)i3 − ( 2 + p 3 ) − k] −
k − pi
2 +p 3 )

2

(−1)k
i

(p − 1)[p(i1 + i2 + i3 + b(I)) −
i
β 2 Qp

2

(i1 +i2 +b(I))+p(p−1)i3 −(

= (−1)k
β 2 Qp

2

3

− i] −

3

2

×

×

β 3 Qp(i1 +i2 +i3 +b(I))− 3 −i [1] + others
(p − 1)[p2 (i1 + i2 + b(I)) + p(p − 1)i3 − ( 2 + p 3 ) − k] − 2
×
k − p(p − 1)(i1 + i2 + i3 + b(I)) − p 3

(i1 +i2 +b(I))+p(p−1)i3 −(

for i = (p − 1)(i1 + i2 + i3 + b(I)) −
Therefore,
y = (−1)

2 +p 3 )−k+i

2 +p 3 )−k+i

β 3 Qi1 +i2 +i3 +b(I) [1] + others,

3.

2 +2 3

β 1 Qj1 β 2 Qj2 β 3 Qj3 [1] + others.

As σ ◦k ◦ β Qj [1] = β Qj (σ ◦k ), then σ ◦k ◦ y can be written in the needed form.
Lemma 4.6. The function mapping I = ( 1 , i1 + b(I), · · · , n , pn−1 (i1 + · · · + in + b(I)) −
∆n n ), 2i1 + b(I) > k, to admissible string J = ( 1 , j1 , · · · , n , jn ), with exc(J) > k, given as in
Lemma 4.5, is a bijection.
Proof. It is immediate.
From Lemma 4.5, the set of elements σ ◦k ◦ E( 1 ,i1 +b(I)) ◦ · · · ◦
E( n ,pn−1 (i1 +···+in +b(I))−∆n n ) belongs to the indecomposable quotient (with respect to the

star product) QH∗ QS k and it is linear independent.
Moreover, the degree of
Proof of Theorem 4.4.

σ ◦k ◦ E(

1 ,i1 +b(I))

◦ · · · ◦ E(

n−1 (i +···+i +b(I))−∆
n ,p
1
n
n n)

is equal to the degree of β 1 Qj1 · · · β n Qjn (σ ◦k ).
Finally, from Lemma 4.6, we obtain that this set generates QH∗ QS k in each degree.
Thus, the elements E( ,s) = β Qs [1] and σ generate {H∗ QS k }k≥0 as a Hopf ring. The problem
is to find a complete set of relations. It is solved by investigating the structure of the dual of
Bk [n].
Let E (s) ∈ H∗ QS 0 [[s]], = 0, 1, be defined by
E(0,k) sk ,

E 0 (s) =
k≥0

E(1,k) sk .

E 1 (s) =

k≥1


Page 14 of 22

PHAN HOÀNG CHƠN

Since the coproduct on the E(
H2k(p−1)− k (BV1 )GL1 ,

arises from the coproduct on [u k v [k(p−1)−

k ,k)

k]

] in

E(0,i) ⊗ E(0,j) ,

ψ(E(0,k) ) =
i+k=j

(E(0,i) ⊗ E(1,j) + E(1,i) ⊗ E(0,j) ).

ψ(E(1,k) ) =
i+j=k

Therefore,
ψ(E 0 (s)) = E 0 (s) ⊗ E 0 (s);

ψ(E 1 (s)) = E 0 (s) ⊗ E 1 (s) + E 1 (s) ⊗ E 0 (s).
For x ∈ H∗ QS k we define Q0 (s)x, Q1 (s)x ∈ H∗ QS k [[s]] as follows
Qk xsk ;

Q0 (s)x =
k≥0
0

βQk xsk .

Q1 (s)x =
k≥1

0

1

1

Then we obtain that E (s) = Q (s)[1] and E (s) = Q (s)[1].
A complete set of algebraic relations for {H∗ QS k }k≥0 is given in the following proposition.
Proposition 4.7. For s, t are formal variables, we have relations
E 0 (sp−1 ) ◦ E 0 (tp−1 ) = E 0 (sp−1 ) ◦ E 0 ((s + t)p−1 );

(4.1)

t
;
s+t
t

E 1 (sp−1 ) ◦ E 1 (tp−1 ) = E 1 (sp−1 ) ◦ E 1 ((s + t)p−1 )
;
s+t
E 0 (sp−1 ) ◦ E 1 (tp−1 ) = E 0 (sp−1 ) ◦ E 1 ((s + t)p−1 )

When k = 2i1 + b(I) +
σ

◦k

◦ E(

1,

(4.2)
(4.3)

b(I) > 0 and n ≥ 1,

1 ,i1 +b(I))

◦ · · · ◦ E(

n−1 (i +···+i +b(I))−∆
n ,p
1
n
n n)

= (1 −

1 )y

p

(4.4)

for some y ∈ {H∗ QS k }k≥0 , where σ ◦0 = [1]. In particular,
E(0,0) = [p];
σ

◦2k

◦ E(

,k)

(4.5)

= (1 − )(σ

◦2k

p

) .

(4.6)

Remark 4.8. In the case p = 2 (see [27]) as well as in the Bockstein-nil homology for p

odd (see [15]), the relations (4.2) and (4.3) omit because these relations come from the action
of the Bockstein operation.
In addition, when p = 2 or b(I) = 0 for p odd, the general case of relation (4.4) follows from
n = 1 case, i.e., from relation (4.6). Indeed, using relation (4.6) we get
σ ◦2i1 ◦ E(0,i1 ) ◦ · · · ◦ E(0,pn−1 (i1 +···+in )) = (σ ◦2i1 )
Using the distributivity between the
obtain
(σ ◦2i1 )

p

p

◦ E(0,p(i1 +i2 )) ◦ · · · ◦ E(0,pn−1 (i1 +···+in )) .

product and the ◦ product (see [26, Lemma 1.12]), we

◦ E(0,p(i1 +i2 )) ◦ · · · ◦E(0,pn−1 (i1 +···+in ))
= (σ ◦2i1 ◦ E(0,i1 +i2 ) ◦ · · · ◦ E(0,pn−2 (i1 +···+in )) ) p .

However, for b(I) > 0, the general case of relation (4.4) does not follow from n = 1 case and
relations (4.1)-(4.3). For example, for I = (0, 0, 1, p), in order to prove the relation
σ ◦ E(0,1) ◦ E(1,p−1) = y p , for some y ∈ {H∗ (QS k )}k≥0 ,


MODULAR COINVARIANTS AND THE MOD p HOMOLOGY OF QS k

Page 15 of 22

we must use relation (4.2) to write E(0,1) ◦ E(1,p−1) as a sum of E(0,i) ◦ E(1,j) for i < 1, before

applying relation (4.6). But from relation (4.2), we obtain
E(0,i) ◦ E(1,j) =
m≥0

(p − 1)(i + j − m) − 1
E(0,m) ◦ E(1,i+j−m) .
(p − 1)(i − m)

Applying the relation, we can write
E(0,1) ◦ E(1,p−1) = E(0,0) ◦ E(1,p) + E(0,1) ◦ E(1,p−1) .
It implies E(0,0) ◦ E(1,p) = 0 and E(0,1) ◦ E(1,p−1) is not expressed as a sum of E(0,i) ◦ E(1,j)
for i < 1. In other words, the relation σ ◦ E(0,1) ◦ E(1,p−1) = y p can not follow from (4.6) and
(4.2).
It should be also noted that, for k = 0, if b(I) = 1 = 1, then relation (4.4) becomes to trivial
relation; but if b(I) > 1 = 1 or 1 = 0, the relation is nontrivial.
Proof of Proposition 4.7. It should be noted that the formulas (4.1)-(4.3) can be proved by
using the method in Turner [27], of course, it is more complicated.
(n)
Here we use the multiplicativity of the transfer and the fact that the transfer tr∗ is GLn invariant to show these relations.
(1)
First, we consider the first transfer tr∗ as the element
tr∗ ∈ HomFp (H∗ BV1 , H∗ QS 0 ) ∼
= H∗ QS 0 [[s]] ⊗ E( ).
(1)

(1)

Because tr∗ sends the generator in the degree 2(p − 1)i to E(0,i) , that in the degree 2(p −
1)i − 1 to E(1,i) , and the rest to zero, it is equal to
E 0 (sp−1 ) + s−1 E 1 (sp−1 ).

(2)

Next, the second transfer tr∗
(2)
tr∗

can be considered as the element

∈ HomFp (H∗ BV2 , H∗ QS 0 ) ∼
= H∗ QS 0 [[s, t]] ⊗ E( , ).

By the multiplicativity of the transfer, this element has to be
(E 0 (sp−1 ) + s−1 E 1 (sp−1 )) ◦ (E 0 (tp−1 ) + t−1 E 1 (tp−1 )).
Since the transfer factors through the coinvariant of H∗ BV2 under the action of the general
(2)
linear group GL2 , acting ( 10 11 ) on tr∗ , we obtain
(E 0 (sp−1 ) + s−1 E 1 (sp−1 )) ◦ (E 0 (tp−1 ) + t−1 E 1 (tp−1 ))
= (E 0 (sp−1 ) + s−1 E 1 (sp−1 )) ◦ (E 0 ((s + t)p−1 ) + ( + )(s + t)−1 E 1 ((s + t)p−1 )).
Expanding this equality and comparing the coefficients of “1”, and
follows the formulas
(4.1), (4.2) and (4.3).
From above proof, we observe that the formulas (4.1), (4.2) and (4.3) also hold in
colimBV /CS 0 H∗ (−)[[s, t]], where BV /CS 0 is the category whose objects are homotopy classes
of maps from a classifying space of an elementary abelian p-group to CS 0 , whose morphism are
commutative triangles, and CS 0 denotes the combinatorial model of QS 0 , that is, the disjoint
union of BΣn ’s (see [15, Section 5]).
From Lemma 4.5, for n ≥ 1,
σ ◦k ◦ E(

1 ,i1 +b(I))

= (−1)

◦ · · · ◦E(

2 +···+(n−1)

n

n−1 (i +···+i +b(I))−∆
n ,p
1
n
n n)

β 1 Qj1 · · · β n Qjn (σ ◦k ) +

QK (σ ◦k ),

where exc(K) < exc(I) = 2i1 + b(I).
Since k = 2i1 + b(I) + 1 , the second sum of the formula is trivial.


Page 16 of 22

PHAN HOÀNG CHƠN

If 1 = 1, then exc(I) < k, therefore, the first item is also trivial. Otherwise, if 1 = 0, then
2j1 = deg(β 2 Qj2 · · · β n Qjn (σ ◦k )), therefore, the first item is the p-th power of an element.
Thus, the formula (4.4) is proved.

Since σ is primitive elements with respect to the
For n ≥ 1 and 2i1 + b(I) < k,

Corollary 4.9.
σ

product, we have the following corollary.

◦k

◦ E(

1 ,i1 +b(I))

◦ · · · ◦ E(

n−1 (i +···+i +b(I))−∆
n ,p
1
n
n n)

= 0,

where σ ◦0 = [1].
Let us put E (sp−1 ) = s−1 E (sp−1 ) ∈ H∗ Q0 [[s]], qualities (4.1)-(4.3) can be reduced as
follows.
For s, t are formal variable, then we have relation

Corollary 4.10.

p−1

E (s
1

) ◦ E 2 (tp−1 ) = E 1 (sp−1 ) ◦ E 2 ((s + t)p−1 ),

For A ∈ GL , B ∈ GLk , denote A ⊕ B = ( A0 B0 ) ∈ GL
we have the lemma.

+k

1

≤

2.

and a ⊕ A = ( a0 A0 ) ∈ GL

(4.7)
+1 .

Then

Lemma 4.11. For n ≥ 2, the general linear group GLn = GLn (Fp ) is generated by
{T, Σn , Ta : a ∈ F∗p }, where
T = ( 10 11 ) ⊕ In−2 ,

Ta = a ⊕ In−1 .

Theorem 4.12. The homology {H∗ QS k }k≥0 is the coalgebraic ring in Fp [Z] generated by
E(0,i) (i ≥ 0), E(1,j) (j ≥ 1) and σ modulo all the relations implied by Proposition 4.7.
The coproduct is specified by
ψ(σ) = 1 ⊗ σ + σ ⊗ 1;

ψ(E 0 (s)) = E 0 (s) ⊗ E 0 (s);

ψ(E 1 (s)) = E 0 (s) ⊗ E 1 (s) + E 1 (s) ⊗ E 0 (s);

ψ(a ◦ b) = ψ(a) ◦ ψ(b).

The theorem can be show by using the framework of Turner [27] and Eccles. et.al [6], we
mean that we can use the method in [27] to show for k = 0, and then use the bar spectral
sequence (as in [6]) to induct for k > 0. Here, we modify the method of Turner [27] to show
the theorem directly (without using the bar spectral sequence). In order to do this, we need
some notations.
We define elements f 0 (vi , s), f 1 (ui , vi , s) ∈ H∗ BVn [[s]] for any ui , vi by
[k]

vi sk ;

f 0 (vi , s) =

[k−1] k

f 1 (ui , vi , s) =

k≥0

ui vi

s ,

k≥1

f 0 (0, s) = f 0 (vi , 0) = 1.
Then we have
(1)

tr∗ (f 0 (vj , s)) = E 0 (sp−1 );

(1)

tr∗ (f 1 (ui , vi , s)) = E 1 (sp−1 ).

Put f 0 (vi , s) = s−1 f 0 (vi , s) and f 1 (ui , vi , s) = s−1 f 1 (ui , vi , s), then
(1)

tr∗ (f 0 (vj , s)) = E 0 (sp−1 );

(1)

tr∗ (f 1 (ui , vi , s)) = E 1 (sp−1 ).


MODULAR COINVARIANTS AND THE MOD p HOMOLOGY OF QS k

Page 17 of 22

Proof. Let D∗,∗ be the coalgebra generated by E(0,i) ∈ D2i(p−1),0 (i ≥ 0), E(1,j) ∈
D2j(p−1)−1,0 (j ≥ 1) and σ ∈ D1,1 . Apply the Ravenel-Wilson free Hopf ring functor [26] to the
coalgebra D∗,∗ to give H D∗,∗ , the free Fp [Z]-Hopf ring on D∗,∗ . There is a map of coalgebras
D∗,∗ → {H∗ QS k }k≥0 mapping the element E( ,i) to the element E( ,i) ∈ {H∗ QS k }k≥0 . By the
universality, the map extends to a unique map of Hopf rings
h : H D∗,∗ → {H∗ QS k }k≥0 .
Let A∗,∗ be the free Fp [Z]-Hopf ring on D∗,∗ subject to relations arising from Proposition
4.7. Since all relations defined in A∗,∗ hold in {H∗ QS k }k≥0 , the map h induces a unique map
¯ : A∗,∗ → {H∗ QS k }k≥0 .
h
Using Theorem 4.4, we get that this map is surjective. Therefore, it induces a surjection
between indecomposable quotients (with respect to product)
QA∗,k → QH∗ QS k .
In order to prove A∗,∗ ∼
= {H∗ QS k }k≥0 , it is sufficient to prove the induced surjection between
indecomposable quotients is an isomorphism.
We now begin our proof of claim that QA∗,k → QH∗ QS k is an isomorphism. For s =
(s1 , · · · , sn ) being a vector of formal variables and for = ( 1 , · · · , n ), i ∈ {0, 1}, we define
u (s) = f 1 (u1 , v1 , s1 ) · · · f n (un , vn , sn ),
where f 0 (ui , vi , si ) = f 0 (vi , si ), and we define
E (sp−1 ) = E 1 (sp−1
) ◦ · · · ◦ E n (sp−1
n ).
1
Let g : n≥1 H∗ BVn → A∗,0 be the map of Fp -algebras given by u (s) → E (sp−1 ). It is
easy see that g is a surjection.
From Corrollary 4.10 and Lemma 4.11, it is easy to check that g(u A(s)) = E (sp−1 ) =
g(u (s)), for A ∈ GLn . Therefore, g factors through the coinvariants space of the general linear
groups n≥1 (H∗ BVn )GLn .

Moreover, from Proposition 3.4, elements E( 1 ,i1 ,··· , n ,in ) ∈ (H∗ BVn )GLn , for 2i1 + b(I) < 0,
are trivial in B0 [n]∗ . Therefore, from Theorem 3.6 and Proposition 3.7, they can be written as
a combination of elements of the form
[(p−1)(j1 +ω)−ω1 ]

1
[uω
1 v1

n−1

n [p
· · · uω
n vn

(p−1)(j1 +···+jn +ω)−pn−1 ωn ]

],

for ωi = 0 or 1, ω = ω1 + · · · + ωn and 2j1 + ω < 0.
Combining with the fact that g is an algebra homomorphism, we get that the image of
E( 1 ,i1 ,··· , n ,in ) , 2i1 + b(I) < 0, under g can be written as a combination of the elements of
the form E(ω1 ,j1 +ω) ◦ · · · ◦ E(ωn ,pn−1 (j1 +···+jn +ω)−∆n ωn ) , with 2j1 + ω < 0, ω = ω1 + · · · + ωn .
It implies g(E( 1 ,i1 ,··· , n ,in ) ) = 0 for 2i1 + b(I) < 0.
Hence, from Corollary 4.9, g factors through n≥1 B0 [n]∗ . In other words, the diagram
n≥1

p

/ A∗,0

?

g

H∗ BVn


n≥1

g
∗

B0 [n]

is commutative.
For any k ≥ 0, let gk be the composition
g
¯

σ ◦k ◦−

B0 [n] −
→ A∗,0 −−−−→ A∗,k .
n≥1


Page 18 of 22

PHAN HOÀNG CHƠN

When k = 0, g0 is just g¯. Since, from Corollary 4.9, in A∗,k ,
σ ◦k ◦ E(

1 ,i1 +b(I))

◦ · · · ◦ E(

n−1 (i +···+i +b(I))−∆
n ,p
1
n
n n)

= 0, 2i1 + b(I) < k,

by the same above argument, the Fp -map gk factors through
n≥1 Bk [n] and gk is also a
surjection.
For any n ≥ 1, let QA∗,k [n] be the subspace of QA∗,k spanned by all elements σ ◦k ◦
E( 1 ,i1 ) ◦ · · · ◦ E( n ,in ) and let QH∗ QS k [n] be the subspace of QH∗ QS k spanned by all elements
β 1 Qj1 · · · β n Qjn (σ ◦k ).
By Theorem 3.6, in Bk [n]∗ , we have
: 2i1 + b(I) + 1 > k} =
n−1
n−1
[(p−1)(i1 +b(I))− 1 ]
Span{[u1 v1
· · · unn vn[p (p−1)(i1 +···+in +b(I))−p

Span{E(

1 ,i1 ,··· , n ,in )
1

2i1 + b(I) +

1

n]

]:

> k}.

Therefore, we have a surjection
[(p−1)(i1 +b(I))−

S = Span{[u11 v1

1]

· · · unn vn[p

n ≥ 1, 2i1 + b(I) +

1

n−1

(p−1)(i1 +···+in +b(I))−pn−1

n]

]:

> k} → QA∗,k [n].

It implies that, in each degree d, dim(S) ≥ dim(QA∗,k [n]) ≥ dim(QH∗ QS k [n]).
Finally, we observe that, for each degree d,
n
n , in )|2(i1 + 1 )(p
n
n−1
)− n =
n )(p − p

Card{( 1 , i1 , · · · ,
+ 2(in +

= Card{( 1 , i1 , · · · ,
+

n , in )| 1

− 1) −

1

+ ···

d}

+ 2((p − 1)(i1 + b(I)) −

n−1
((p − 1)(i1 + · · · + in + b(I)) −
n + 2p

1)

+ ···

n ) = d}.

∼ QH∗ QS k .
So dim(S) = dim(QA∗,k [n]) = dim(QH∗ QS k [n]). It implies QA∗,k =
The proof is complete.

5. The actions of A and R on H∗ QS k
In this section, using the same method of Turner [27], we describe the action of the mod
p Dyer-Lashof operations as well as of mod p Steenrod operations on the Hopf ring. For
convenience, we write P k instead of P∗k and write their action on the right. For x ∈ H∗ QS k
and formal variable s, we define the formal series
(xβ P k )sk , = 0, 1.

xP (s) =
k≥0

In order to prove the main theorem of this section, we need the following lemma.
Lemma 5.1. There are the following relations:

x ◦ Q (s)(y) = Q (s)(xP 0 (s−1 ) ◦ y) − (−1)degy Q0 (s)(xP 1 (s−1 ) ◦ y);

(5.1)

f 0 (vi , s)P 0 (t) = f 0 (vi , (s + sp t));

(5.2)

f 0 (vi , s)P 1 (t) = f 1 (ui , vi , (s + sp t));

(5.3)

f 1 (ui , vi , s)P 0 (t) = f 1 (ui , vi , (s + sp t)).

(5.4)


MODULAR COINVARIANTS AND THE MOD p HOMOLOGY OF QS k

Page 19 of 22

Proof. From Theorem 4.2, we obtain
x ◦ Q (s)(y)

β Qk+i 

=




xP i ◦ y  sk −
i≥0

k≥

≥ +i

xβP i ◦ y  sk
i≥1




−i

xP i ◦ y  s

β Q 

=

(−1)degy Qk+i 
k≥







xβP i ◦ y  s

(−1)degy Q 

−
≥i

i≥0

−i

.

i≥1

It should be noted that if xP i (respect, xβP i ) is nontrivial then the degree of xP i ◦ y
(respect, xβP i ◦ y) is not less than 2i (respect, 2i + 1). It implies that, when < + i (respect,
< i) then β Q (xP i ◦ y) (respect, Q (xβP i ◦ y)) is trivial.
Therefore, the right hand side of above formula can be written as follows




(xP i ◦ y)s−i  s −

β Q 
≥

≥0

i≥0
0

−1

= Q (s)(xP (s

(xβP i ◦ y)s−i  s

(−1)degy Q 

) ◦ y) − (−1)

i≥1
degy

Q (s)(xP 1 (s−1 ) ◦ y).
0

Hence, the formula (5.1) is proved.
From
[n]

vi β P k =

n − (p − 1)k −
k

[n−(p−1)k− ]

ui vi

we have the formulas (5.2) and (5.3).
Since P k acts trivially on ui for k > 0, then
[n−1]

u i vi

Pk =

n − (p − 1)k − 1
[n−(p−1)k−1]
ui vi
.
k

This implies the last formula.
The main results of the section is the following theorem, which gives a description of the
actions of the Dyer-Lashof algebra and the Steenrod algebra on the Hopf ring.
Theorem 5.2. Let x, y ∈ H∗ QS k and let s, t, t1 , t2 , · · · be formal variables; tp−1 =
k
, tp−1
n ), = ( 1 , · · · , n ). The following hold in H∗ QS [[s, t, t1 , t2 , · · · ]].

(tp−1
,···
1

[n]P (s) = (1 − )[n].
0

p−1

0

p−1

1

p−1

E (s
E (s
E (s

(5.5)

0

0

p

p−1

).

(5.6)

1

1

p

p−1

).

(5.7)

0

1

p

p−1

).

(5.8)

)P (t) = E ((s − s t)
)P (t) = E ((s − s t)
)P (t) = E ((s − s t)

E 1 (sp−1 )P 1 (t) = 0.

p−1

Q (s
1

(5.9)

(x y)P (s) = (−1)

deg y

(x ◦ y)P (s) = (−1)

deg y

0

1

(5.10)

0

0

1

(5.11)

xP (s) yP (s) + (xP (s)) yP (s).
xP (s) ◦ yP (s) + (xP (s)) ◦ yP (s).

Q (s)[n] = [n] ◦ E (s).
)E ((st)p−1 ) = (1 − tˆp−1 )[E
2

0

(5.12)

(stˆ)p−1 ◦ E 1 (sp−1 )
+ 1 (1 − 2 )E 1 (stˆ)p−1 ◦ E 0 (sp−1 )].
2

Q (s)(x y) = Q (s)x Q0 (s)y + (−1)
Q (s)([n] ◦ y) = [n] ◦ Q (s)y.

deg y

Q0 (s)x Q1 (s)y.

(5.13)
(5.14)
(5.15)


Page 20 of 22

PHAN HOÀNG CHƠN

p−1

Q (sp−1 )(E ((st)p−1 )) = (1 − tˆ

)[E ((sˆt)p−1 ) ◦ E (sp−1 )

n

(1 − i )E i ((sˆt)p−1 ) ◦ E 0 (sp−1 )].

+

(5.16)

i=1
k
Here we denote by tˆ = k≥0 tp , tˆi =
obtained from by replacing i by 1.

pk
k≥0 ti ,

p−1
, · · · , tˆp−1
tˆ
= (tˆp−1
n ), and
1

i

the vector

Proof. The first equality is immediate by degree.
(1)
(1)
Since tr∗ (v [n] P k ) = (−1)k tr∗ (v [n] )P k , then equalities (5.6)-(5.9) are implied from (5.2),
(5.3) and (5.4).
Since the coproduct of P (s) is given by
ψ(P (s)) = P (s) ⊗ P 0 (s) + P 0 (s) ⊗ P 1 (s),
the formulas (5.10) and (5.11) come from the Cartan formula.
Letting y = [1] in (5.1) to obtain
x ◦ Q (s)[1] = Q (s)(xP 0 (s−1 )) − Q0 (s)(xP 1 (s−1 )).

(5.17)

Letting x = [n] in above equality and combining with (5.5), we obtain (5.12).
Replace x = E (up−1 ) in (5.17), we get
E (up−1 ) ◦ Q (s)[1] = Q (s)(E (up−1 )P 0 (s−1 ))
− Q0 (s)(E (up−1 )P 1 (s−1 )).
Combining with (5.6)-(5.9), we give
E (up−1 ) ◦ Q (s)[1] = (1 − up−1 t−1 )− [Q (s)(E (u − up s−1 )p−1 )
− (1 − )Q0 (s)(E 1 (u − up s−1 )p−1 )].
From (5.18), letting

= 0 and

(5.18)

= 1, one gets

E 1 (up−1 ) ◦ Q0 (s)[1] = (1 − up−1 t−1 )−1 Q0 (s)(E 1 (u − up s−1 )p−1 ).
These formulas imply (replacing s by sp−1 )
Q 1 (sp−1 )E 2 ((u − up s1−p )p−1 )
= (1 − up−1 s1−p )[E 2 (up−1 ) ◦ Q 1 (sp−1 )[1] +
By letting t = u/s − (u/s)p with noting that tˆ =
in the form
Q 1 (sp−1 )E 2 ((st)p−1 )
= (1 − tˆp−1 )[E

2

1 (1

−

2 )E

1

(up−1 ) ◦ Q0 (sp−1 )[1]].

k

k≥0

(stˆ)p−1 ◦ E 1 (sp−1 ) +

tp = u/s, it is easy to write the equality

1 (1

−

2 )E

1

(stˆ)p−1 ◦ E 0 (sp−1 )].

So (5.13) is proved. The equality (5.14) is just the Cartan formula.
In order to prove (5.15), to replace x = [n] in (5.1) with noting that [n]P 1 (s) = 0, we obtain
[n] ◦ Q (s)y = Q (s)([n]P 0 (s−1 ) ◦ y).
Using (5.5) we have (5.15).
Since (n − 1)-fold coproduct of P (s) is given by
ψ n−1 (P 0 (s)) = P 0 (s) ⊗ · · · ⊗ P 0 (s),
and
ψ n−1 (P 1 (s)) = P 1 (s) ⊗ · · · ⊗ P 0 (s) + · · · + P 0 (s) ⊗ · · · ⊗ P 1 (s),


MODULAR COINVARIANTS AND THE MOD p HOMOLOGY OF QS k

Page 21 of 22

the last formula follows from formula (5.13) and the Cartan formula.
The proof is complete.
As discussion in the introduction, the category of A-H∗ QS 0 -coalgebraic modules and the one
of A-R-allowable Hopf algebra also play important role in the study of the mpd p homology
of the infinite loop spaces. We will investigate these categories and the relationship between
them elsewhere.
Acknowledgements

The author would like to thank Lê Minh Hà and J. Peter May for many fruitful discussion,
and Takuji Kashiwabara for his comments on an earlier version of this paper. The paper
was completed while the author was visiting the Vietnam Institute for Advanced Study in
Mathematics (VIASM). He thanks the VIASM for support and hospitality.

References
1. S. Araki and T. Kudo, Topology of Hn -spaces and H-squaring operations, Memoirs of the Faculty of
Science 10 (1956), 85–120.
2. J. M. Boardman, R. L. Kramer, and W. S. Wilson, The periodic Hopf ring of connective Morava K-theory,
Forum Math. 11 (1999), no. 6, 761–767. MR 1725596 (2000k:55009)
3. F. R. Cohen, T. J. Lada, and J. P. May, The homology of iterated loop spaces, Lecture Notes in
Mathematics, Vol. 533, Springer-Verlag, Berlin, 1976. MR 0436146 (55 #9096)
4. L. E. Dickson, A fundamental system of invariants of the general modular linear group with a solution of
the form problem, Trans. Amer. Math. Soc. 12 (1911), no. 1, pp. 75–98 (English).
5. E. Dyer and R. K. Lashof, Homology of iterated loop spaces, Amer. J. Math. 84 (1962), no. 1, pp. 35–88
(English).
6. P. J. Eccles, P. R. Turner, and W. S. Wilson, On the Hopf ring for the sphere, Math. Z. 224 (1997), no. 2,
229–233. MR 1431194 (98e:55024)
7. J. H. Gunawardena, J. Lannes, and S. Zarati, Cohomologie des groupes symétriques et application de
Quillen, Advances in homotopy theory (Cortona, 1988), London Math. Soc. Lecture Note Ser., vol. 139,
Cambridge Univ. Press, Cambridge, 1989, pp. 61–68. MR 1055868 (91d:18013)
8. L. M. Hà and K. Lesh, The cohomology of symmetric groups and the Quillen map at odd primes, J. Pure
Appl. Algebra 190 (2004), no. 1-3, 137–153.
9. J. R. Hunton and P. R. Turner, Coalgebraic algebra, J. Pure Appl. Algebra 129 (1998), no. 3, 297–313.
MR 1631257 (99g:16048)
10. D. S. Kahn and S. B. Priddy, The transfer and stable homotopy theory, Math. Proc. Cambridge Philos.
Soc. 83 (1978), no. 1, 103–111.
11. T. Kashiwabara, Hopf rings and unstable operations, J. Pure Appl. Algebra 94 (1994), no. 2, 183–193.
MR 1282839 (95h:55005)
, Sur l’anneau de Hopf H∗ (QS 0 ; Z/2), C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 9,

12.
1119–1122. MR 1332622 (96g:55010)
, Homological algebra for coalgebraic modules and mod p K-theory of infinite loop spaces, K13.
Theory 21 (2000), no. 4, 387–417, Special issues dedicated to Daniel Quillen on the occasion of his sixtieth
birthday, Part V. MR 1828183 (2002e:55007)
14.
, Coalgebraic tensor product and homology operations, Algebr. Geom. Topol. 10 (2010), no. 3,
1739–1746. MR 2683751 (2011g:16060)
15.
, The Hopf ring for Bockstein-nil homology of QS n , Journal of Pure and Applied Algebra 216
(2012), no. 2, 267 – 275.
16. T. Kashiwabara, N. Strickland, and P. Turner, The Morava K-theory Hopf ring for BP , Algebraic topology:
new trends in localization and periodicity (Sant Feliu de Guíxols, 1994), Progr. Math., vol. 136, Birkh¨
auser,
Basel, 1996, pp. 209–222. MR 1397732 (98c:55006)
17. N. E. Kechagias, Extended Dyer-Lashof algebras and modular coinvariants, manuscripta mathematica 84
(1994), no. 1, 261–290.
, Adem relations in the Dyer-Lashof algebra and modular invariants, Algebraic & Geometric
18.
Topology 4 (2004), 219–241.
19. R. L. Kramer, The periodic Hopf ring of connective Morava K-theory, ProQuest LLC, Ann Arbor, MI,
1991, Thesis (Ph.D.)–The Johns Hopkins University. MR 2685728
20. I. Madsen, On the action of the Dyer-Lashof algebra in H∗ (G), Pacific J. Math. 60 (1975), no. 1, 235–275.
MR 0388392 (52 #9228)
21. I. Madsen and R. J. Milgram, The classifying spaces for surgery and cobordism of manifolds, Annals of
Mathematics Studies, vol. 92, Princeton University Press, Princeton, N.J., 1979. MR 548575 (81b:57014)


Page 22 of 22 MODULAR COINVARIANTS AND THE MOD p HOMOLOGY OF QS k

22. J. P. May, Homology operations on infinite loop spaces, Algebraic topology (Proc. Sympos. Pure Math.,
Vol. XXII, Univ. Wisconsin, Madison, Wis., 1970), Amer. Math. Soc., Providence, R.I., 1971, pp. 171–185.
MR 0319195 (47 #7740)
23. D. S. C. Morton and N. Strickland, The Hopf rings for KO and KU , Journal of Pure and Applied Algebra
166 (2002), no. 3, 247 – 265.
24. H. Mùi, Modular invariant theory and cohomology algebras of symmetric groups, J. Fac. Sci. Univ. Tokyo
Sect. IA Math. 22 (1975), no. 3, 319–369. MR 0422451 (54 #10440)
25. D. Quillen, The spectrum of an equivariant cohomology ring. I, II, Ann. of Math. (2) 94 (1971), 549–572;
ibid. (2) 94 (1971), 573–602. MR 0298694 (45 #7743)
26. D. C. Ravenel and W. S. Wilson, The Hopf ring for complex cobordism, Journal of Pure and Applied
Algebra 9 (1977), 241–280.
27. P. R. Turner, Dickson coinvariants and the homology of QS 0 , Math. Z. 224 (1997), no. 2, 209–228. MR
1431193 (98e:55023)
28. W. S. Wilson, The Hopf ring for Morava K-theory, Publ. Res. Inst. Math. Sci. 20 (1984), no. 5, 1025–1036.
MR 764345 (86c:55008)

Phan Hoàng Chơn
Department of Mathematics and
Application, Saigon University, 273 An
Duong Vuong, District 5, Ho Chi Minh
city, Vietnam.