ALGEBRAS OF OPERATIONS IN KTHEORY
FRANCIS CLARKE, MARTIN CROSSLEY, AND SARAH WHITEHOUSE
Abstract. We describe explicitly the algebras of degree zero op
erations in connective and periodic plocal complex Ktheory. Op
erations are written uniquely in terms of certain infinite linear com
binations of Adams operations, and we give formulas for the prod
uct and coproduct structure maps. It is shown that these rings of
operations are not Noetherian. Versions of the results are provided
for the Adams summand and for real Ktheory.
Contents
1. Introduction 1
2. Degree zero operations in connective Ktheory 3
3. The ring structure of k0(k)(p) 7
4. The Adams splitting and the Adams summand 10
5. Operations in padic Ktheory 12
6. Operations in periodic Ktheory 14
7. The relation between the connective and periodic cases 19
8. 2local operations 20
9. Operations in KOtheory 22
10. Appendix: polynomial identities 23
References 25
1. Introduction
The complex Ktheory of a space or spectrum may be usefully en
dowed with operations. The most wellknown of these are the Adams
operations j arising out of the geometry of vector bundles. As orig
inally constructed by Adams, these are unstable operations. A stable
operation is given by a sequence of maps, commuting with the Bott pe
riodicity isomorphism. For an Adams operation j to be stable requires
j to be a unit in the coefficient ring one is working over. Integrally,
we only have 1 and 1, corresponding to the identity and complex
conjugation. Following work of Adams, Harris and Switzer [3] in which
____________
Date: 21stJanuary 2004.
2000 Mathematics Subject Classification. Primary: 55S25; Secondary: 19L64,
11B65.
Key words and phrases. Ktheory operations, Gaussian polynomials.
1
2 F. CLARKE, M. D. CROSSLEY, AND S. WHITEHOUSE
the structure of the dual object, the algebra of cooperations, was de
termined, Adams and Clarke [2] showed that there are uncountably
many integral stable operations. It is remarkable that, after more than
forty years of topological Ktheory, noone knows explicitly any inte
gral stable operations, apart from linear combinations of the identity
and complex conjugation.
In this paper we give a new description of Ktheory operations in
the plocal setting. This is considerably closer to the integral situation
and more subtle than the rather better understood case of padic co
efficients; see [7, 14]. In plocal connective Ktheory, the degree zero
operations k0(p)(k(p)) form a bicommutative bialgebra, which we will
denote by k0(k)(p). Note, however that this is not the plocalisation
of the bialgebra k0(k) of integral operations, but is isomorphic to the
completed tensor product k0(k) bZ(p). In the periodic theory, the cor
responding object K0(K)(p)= K0(p)(K(p)) = K0(K) bZ(p)is a bicom
mutative Hopf algebra, as it also possesses an antipode. We provide
explicit descriptions for both of these algebras of operations, together
with formulas for all their structure maps. These results build on our
recent paper [8], in which we gave new additive bases for the ring of
cooperations in plocal Ktheory. Such an understanding of the degree
zero operations is sufficient to determine the whole of K*(K)(p)and
the torsionfree part of k*(k)(p); see [13 ].
Since we have a stable Adams operation fffor each ff 2 Zx(p), and
since these multiply according to the formula ff fi= fffi, the group
ring Z(p)[Zx(p)] is a subring of the ring of operations. Our results express
operations in terms of certain infinite sums involving Adams operations
and thus describe the ring of operations as a completion of the group
ring. This idea is certainly not new; Johnson [11 ] also has basis ele
ments for plocal operations of this form. However, our description has
considerable advantages in the form of explicit formulas, which in the
connective case are particularly nice. We note that Madsen, Snaith
and Tornehave [14 ] have also considered operations defined as infinite
sums of Adams operations, but for them the padic context is essential.
Our results obtained will be used in a later paper to give a simplifi
cation of Bousfield's work in [6] describing the K(p)local category.
We now outline the structure of the paper.
Sections 2 and 3 are concerned with the case of operations in con
nective Ktheory. In Section 2 we describe the bialgebra k0(k)(p)in an
explicit form which enables us to also describe the structure maps. We
also give formulas for the action on the coefficient ring and on the Hopf
bundle over CP 1. In Section 3 we show that, as a ring, k0(k)(p)is not
Noetherian, and we characterise its units. We also indicate how it can
be considered as a completion of a polynomial ring.
ALGEBRAS OF OPERATIONS IN KTHEORY 3
In Section 4 we consider the idempotents in connective Ktheory
which were introduced by Adams, and we show how the results of
Sections 2 and 3 extend to the algebra of operations on the Adams
summand. We prove in Section 5 that the ring of operations on the
padic Adams summand is a power series ring.
In Section 6 we show how the results of Sections 24 generalise to
periodic Ktheory, and in Section 7 we discuss the relation between the
connective and periodic cases.
In Sections 8 and 9 we work over the prime 2. We outline how our
results from the preceding sections need to be adapted, and we consider
operations in KOtheory.
Finally, in an appendix we give some general relations among polyno
mials which underpin a number of the formulas given in the preceding
sections.
Unless otherwise stated, p is assumed to be an odd prime. Having
chosen p, we fix q to be an integer which is primitive modulo p2, and
hence primitive modulo any power of p.
The third author acknowledges the support of a Scheme 4 grant from
the London Mathematical Society.
2. Degree zero operations in connective Ktheory
For each nonnegative integer n, we define `n(X) 2 Z[X] by
n1Y
`n(X) = (X  qi),
i=0
where q, as stated in the Introduction, is primitive modulo p2. The
notation derives from [9]. Generalisations of these polynomials are
considered later in this paper.
The Gaussian polynomials in the variable q (or qbinomial coeffi
cients) may be defined as
~ ~
n `j(qn)
= ______.
j `j(qj)
Definition 2.1. Define elements 'n 2 k0(k)(p), for n > 0, by
'n = `n( q),
where q is the Adams operation.
Thus, for example, '0 = 1, '1 = q  1 and '2 = ( q  1)( q  q).
Theorem 2.2. The elements of k0(k)(p)can be expressed uniquely as
infinite sums X
an'n,
n>0
where an 2 Z(p).
4 F. CLARKE, M. D. CROSSLEY, AND S. WHITEHOUSE
Proof. The bialgebra k0(k)(p)= K0(k)(p)is the Z(p)dual of the bial
gebra K0(k)(p); see [7, 11 ]. In Proposition 3 of [8] we gave a basis
for K0(k)(p), consisting of the polynomials fn(w) = `n(w)=`n(qn), for
n > 0. The theorem follows from the fact (which is implicit in [9]) that
the 'n are dual to this basis. To see this, recall that a coalgebra admits
an action of its dual which, in the case of K0(k)(p), is determined by
r . f (w) = f (rw). By a simple induction on m, this implies that
'm . fn(w) = qm(mn) wm fnm (w),
where fi(w) is understood as 0 if i < 0, so that 'm .fn(w) = 0 if n < m.
The Kronecker pairing can be recovered from this action by evaluating
at w = 1, hence
(
ff m(mn) 1, if m = n,
'm , fn(w) = q fnm (1) =
0, otherwise.
Remark 2.3. Operations in Ktheory are determined by their action
on coefficientsP[10 ]. It is therefore instructive to see how an infinite
sum n>0 an'nPacts. Since q acts on ß2i(k(p)) as multiplication by qi,
we see that n>0 an'n acts on the coefficient group ß2i(k(p)) as multi
plication by
Xi
an`n(qi).
n=0
The sum is finite since `n(qi) = 0 for n > i.
In particular, the augmentationP" : k0(k)(p) ! Z(p) given by the
action on ß0(k(p)), satisfies " n>0 an'n = a0.
P n nj n
It is easy to see by induction that `n(X) = j=0(1)njq( 2 ) j Xj;
see [4, (3.3.6)] and also [8, Proposition 8]. Hence we can express each
'n explicitly as a linear combination of Adams operations.
Proposition 2.4. For all n > 0,
Xn nj ~ ~
n qj
'n = (1)njq( 2 ) .
j=0 j
Conversely, our proof of Theorem 2.2 shows how to express all the
stable Adams operations in terms of the 'n.
Proposition 2.5. If j 2 Zx(p),
X `n(j)
j = _______n'n.
n>0 `n(q )
ALGEBRAS OF OPERATIONS IN KTHEORY 5
In particular, for i 2 Z,
~ ~
i X i
q = 'n.
n>0 n
Note that this is a finite sum for i > 0.
Additive operations in Ktheory are determined by their action on
the Hopf bundle over CP 1; see [5]. Writing
k0(CP 1)(p)= K0(CP 1)(p)= Z(p)[[t]],
where 1 + t is the Hopf bundle, we have the following formula for the
action of k0(k)(p)on the Hopf bundle.
Proposition 2.6. For all n > 0,
_ ` '~ ~!
X Xn nj qj n
'n(1 + t) = (1)njq( 2 ) ti.
i>n j=0 i j
Proof. Since r acts on line bundles by raising them to the rth power,
Proposition 2.4 shows that
Xn nj ~ ~
n qj
'n(1 + t) = (1)njq( 2 ) (1 + t) .
j=0 j
The formula now follows by using the binomial expansion and reversing
the order of summation. That the coefficient of ti in 'n(1 + t) is zero
for i < n can be proved by a simple induction, using the identity
'n+1 = ( q  qn)'n.
The product structure on k0(k)(p) is determined by the following
formula.
Proposition 2.7.
min(r,s)X r s r+s r s
`i(q )`i(q ) X `r+sj(q )`r+sj(q )
'r's = ____________i'r+si = ____________________r+sj'j.
i=0 `i(q ) j=max(r,s) `r+sj(q )
Proof. This result is essentially a fact about the polynomials `n(X).
Since the algebra of these polynomials underlies many of our results,
we have gathered together the relevant facts, in appropriate general
ity, in an appendix (Section 10). In particular, we need to use here
Proposition 10.4 with m = 0, X = q and c = q = (1, q, q2, . .)..
To show that the coefficient Ar,js(q, q[s]) given by that proposition
is equal to `r+sj(qr)`r+sj(qs)=`r+sj(qr+sj), it is only necessary to
verify that this expression satisfies the recurrence (10.5), i.e., taking
i = r + s  j that
`i+1(qr+1)`i+1(qs)_ r+si r `i(qr)`i(qs) `i+1(qr)`i+1(qs)
= (q  q )____________+ ________________.
`i+1(qi+1) `i(qi) `i+1(qi+1)
6 F. CLARKE, M. D. CROSSLEY, AND S. WHITEHOUSE
Using the identities
`i+1(qs) = `i(qs)(qs  qi) and `i+1(qr+1) = `i(qr)(qr+i+1  qi),
this is easy to check.
This proposition clarifies how it is that infinite sums can be multi
plied without producing infinite coefficients:
0 1
_ ! _ !
X X X B X `r+sj(qr)`r+sj(qs)C
ar'r bs's = @ arbs____________________r+sjA'j,
r>0 s>0 j>0 r,s6j `r+sj(q )
r+s>j
where the important point is that the inner summations are finite.
The multiplicative structure of k0(k)(p)is very intricate; we study it
in more detail in Section 3.
The following generalisation of Proposition 2.7 is used in Section 3.
Proposition 2.8. For all r, s > m,
r+smX
'r's = 'm cm,jr,s'j,
j=max(r,s)
where the coefficients are given by
`r+sj(qr)`r+sj(qs)
cm,jr,s= _____________________________________________.
q(r+sjm)m `r+sjm (qr+sjm )`m (qr)`m (qs)
Proof. Interchanging the role of r and s if necessary, we may assume
that s > r. Then Proposition 10.4 provides the given expansion for
'r's with cm,jr,s= Arm,js(q[m], q[s]). The formula for cm,jr,sholds since
this expression satisfies the recurrence (10.5).
The multiplication formula for the fn(w) (Proposition 7 of [8]) leads
by duality to the following result.
Proposition 2.9. The coproduct satisfies
X `r+s(qn)
'n = ____________rs'nr 'ns.
r,s>0 `r(q )`s(q )
r+s6n
The bounds in this summation ensure that the formula determines a
map from k0(k)(p)into the completed tensor product k0(k)(p)bk0(k)(p),
whose elements may be written as doubly infinite sums of the 'i 'j.
It is unreasonable to expect a coproduct to map into the usual tensor
product in this setting, so we will take the terms coalgebra, bialgebra
and Hopf algebra to mean structures where the coproduct maps into
the completed tensor product.
ALGEBRAS OF OPERATIONS IN KTHEORY 7
3. The ring structure of k0(k)(p)
In this section we write I for the augmentation ideal of the algebra
k0(k)(p)of stable operations of degree 0 in plocal connective Ktheory.
Thus I=I2 is the module of indecomposables. The ring of padic integers
is denoted by Zp.
Theorem 3.1.
(1) The ring k0(k)(p)is not Noetherian.
(2) Its module of indecomposables is isomorphic to Zp.
To prove this theorem, we define a sequence of ideals of k0(k)(p).
Definition 3.2. For m > 0 let
X
Bm = an'n : an 2 Z(p) .
n>m
Thus B0 B1 B2 . .,.with B0 = k0(k)(p)and B1 = I. It is
clear from Remark 2.3 that Bm consists of those operations which act
as zero on the coefficient groups ß2i(k(p)) for 0 6 i < m. Thus this
filtration is independent of our choice of q.
Proposition 3.3. For each 0 6 n 6 m, we have BnBm = 'nBm . In
particular, Bm is an ideal of k0(k)(p).
Proof. Clearly 'nBm PBnBm , since 'n 2 Bn. P
Suppose that ff = r>n ar'r 2 Bn and fi = s>m bs's 2 Bm , then,
using Proposition 2.8,
X
fffi= arbs'r's
r>n
s>m
X r+snX
= arbs'n cn,jr,s'j
r>n j=max(r,s)
s>m
X ` X '
= 'n arbscn,jr,s'j,
j>m r,s6j
r+s>j+n
which belongs to 'nBm , as the inner summation is finite.
Lemma 3.4. For each m > 1, the quotient Bm ='1Bm is isomorphic
to Zp.
Proof. Define a Z(p)module homomorphism ßm : Bm ! Zp by
_ !
X X
ßm : an'n 7! an(1  qm )(1  qm+1 ) . .(.1  qn1 ).
n>m n>m
Note that the sum does indeed converge padically.
8 F. CLARKE, M. D. CROSSLEY, AND S. WHITEHOUSE
By Proposition 2.7, '1'n = (qn 1)'n +'n+1 . Hence ßm ('1'n) = 0,
if n > m, and kerßm P '1Bm .
Now supposePff = n>m an'n 2 ker ßm . We will show that there
exists fi = n>m bn'n 2 Bm such that ff = '1fi. This is equivalent to
showing that the equations
(3.5) bn1 + (qn  1)bn = an (n > m)
may be solved for { bn 2 Z(p): n > m }, where bn = 0 for n < m.
Suppose that we have found br 2 Z(p)satisfying (3.5)for m 6 r < n.
It follows then that
0 = ßm (ff)= (an  bn1 )(1  qm ) . .(.1  qn1 )
+ an+1 (1  qm ) . .(.1  qn) + . ...
Thus if M = p (1  qm ) . .(.1  qn1 ) and N = p(1  qn),
0 (an  bn1 )pM mod pM+N ,
so that an bn1 mod pN , and (3.5) may be solved for bn 2 Z(p).
This shows that ker ßm = '1Bm ; it remains to prove that ßm is
surjective.
Let M = p (1  q) . .(.1  qm1 ) and choose R such that m 6
pR1 (p  1). For any r > R we can write
pr1(p1)Y pr1
____M
(1  qj) = urp p1 ,
j=m
where ur 2 Zx(p).
If x 2 Zp, by grouping the padic digits of x we can write
pR1_M pR+11_M pr1_M
x = x0 + xR p p1 + xR+1 p p1 + . .+.xrp p1 + . .,.
where xr 2 N for r = 0, and for r > R. Now let
X xr
ff = x0'm + ___'pr1(p1)+1.
r>R ur
It is clear that ff 2 Bm and ßm (ff) = x.
Proposition 3.6. For m > 1, the ideal Bm is not finitely generated.
Proof. By Proposition 2.8, k0(k)(p)acts on the quotient Bm ='1Bm = Zp
via the augmentation " : k0(k)(p)! Z(p)and the inclusion Z(p) Zp.
Thus if Bm were a finitely generated k0(k)(p)ideal, then Zp would be
a finitely generated Z(p)module. But a finite subset of Zp can only
generate a countable Z(p)submodule of Zp.
Proof of Theorem 3.1. Part (1) follows immediately. For part (2) we
note that Proposition 3.3 shows that I2 = '1B1, so that I=I2 =
B1='1B1 ~=Zp by Lemma 3.4.
ALGEBRAS OF OPERATIONS IN KTHEORY 9
It is interesting to note how far the augmentation ideal I is from
being generated by '1 = q  1. Before carrying out the completion
giving k0(k)(p), the augmentation ideal in Z(p)[ q  1] is principal, and
it is so again after pcompletion; see Section 5. In contrast, we have
Corollary 3.7. The quotient I=<'1> is isomorphic to Zp=Z(p), where
<'1> is the ideal of k0(k)(p)generated by '1.
Proof. It is clear that k0(k)(p)= Z(p)+ I.
We remark that the abelian group Zp=Z(p)is torsion free and divisi
ble. It is thus a Qvector space.
The intersection of the ideal Bm with the polynomial subalgebra
of k0(k)(p)generated by q is the principal ideal generated by 'm =
`m ( q). Theorem 2.2 exhibits k0(k)(p)as the completion of Z(p)[ q]
with respect to the filtration by these ideals. But note that this filtra
tion is not multiplicative, in the sense of [16 ], and hence there is no
associated graded ring. It is for this reason that the completion fails
to be Noetherian, and contains zero divisors, as we see in Section 4.
Of course the Adams operations r, where r is a plocal unit, gen
erate inside k0(k)(p)a copy of the group ring Z(p)[Zx(p)] which is dense
in the Bm filtration topology. Thus k0(k)(p)is naturally a completion
of Z(p)[Zx(p)].
In the following theorem we identify the units of k0(k)(p)in terms of
our basis. This formulation is closely related to Theorem 1 of [12 ].
P
Theorem 3.8. The element n>0an'n is a unit in the ring k0(k)(p)
P i
if and only if n=0an`n(qi) is a plocal unit for i = 0, 1, . .,.p  2.
P P i
Proof. If ff = n>0 an'n is a unit, then, by Remark 2.3, n=0 an`n(qi)
represents the action of ff on ß2i(k(p)), and so must be invertible for
all i. P
Conversely, assume in=0an`n(qi) 2 Zx(p)for i = 0, 1, . .,.p  2, then,
since `n(qi) `n(qj) mod p if i j mod p1 and `n(qi) = 0 if n > i,
this holds for all i > 0.
Now suppose, inductively, that we have found b0, b1, . .,.bi1 2 Z(p)
such that
_ !
X
an'n (b0 + b1'1 + . .+.bi1'i1) 2 1 + Bi.
n>0
Then, using Proposition 2.7, we see that, for any bi in Z(p),
_ !
X
an'n (b0 + b1'1 + . .+.bi'i) 2 1 + Bi,
n>0
10 F. CLARKE, M. D. CROSSLEY, AND S. WHITEHOUSE
with the coefficient of 'i having the form
i P j
bi in=0an`n(qi) + terms involving b1, . .,.bi1.
Thus, by our assumption, it is possible to chose bi 2 Z(p)so that
_ !
X
an'n (b0 + b1'1 + . .+.bi'i) 2 1 + Bi+1.
n>0
Repeating this process ad infinitum yields the required inverse.
4. The Adams splitting and the Adams summand
Idempotent operations which split plocal Ktheory (for p odd) into
p  1 summands were constructed in [1]. We show how to write the
connective version of these idempotents in terms of our basis elements.
Proposition 4.1. If ff 2 { 0, 1, . .,.p  2 }, the Adams idempotent
eff2 k0(k)(p)is given by
X
eff= cn,ff'n,
n>0
where ~ ~
1 X ni (ni) n
cn,ff= _______ (1) q 2 ,
`n(qn) i
the summation being over all integers i for which 0 6 i 6 n and i ff
mod p  1.
Proof. If an operation ' 2 k0(k)(p)acts on the coefficient group ß2i(k(p))
by multiplication by ~i, then <', wi> = ~i. Hence Proposition 8 of [8]
shows that
~ ~
ff 1 Xn ni ni n
', fn(w) = _______ (1) q( 2 ) ~i.
`n(qn) i=0 i
The result now follows by duality, since eff acts as the identity on
ß2i(k(p)) if i ff mod p  1, and as zero otherwise.
The spectra K(p)and k(p)are each split by Adams's idempotents into
p  1 suspensions of a multiplicative spectrum which we denote by G
and g, respectively.1 The coefficient ring G* can be identified with the
subring Z(p)[up1, up+1 ] Z(p)[u, u1] = ß*(K(p)), and g* is identified
with Z(p)[up1] Z(p)[u] = ß*(k(p)).
There is an algebra isomorphism ' : g0(g) ! e0k0(k)(p), under which
q 2 g0(g) maps to e0 q, but e0k0(k)(p) is not a subbialgebra of
k0(k)(p). However, composing the projection k0(k)(p)! e0k0(k)(p)with
the inverse of ', exhibits g0(g) as a quotient bialgebra of k0(k)(p). Thus
g0(g) is a summand of k0(k)(p)as an algebra, but not as a bialgebra.
____________
1The notations E(1) and e(1), or L and l, are also used.
ALGEBRAS OF OPERATIONS IN KTHEORY 11
Note that if p > 3, then, in contrast to the padic case [15 , 7], the
algebra k0(k)(p) is not isomorphic to g0(g) bZ(p)[Cp1]. This can be
seen by considering the action on the coefficient ring which shows that
k0(k)(p)contains no elements of order p  1.
The results of Section 2 have analogues for the algebra g0(g) of degree
zero stable operations on the Adams summand. We need first to adapt
the ideas of [8] to provide a basis for G0(g). Let z = wp1 2 G0(g).
Recalling that we have chosen q to be primitive modulo p2, we write
q^= qp1 and let
n1Y
^`n(X) = (X  ^qi).
i=0
Proposition 4.2. A Z(p)basis for G0(g) is given by the elements
^`(z)
f^n(z) = ___n___, for n > 0.
^`n(^qn)
Proof. We have
G0(g) = f (z) 2 Q[z] : f (1 + pZ(p)) Z(p) .
The multiplicative group 1 + pZp is topologically generated by ^q, and
p(^qn 1) = 1 + p(n) for all n > 1. The rest of the proof parallels
that of [8, Proposition 3].
We now identify the dual basis for the algebra g0(g) = G0(g).
Definition 4.3. Define ^'n2 g0(g), for n > 0, by
n1Y
^'n= ^`n( q) = ( q  ^qi).
i=0
To avoid any misunderstanding, we emphasise that the q indexing the
Adams operation is hatless.
Theorem 4.4 ([13 , Theorem 2.2]). The elements of g0(g) can be ex
pressed uniquely as infinite sums
X
an'^n,
n>0
where an 2 Z(p).
Proof. The proof is just as for Theorem 2.2: g0(g) = G0(g) is Z(p)dual
to the bialgebra G0(g), and the '^n are dual to the f^n(z), the action
being given by ^'m . ^fn(z) = ^qm(mn)zm ^fnm(z).
The formulas for the product and coproduct of the ^'nin g0(g) are,
of course, just like those of Section 2 for the 'n, but with the primitive
element q replaced by ^q. Similarly the proof of Theorem 3.1 generalises
to show that g0(g) is not Noetherian.
12 F. CLARKE, M. D. CROSSLEY, AND S. WHITEHOUSE
The proof of Theorem 3.8 simplifies in the split context, since ^`n(^qi)
is divisible by p for all i > 0 and all n > 0. Hence we have
Theorem 4.5. An element of g0(g) is a unit if and only if its augmen
tation is a unit.
This result was proved by Johnson in [12 ]; it shows that g0(g) is a
local ring. Johnson also showed in that paper that g0(g) is an integral
domain.
5. Operations in padic Ktheory
The padic completion of g0(g), which we denote by g0(g)p, is the
ring of degree zero operations in padic Ktheory. In fact in this case
the algebra of operations in the connective theory does not differ from
the algebra of operations in the periodic theory; see [11 ] and Section 7
below. We give here an algebraic proof of the result due to Clarke [7]
and Mitchell [15 ] that g0(g)p is a power series ring.
Theorem 5.1. g0(g)p is equal to the power series ring over Zp gener
ated by q  1.
Proof. Retaining the notation of Section 4, we let s(n, i), S(n, i) 2 Z(p)
be such that, for n > 1,
Xn Xn
^`n(X) = s(n, i)(X  1)i, and (X  1)n = S(n, i)^`i(X).
i=1 i=1
These constants are analogues of the Stirling numbers of the first and
second kinds, respectively. (In the notation of Section 10, s(n, i) =
An,i(^q, 1) and S(n, i) = An,i(1, ^q), where ^qis the sequence (^qi1)i>1 and
1 is the constant sequence (1)i>1.) It is clear that s(n, n) = S(n, n) = 1,
s(n, 1) = (1^q)(1^q2) . .(.1^qn1), and S(n, 1) = (^q1)n1 . Moreover
the two lower triangular matrices s(n, i) n,i>1 and S(n, i) n,i>1 are
mutually inverse.
In these cases, the recurrence of Proposition 10.2 becomes
s(n + 1, i)= s(n, i  1)  (^qn 1)s(n, i),
and S(n + 1, i)= S(n, i  1) + (^qi 1)S(n, i).
Since ^qn1 is divisible by p for all n, it follows easily that p s(n, i) >
n  i and p S(n, i) > n  i.
Let (p, Y ) denote the maximal ideal of Zp[[Y ]]. Since ^`n(Y + 1) =
Y (Y + 1  ^q) . .(.Y + 1  ^qn1), we have `^n(Y + 1) 2 (p, Y )n for
all n. As a result, there are ring homomorphisms forming the following
ALGEBRAS OF OPERATIONS IN KTHEORY 13
commutative diagram
Zp[X]= `^n(X)  ! Zp[[Y ]]=(p, Y )n
x x
? ?
? ?
Zp[X]= `^n+1(X)  ! Zp[[Y ]]=(p, Y )n+1
in which the horizontal maps are defined by X 7! Y + 1. In the limit
there is a ring homomorphism
n
g0(g)p = lim Zp[X]= `^n(X) ! lim Zp[[Y ]]=(p, Y ) = Zp[[Y ]],
which, we will show, is an isomorphism. The variable X corresponds
to q, and thus Y to q  1.
The kernel of Zp[X]= `^n(X) ! Zp[[Y ]]=(p, Y )n, which we shall de
note by In, is the free Zpmodule generated by the elements pni(X 
1)i : 0 6 i 6 n  1 , where g(X) denotes the coset of g(X) in the
quotient ring Zp[X]= `^n(X) . Under the homomorphism In+1 ! In,
the element pn+1i(X  1)i maps to p pni(X  1)i , if i < n, but
p(X  1)n maps to
n1X
s(n, j) nj j
 p_______njp (X  1) .
j=1 p
Note here that s(n, j)=pnj 2 Z(p)by the remarks above. This shows
that the image of In+1 lies in pIn, and hence, since no nonzero element
of In is infinitely divisible by p, that lim In = 0. Thus g0(g)p maps
injectively into Zp[[Y ]].
To prove that it does so surjectively we define maps Zp[[Y ]] !
Zp[X]= `^n(X) by
_ !
X n1X X1
crY r7! c0 + S(j, i)cj ^`i(X)].
r>0 i=1 j=i
It is here, of course, that we need to be working over Zp. The conver
gence of the infinite series is guaranteed since pji divides S(j, i).
It will turn out that we have a ring homomorphism. But at this
point we need only to know that it is a homomorphism of Zpmodules,
and this is trivial. It is also trivial that the maps factor through the
projection
Zp[X]= `^n+1(X) ! Zp[X]= `^n(X)
and so define a Zpmodule homomorphism
0
Zp[[Y ]] ! lim Zp[X]= `^n(X) = g (g)p.
14 F. CLARKE, M. D. CROSSLEY, AND S. WHITEHOUSE
To verify that this is the inverse of the ring homomorphism constructed
earlier we need to verify that the composition
n
Zp[[Y ]] ! Zp[X]= `^n(X) ! Zp[[Y ]]=(p, Y )
coincides with the natural map. ButPsince S(j, i) 0 mod pn for
j > i + n, the composition sends r>0crY rto
n1Xi+n1XiX
c0 + S(j, i)s(i, k)cj[Y k]
i=1 j=i k=1
n1X2n2X min(j,n1)X
= c0 + S(j, i)s(i, k)cj[Y k],
k=1 j=k i=max(k,jn+1)
where [Y k] denotes the coset of Y kin Zp[[Y ]]=(p, Y )n, and we note that
[Y k] = 0 for k > n.
Now [Y k] has order pnk in Zp[[Y ]]=(p, Y )n, and S(j, i)s(i, k) is di
visible by pjk. PThis means that if j > n, then S(j, i)s(i, k)[Y k] = 0
and the image of r>0crY ris
n1Xn1XXj n1X
c0 + S(j, i)s(i, k)cj[Y k] = c0 + ck[Y k],
k=1 j=k i=k k=1
P j
since i=kS(j, i)s(i, k) = ffij,k. This completes the proof.
6. Operations in periodic Ktheory
We now turn our attention to the periodic case. We let K0(K)(p)de
note the algebra of degree zero operations in plocal periodic Ktheory.
We show in this section how the results of Sections 24 extend to this
context. Here the degree zero operations determine all stable opera
tions, and K0(K)(p)is a Hopf algebra so we determine the antipode as
well as the other parts of the structure.
For each nonnegative integer n, we define the polynomial n(x) by
Yn
n(X) = (X  ~qi),
i=1
where ~qiis the ith term of the sequence
(1)ibi=2c
~q= (1, q, q1 , q2, q2 , q3, q3 , q4, . .).= q i>1,
i.e., n(X) = `n(X; ~q) in the notation of the appendix.
Definition 6.1. Define elements n 2 K0(K)(p), for n > 0, by
n = n( q).
Thus, for example, 0 = 1, 1 = q  1, 2 = ( q  1)( q  q) and
3 = ( q  1)( q  q)( q  q1 ).
ALGEBRAS OF OPERATIONS IN KTHEORY 15
Theorem 6.2. The elements of K0(K)(p)can be expressed uniquely as
infinite sums X
an n,
n>0
where an 2 Z(p).
Proof. The proof is analogous to that of Theorem 2.2. The polynomi
als Fn(w) = wbn=2cfn(w) form a Z(p)basis for K0(K)(p)according to
Corollary 6 of [8], and the n are, modulo multiplication by units, dual
to this basis. In fact the Kronecker pairing satisfies
(
ff qnbn=2c, if n = j,
n, Fj(w) =
0, otherwise.
As in the proof of Theorem 2.2, this follows from a study of the action
of the dual on K0(K)(p), but the details are a little more complicated
and are given in the following result.
Lemma 6.3.
(1) n . Fj(w) = 0(if j < n;
qnk wk , if n = 2k,
(2) n . Fn(w) =
qnk wk+1, if n = 2k + 1;
(3) n . Fj(w) is divisible by fjn(w) for j > n.
Proof. (1) Recall that q acts as q . f (w) = f (qw). If j < n and
 bj=2c 6 i 6 dj=2e, then q qi is a factor of n, so that n . wi = 0.
But Fj(w) is a Laurent polynomial of codegree  bj=2c and degree
j  bj=2c = dj=2e.
(2) By the proof of (1), all monomials occurring in Fn(w) are an
nihilated by n except one. If n = 2k, this is the lowest degree mono
2k
mial wk , whose coefficient is f2k(0) = q( 2)=`2k(q2k), and we have
k
2k . F2k(w) = 2k . f2k(0)w
2k 2k(qk ) k
= q( 2)_________w
`2k(q2k)
2 k
= q2k w .
If n = 2k + 1, it is the highest degree monomial wk+1 which must be
considered. The leading coefficient is 1=`2k+1(q2k+1), and
2k+1(qk+1) k+1
2k+1 . F2k+1(w) = ____________w
`2k+1(q2k+1)
= q(2k+1)kwk+1.
(3) The proof is by (finite) induction on n. Note that 0 . Fj(w) =
Fj(w) is certainly divisible by fj(w). Now assume that n . Fj(w) =
16 F. CLARKE, M. D. CROSSLEY, AND S. WHITEHOUSE
fjn(w)Gn,j(w) for some Laurent polynomial Gn,j(w). Then, since
n+1 = ( q  qi) n for some i,
n+1 . Fj(w) = ( q  qi) . fjn(w)Gn,j(w)
= fjn(qw)Gn,j(qw)  qifjn(w)Gn,j(w).
But this is zero for w = 1, q, q2, . .,.qjn2 and therefore divisible
by fjn1 (w).
P
Remark 6.4. The action of the infinite sum n>0an n on the coef
ficient group ß2i(K(p) is multiplication by the finite sum
X2i
an n(qi).
n=0
P
Clearly the augmentation sends n>0an n to a0.
We have the following analogue of Proposition 2.4.
Proposition 6.5. For all n > 0,
Xn ~ ~
n qj
n = (1)njqe(n,j) ,
j=0 j
where (
(n  j)(j  1)=2, if n is even,
e(n, j) =
(n  j)j=2, if n is odd.
Proof. Assuming, inductively, that the result holds for even n = 2k,
j
since 2k+1 = 2k( q  qk ), the coefficient of q in 2k+1 is
` ~ ~ ~ ~'
2k k(2kj)(j1)=22k
(1)j+1 q(2kj+1)(j2)=2 + q
j  1 j
` ~ ~ ~ ~ '
2k 2k
= (1)j+1q(2k+1j)j=2 q2kj+1 +
j  1 j
~ ~
2k + 1
= (1)j+1q(2k+1j)j=2 .
j
The argument to show that the odd case implies the next even case is
similar.
Conversely, the proof of Theorem 6.2 yields
Proposition 6.6. If j 2 Zx(p),
X `n(j)
j = qnbn=2cjbn=2c_______n n.
n>0 `n(q )
In particular, for i 2 Z,
~ ~
i X (ni)bn=2ci
q = q n.
n>0 n
ALGEBRAS OF OPERATIONS IN KTHEORY 17
Note that this is a finite sum for i > 0.
We now consider the antipode Ø of the Hopf algebra K0(K)(p). In
the dual K0(K)(p)the antipode is given by w 7! w1 (see [3]), while
1
Ø j = j .
Proposition 6.7. The antipode in K0(K)(p)is determined by
_ ~ ~ ~ ~ !
X X n n i
Ø n = (1)niq(i+j)bj=2c+e(n,i) j,
j>2b(n1)=2c+1 i=0 i j
where e(n, i) is defined in Proposition 6.5.
Proof. Proposition 6.5 shows that
Xn ~ ~
n e(n,i)qi
Ø n = (1)ni q ,
i=0 i
and, by Proposition 6.6,
~ ~
i X (i+j)bj=2ci
q = q j.
j>0 j
The required formula, with summation over j > 0, now follows by
substitution.
To see that in fact the coefficients are zero until j = 2 b(n  1)=2c+1
we note that by duality the expansion of Ø n can also be obtained
from expressing Fj(w1 ) as a linear combination of the Fn(w). The
remarks in the proof of Lemma 6.3 (1) show that the Laurent polyno
mial Fj(w1 ) can be written as a Z(p)linear combination of the Fn(w)
for n 6 j + 1 if j is odd, and for n 6 j if j is even. Thus we may
write Ø n is an infinite linear combination of the j for j > n if n is
odd, and for j > n  1 if n is even.
The same approach, using Propositions 6.5 and 6.6, yields the fol
lowing formulas for the product and coproduct.
Proposition 6.8. For all r, s > 0,
Xr+s
r s = Akr,s k,
k=max(r,s)
where
Xr Xs ~ ~~ ~~ ~
r s i + j
Akr,s= (1)r+sijqe(r,i)+e(s,j)+(kij)bk=2c .
i=0 j=0 i j k
18 F. CLARKE, M. D. CROSSLEY, AND S. WHITEHOUSE
Proposition 6.9. The coproduct in K0(K)(p)satisfies
n =
8 X
>> Cr,s nr ns, if n is even,
>> n
< r,s>0
r+s6nX X
>> Cr,s nr ns + Cr,s nr ns, if n is odd,
>> n n
: r,s>0 r,s>2
r+s6n r+s=n+1
where
Cr,sn=
min(r,s)X ~ ~~ ~~ ~
n n  k n  k
(1)kqe(n,nk)+(kr)b(nr)=2c+(ks)b(ns)=2c .
k=0 k n  r n  s
The results of Section 3 also apply in the periodic case.
Theorem 6.10.
(1) The ring K0(K)(p)is not Noetherian.
(2) Its module of indecomposables is isomorphic to Zp.
Proof. The proof exactly parallels that of Theorem 3.1. We define the
family of ideals of K0(K)(p)
X
Am = an n : an 2 Z(p) .
n>m
Then we have AnAm = nAm if 0 6 n 6 m. Just as is Section 3, we
show that, for m > 1, Am = 1Am ~= Zp, and thus Am is not finitely
generated.
We can give criteria analogous to those of Theorem 3.8 for when a
general element of K0(K)(p)is a unit, but we omit the details.
We can now generalise the results of Section 4 to the periodic case.
Proposition 6.11. If ff 2 { 0, 1, . .,.p  2 }, the Adams idempotent
Eff2 K0(K)(p)is given by
X
Eff= Cn,ff n,
n>0
where ~ ~
1 X dn=2einbn=2c+(dn=2ei) n
Cn,ff= _______ (1) q 2 ,
`n(qn) bn=2c + i
the summation being over all integers i for which n  1 < 2i 6 n + 1
and i ff mod p  1.
Recall that we let ^q= qp1, where q is primitive modulo p2.
ALGEBRAS OF OPERATIONS IN KTHEORY 19
Definition 6.12. Let
Yn
^ n= ( q  ^q(1)ibi=2c).
i=1
Theorem 6.13. The elements of G0(G) can be expressed uniquely as
infinite sums X
an ^n,
n>0
where an 2 Z(p).
It is possible to write down formulas generalising those above for the
antipode, product and coproduct in G0(G), but we omit the details.
We can also show easily that G0(G) is a nonNoetherian local ring.
7. The relation between the connective and periodic
cases
We consider now the relation between the connective and periodic
cases, focussing on the nonsplit setting, although it is clear that similar
results hold for the relation between g0(g) and G0(G).
The covering map k ! K leads to an inclusion
K0(K)(p),! K0(k)(p)= k0(k)(p),
which is described by the following formula.
Proposition 7.1. For n > 0,
Xn _ X n ~ ~ ~ ~!
n j
n = (1)njqe(n,j) 'i,
i=bn=2c+1 j=i j i
where e(n, j) is as defined in Proposition 6.5.
Proof. We obtain the required formula by combining Propositions 6.5
and 2.5, but with a summation from i = 0 to n.
To see that the coefficients are zero for i = 0, . .,.bn=2c, we note that
the polynomial n(X) is divisible by `bn=2c+1(X), and the quotient is
`dn=2e1 X, (q1 , q2 , . .)., in the notation of Section 10. Writing this
quotient as a linear combination of the `r X, (qbn=2c+1, qbn=2c+2, . .).,
and substituting X = q, gives rise to the formula
Xn
n = An,ibn=2c1(q1 , q2 , . .)., (qbn=2c+1, qbn=2c+2, . .).'i.
i=bn=2c+1
An arbitrary element of K0(K)(p)can then be written as
_ _ ~ ~ ~ ~! !
X X X2i Xn n j
an n = (1)njqe(n,j) an 'i.
n>0 i>0 n=i j=i j i
20 F. CLARKE, M. D. CROSSLEY, AND S. WHITEHOUSE
Note that the inner summations are finite.
Since they are polynomials in q, the basis elements 'i lie in the
image of the inclusion and can be expressed in terms of our basis
for K0(K)(p)as follows.
Proposition 7.2.
Xn ~ ~
n  di=2e  1
'n = qb(i+3)=2c(ni)`ni(qi1) i.
i=0 n  i
Proof. It is merely necessary to verify that the given coefficients sat
isfy the recurrence given by Proposition 10.2 for An,i(q, ~q), and this is
routine.
Note that the summation here runs from i = 0. It is not the case that
the coefficients are zeroPfor i small enough. So any attempt to write an
arbitrary infinite sum an'n in k0(k)(p)in terms of the i will lead
to infinite sums for the coefficients. This reflects the fact that the map
K0(K)(p),! k0(k)(p)is a strict monomorphism. However, if we com
plete padically, the highly pdivisible factor `ni(qi1) ensures that
these inner sums converge. Thus we recover the fact that, by contrast,
K0(K)p ! k0(k)p is an isomorphism, as discussed in [7] and [11 ].
8. 2local operations
Here we provide a description of operations in 2local Ktheory. This
is a little more complicated than for odd primes, essentially because in
stead of the single `generator' q one has to deal with both 3 and 1.
Throughout this section and the next, the variable q occurring im
plicitly in the polynomial `n(X) will be set equal to 9, and we let
`~n(X) = Q n1i=0(X  32i+1). Thus, in the notation of the appendix,
`n(X) = `n(X; a) and ~`n(X) = `n(X; b), where a is the sequence of
even powers of 3 and b is the sequence of odd powers of 3. These
choices are related to the fact that { 3i : i > 0 } is dense in Zx2;
see [8].
Definition 8.1. Define elements in 2 k0(k)(2), for n > 0, by
i2m+1 = ( 1  1)~`m( 3),
Xm m
`i(3)`i(9 )
i2m = `m ( 3) + ___________ii2m2i+1 .
i=1 2`i(9 )
Theorem 8.2. The elements of k0(k)(2)can be expressed uniquely as
infinite sums X
anin,
n>0
where an 2 Z(2).
ALGEBRAS OF OPERATIONS IN KTHEORY 21
Proof. The proof mirrors that of Theorem 2.2. We show that the given
elements form the dual basis to the basis { fn(2)(w) : n > 0 } obtained
for K0(k)(2)in [8, Proposition 20]. This may be proved by induction
arguments using the following formulas, describing the action of k0(k)(2)
on K0(k)(2).
1 . f2(2)m(w)= f2(2)m(w),
1 . f2(2)m+1(w)= f2(2)m(w)  f2(2)m+1(w),
3 . f2(2)m(w)= 9m f2(2)m(w) + f2(2)m2(w),
3 . f2(2)m+1(w)= 32m+1 f2(2)m+1(w)  9m f2(2)m(w) + f2(2)m1(w).
Remark 8.3. The operation in acts on the coefficient group ß2i(k(2))
as multiplication by the values given in the following table.

n = 2m n = 2m + 1
_______________________________
i even `m (3i) 0

i odd ~`m(3i) 2~`m(3i)
Thus in acts as zero on ß2i for all i < n.
The following proposition gives product formulas.
Proposition 8.4.
min(m,n)X i j
i1
i2m i2n= dim,n i2m+2n2i  3___2i2m+2n+12i ,
i=0
min(m,n)X
i2m+1 i2n= 3idim,ni2m+2n+12i ,
i=0
min(m,n)X
i2m+1 i2n+1= 2 3idim,ni2m+2n+12i ,
i=0
`i(9m )`i(9n)
where dim,n= _____________.
`i(9i)
Proof. These formulas are proved by long but straightforward induction
arguments.
Since i2n+1 = i1i2n, the following proposition completely determines
the coproduct.
22 F. CLARKE, M. D. CROSSLEY, AND S. WHITEHOUSE
Proposition 8.5.
i1 = i1 1 + i1 i1 + 1 i1,
X `r+s(9n)
i2n = ____________rsi2n2r i2n2s.
r,s>0 `r(9 )`s(9 )
r+s6n
Proof. These formulas may be deduced from the product formula for
the dual basis of cooperations just as in the proof of Proposition 2.9.
In principle, similar methods will give a description of the periodic
case K0(K)(2). However, this will be even more complicated, and we
omit the details.
9. Operations in KOtheory
We consider ko and KO localised at p = 2. (For p odd, these spectra
split as (p  1)=2 copies of g and G, respectively.) Proofs are omitted
since the arguments are just the same as those in [8] and in earlier
sections of this paper.
Just as in Section 8, the variable q used implicitly in the polynomials
`n(X) and n(X) is set equal to 9.
Definition 9.1. Let x = w2 = u2v2 2 KO0(ko); see [3]. Let
`n(x)
hn(x) = _______.
`n(9n)
Proposition 9.2.
(1) { hn(x) : n > 0 } is a Z(2)basis for KO0(ko)(2).
(2) { xbn=2chn(x) : n > 0 } is a Z(2)basis for KO0(KO)(2).
Theorem 9.3.
(1) The elements of ko0(ko)(2)can be expressed uniquely as infinite
sums X
an`n( 3),
n>0
where an 2 Z(2).
(2) The elements of KO0(KO)(2)can be expressed uniquely as in
finite sums X
an n( 3),
n>0
where an 2 Z(2).
Corollary 9.4. There is an isomorphism of bialgebras
k0(k)(2)=< 1  1> ~=ko0(ko)(2).
ALGEBRAS OF OPERATIONS IN KTHEORY 23
Proof. We have seen that the ideal I generated by i1 = 1  1 is a
coideal. Note that i2n `n( 3) modulo I. By Theorem 9.3 the `n( 3)
form a basis of ko0(ko)(2), and the product and coproduct are respected.
The methods of Section 5 may be adapted to show that the 2adic
completion ko0(ko)2 is the power series ring over Z2 generated by 31.
10. Appendix: polynomial identities
Given two sequences a = (ai)i>1 and b = (bi)i>1 of elements of a
commutative ring R, let
Yn Yn
`n(X; a) = (X  ai) and `n(X; b) = (X  bi).
i=1 i=1
These polynomials provide two bases for R[X] as an Rmodule and can
therefore be written in terms of each other.
Definition 10.1. Define An,r(a, b) 2 R by
Xn
`n(X; a) = An,r(a, b)`r(X; b).
r=0
Together with the identities A0,0(a, b) = 1, A0,r(a, b) = 0 for r > 0,
and An,1(a, b) = 0 for n > 0, the coefficients An,r(a, b) are determined
by the following recurrence, which is used repeatedly in this paper.
Proposition 10.2. For n, r > 0,
An+1,r(a, b) = (br+1  an+1 )An,r(a, b) + An,r1(a, b).
Proof. Since `n+1 (X; a) = (X  an+1 )`n(X; a),
Xn
`n+1 (X; a) = An,r(a, b)(X  an+1 )`r(X; b)
r=0
Xn
= An,r(a, b) `r+1(X; b) + (br+1  an+1 )`r(X; b) ,
r=0
and the result follows using An,1(a, b) = An,n+1(a, b) = 0.
In most cases which we have considered, direct substitution in the
recurrence allows considerable simplification. However it is possible to
give an explicit formula for the coefficients An,r(a, b) as polynomials in
the ai and bj.
Proposition 10.3. For n, r > 0,
X Y
An,r(a, b) = (bff(i,J) ai),
J {1,...,n}16i6n
J=r i=2J
fi fi
where oe(i, J) = 1 + fi{ j 2 J : j < i }fi.
24 F. CLARKE, M. D. CROSSLEY, AND S. WHITEHOUSE
Proof. Let
X Y
Aen,r= (bff(i,J) ai).
J {1,...,n}16i6n
J=r i=2J
We have Ae0,0= 1, Ae0,r= 0 for r > 0, and Aen,1 = 0 for n > 0. It
is therefore only necessary to verify that Aen,rsatisfies the recurrence
Aen+1,r= (br+1  an+1 )Aen,r+ eAn,r1of Proposition 10.2.
Break the sum defining eAn+1,rinto two parts by dividing the subsets
J {1, . .,.n + 1} such that J = r according to whether n + 1 2 J or
not. If n+1 =2J, then J {1, . .,.n} and the corresponding summand
in Aen,roccurs in Aen+1,r multiplied by the factor (br+1  an+1 ) since
oe(n + 1, J) = r + 1.
If n + 1 2 J, let I = J r {n + 1} {1, . .,.n}, then oe(i, J) = oe(i, I)
for all i =2J, and
Y Y
(bff(i,J) ai) = (bff(i,I) ai)
16i6n+1 16i6n
i=2J i=2I
is a summand in both Aen+1,rand Aen,r1.
It is clear that in both cases the process is reversible.
We show finally how essentially the same coefficients arise in for
mulas for products of the `n(X, a). Given a sequence c = (ci)i>1, we
write c[m] for the shifted sequence (cm+i )i>1.
Proposition 10.4. If r > m > 0 and s > 0, then
r+smX
`r(X; c)`s(X; c) = `m (X; c) Arm,js(c[m], c[s])`j(X; c).
j=s
Proof. Setting a = c[m] and b = c[s] in Definition 10.1, we have
r+smX
`rm (X; c[m]) = Arm,js(c[m], c[s])`js(X; c[s]).
j=s
Multiplying by `m (X; c)`s(X; c), and using the identities `r(X; c) =
`m (X; c)`rm (X; c[m]) and `j(X; c) = `s(X; c)`js(X; c[s]), now gives
the result.
Writing Am,jr,s= Arm,js(c[m], c[s]) for the coefficient in the above
expansion for `r(X; c)`s(X; c), the recurrence of Proposition 10.2 takes
the form
(10.5) Am,jr+1,s= (cj+1  cr+1)Am,jr,s+ Am,j1r,s.
ALGEBRAS OF OPERATIONS IN KTHEORY 25
References
[1]J. F. Adams, Lectures on generalised cohomology, in: Lecture Notes in Math
ematics, Vol. 99, Springer, Berlin, 1969.
[2]J. F. Adams and F. W. Clarke, Stable operations on complex Ktheory, Illino*
*is
J. Math. 21 (1977) 826829.
[3]J. F. Adams, A. S. Harris, and R. M. Switzer, Hopf Algebras of Cooperations
for Real and Complex Ktheory, Proc. London Math. Soc. (3) 23 (1971) 385
408.
[4]G. E. Andrews, The Theory of Partitions, Encyclopedia of Mathematics and its
Applications, Vol. 2, AddisonWesley Publishing Co., Reading, Mass.London
Amsterdam, 1976.
[5]M. J. Boardman, D. C. Johnson and W. St. Wilson, Unstable operations in
generalized cohomology, in: Handbook of Algebraic Topology, NorthHolland,
Amsterdam, 1995, pp. 687828.
[6]A. K. Bousfield, On the homotopy theory of Klocal spectra at an odd prime,
Amer. J. Math. 107 (1985) 895932.
[7]F. Clarke, Operations in Ktheory and padic analysis, Groupe d''Etude
d'Analyse Ultram'etrique, Paris, 1986/87, 12 pages; http://wwwmaths.swan.
ac.uk/staff/fwc/research.html#ktheory.
[8]F. Clarke, M. D. Crossley and S. Whitehouse, Bases for cooperations in K
theory, KTheory 23 (2001) 237250.
[9]E. C. Ihrig and M. E. H. Ismail, A qumbral calculus, J. Math. Anal. Appl. *
*84
(1981) 178207.
[10]K. Johnson, The action of the stable operations of complex Ktheory on coef
ficient groups, Illinois J. Math. 28 (1984) 5763.
[11]K. Johnson, The relation between stable operations for connective and non
connective plocal complex Ktheory, Canad. Math. Bull. 29 (1986) 246255.
[12]K. Johnson, The algebra of stable operations for plocal complex Ktheory,
Canad. Math. Bull. 30 (1987) 5762.
[13]W. Lellmann, Operations and cooperations in oddprimary connective K
theory, J. London Math. Soc. 29 (1984) 562576.
[14]I. Madsen, V. Snaith and J. Tornehave, Infinite loop maps in geometric topo*
*l
ogy, Math. Proc. Camb. Phil. Soc. 81 (1977) 399430.
[15]S. A. Mitchell, On padic topological Ktheory, in: Algebraic Ktheory and
Algebraic Topology (Lake Louise, AB, 1991), Kluwer Acad. Publ., Dordrecht,
pp. 197204.
[16]D. G. Northcott, Lessons on Rings, Modules and Multiplicities, Cambridge
University Press, London, 1968.
Department of Mathematics, University of Wales Swansea, Swansea
SA2 8PP, Wales
Email address: F.Clarke@Swansea.ac.uk
Department of Mathematics, University of Wales Swansea, Swansea
SA2 8PP, Wales
Email address: M.D.Crossley@Swansea.ac.uk
Department of Pure Mathematics, University of Sheffield, Sheffield
S3 7RH, England
Email address: S.Whitehouse@sheffield.ac.uk