EXTENSIONS OF FILTERED ~-RING STRUCTURES OVER
THE DUAL NUMBER RING
DONALD YAU
Abstract. We study problems related to the existence and uniqueness
of filtered ~-ring structures over the truncated polynomial ring Z[x]=x3
that extend a given filtered ~-ring structure over Z[x]=x2.
1. Introduction
A ~-ring is, roughly speaking, a commutative ring R with unit together
with operations ~i on it that act like the exterior power operations. It is
widely used in Algebraic Topology, Algebra, and Representation Theory.
For example, the complex representation ring R(G) of a group G is a ~-
ring, where ~i is induced by the map that sends a representation to its ith
exterior power. Another example of a ~-ring is the complex K-theory of a
topological space X. Here, ~i arises from the map that sends a complex
vector bundle j over X to the ith exterior power of j. In the algebra side,
the universal Witt ring W(R) of a commutative ring R is a ~-ring. In fact,
in some sense, it is the universal ~-ring that can be associated to R.
The ~-operations impose severe restrictions on the structure of the un-
derlying ring, and not every commutative ring can be given the structure of
a ~-ring. Given an arbitrary commutative ring R with unit, it is, therefore,
natural to ask whether it admits a ~-ring structure. If so, how many ~-ring
structures does it admit and is there a natural structure on the moduli set of
~-ring structures? If R is a ~-ring and the underlying ring of R is a quotient
of another ring S, can the ~-ring structure on R be extended to one on S?
If so, in how many ways can it be done and what is the structure of the
moduli set of such extensions?
Some of these problems have been studied before. In [2] Clauwens showed
that over the integral polynomial ring Z[x], there are essentially only two
~-ring structures. As far as the author is aware, this is the first paper in the
literature in which the question of how many ~-ring structures a given ring
admits is studied explicitly. In contrast to the polynomial case, the author
[6] showed the existence of uncountably many pairwise non-isomorphic ~-
ring structures on the power series ring Z[[x1, . .,.xk]], k 1. In fact, the
____________
Date: April 27, 2004.
2000 Mathematics Subject Classification. 55S25,16E20,16W99.
1
2 DONALD YAU
uncountably many ~-ring structures in [6] are the K-theory of some spaces
that are closely related to the classifying spaces of compact connected Lie
groups. It is probably not true that every ~-ring structure over the power
series ring arises this way.
Sometimes it is more natural to think about filtered ~-ring, which is just
a ~-ring whose underlying ring is filtered by a decreasing sequence of ideals
that are closed under the ~-operations. For example, the K-theory of a CW
complex is naturally a filtered ~-ring. The uncountably many ~-rings in [6]
mentioned in the previous paragraph are filtered ~-ring structures over the
power series ring with a certain filtration, and they are non-isomorphic as
filtered ~-rings. Since a filtered ring R has a topology on it, one is tempted
to ask if the moduli set of filtered ~-ring structures on it has a natural
topology that is, in some sense, compatible with the topology on R. This
is indeed the case, as was shown by the author in [5]. Another result in
[5] classifies the set of isomorphism classes of filtered ~-ring structures on
the dual number ring Z(J)[x]=x2 with the x-adic filtration over the J-local
integers, where J is an arbitrary set of primes. They are in bijection with
the set of sequences {bp}, p primes, where bp is divisible by p in Z(J).
The main purpose of this note is to continue the work in [5] and study ex-
tensions of a filtered ~-ring structure R over the dual number ring Z(J)[x]=x2
to the truncated polynomial ring Z(J)[x]=x3. A necessary and sufficient con-
dition for such an extension to exist is that, for all odd primes q, the linear
coefficient bq of _q(x) in R is congruent to its square b2qmodulo a certain
power of 2, depending on b2 but not q (Theorem 3.1). In particular, such
an extension always exists if either 2 is invertible (Corollary 3.2) or there is
exactly one factor of 2 in b2 (Corollary 3.5). It also implies that there are
uncountably many non-isomorphic extensions when b2 is equal to 0 (Exam-
ple 3.3) and that there are uncountably many filtered ~-ring structures on
Z[x]=x2 that do not admit any extension (Example 3.4).
Next we study when two extensions are isomorphic. Theorem 3.6 gives a
necessary and sufficient condition for two extensions of R to be isomorphic.
One formulation of it only depends on the coefficients in the Adams oper-
ation _2 when applied to x. There are several interesting consequences of
this result. First, over the 2-local integers, if b2 has exactly one factor of *
*2,
then R admits a unique extension to a filtered ~-ring structure on Z(2)[x]=x3
(Corollary 3.7). This last statement is not true over the integers (Example
3.8), but the additional assumption, b2 = 2, would make it true (Corollary
3.9). Moreover, over the integers, as long as b2 is non-zero, there are always
at most finitely many extensions (Corollary 3.11). A version of this result,
with the integers replaced by the J-local integers, is given in Corollary 3.10.
Finally, we note that, although integrally the moduli set of isomorphism
classes of extensions is finite, it could be made arbitrarily large by taking
different filtered ~-ring structures (Example 3.12).
EXTENSIONS OF FILTERED ~-RINGS 3
In the following section, we will review some basic results about ~-rings
and observe that many truncated power series rings have uncountably many
non-isomorphic filtered ~-ring structures (Theorem 2.2).
Here is a list of some of the notations that are used in this note.
o Z(J): The J-local integers, where J is a set of primes.
o rZ(J), r 2 Z(J): The subgroup of Z(J)consisting of elements of the
form rs for some s 2 Z(J).
o Zx(J): The invertible elements in Z(J).
o `p(r), p prime, r rational: `p(n) is the largest integer for which p`p(n)
divides n, when n is an integer; `p(r) = `p(m) - `p(n) if r = m=n.
By convention, `p(0) = -1 for all primes p.
2. Filtered ~-rings
The purpose of this section is to give a brief account of the basic defini-
tions about ~-rings and their filtered analogues. The result in [5] about the
classification of filtered ~-ring structures over the dual number ring is then
recalled. This section ends with a result that says that many truncated
power series rings have uncountably many non-isomorphic filtered ~-ring
structures.
The reader may refer to [1, 3] for more discussions about the basic alge-
braic properties of ~-rings. We should point out that what we call a ~-ring
here is referred to as a "special" ~-ring in [1].
By a ~-ring, we mean a commutative ring R with unit that is equipped
with functions
~i:R ! R (i 0),
called ~-operations. These functions are required to satisfy the following
conditions. For any integers i, j 0 and elements r and s in R:
o ~0(r) = 1.
o ~1(r) = r.
o ~i(1) = 0 forPi > 1.
o ~i(r + s) = ik=0~k(r)~i-k(s).
o ~i(rs) = Pi(~1(r), . .,.~i(r); ~1(s), . .,.~i(s)).
o ~i(~j(r)) = Pi,j(~1(r), . .,.~ij(r)).
The Pi and Pi,jare certain universal polynomials with integer coefficients,
and they are defined using the elementary symmetric polynomials. See the
references mentioned above for their exact definitions. A map f of ~-rings is
a ring map of the underlying rings that is compatible with the ~-operations,
in the sense that f~i= ~if for all i.
4 DONALD YAU
There are some very useful operations inside a ~-ring, the so-called Adams
operations
_n :R ! R (n 1).
They are defined by the Newton formula:
_n(r) - ~1(r)_n-1 (r) + . .+.(-1)n-1~n-1(r)_1(r) + (-1)nn~n(r) = 0.
The Adams operations have the following properties:
o All the _n are ring maps.
o _1 = Id.
o _m _n = _mn = _n_m .
o _p(r) rp (mod pR) for each prime p and element r in R.
A ~-ring map f is compatible with the Adams operations, in the sense
that f_n = _nf for all n.
As one observes in the Newton formula, one can almost retrieve the ~-
operations from the Adams operations. This is, in fact, possible provided
that R is Z-torsionfree. More explicitly, a theorem of Wilkerson [4] says that
if R is a Z-torsionfree ring equipped with ring endomorphisms _n (n 1)
satisfying the last three conditions above, then there exists a unique ~-ring
structure on R whose Adams operations are exactly the given _n. We will
refer to this as Wilkerson's Theorem. In particular, over the truncated
polynomial ring Z(J)[x]=xn, a ~-ring structure is specified by polynomials
_p(x), p primes, such that
_p(_q(x)) = _q(_p(x))
and
_p(x) xp (mod pZ(J)[x]=xn).
By a filtered ring, we mean a commutative ring R with unit together with
a decreasing sequence of ideals
R = I0 I1 I2 . . .
such that In Im Im+n . A filtered ~-ring is a ~-ring R which is also a
filtered ring in which each ideal In is closed under ~i for i 1. Suppose
that R and S are two filtered ~-rings. Then a filtered ~-ring map
f :R ! S
is a ~-ring map that also preserves the filtration ideals, in the sense that
f(In ) In for all n.
We can now recall the classification of filtered ~-ring structures over the
dual number ring Z(J)[x]=x2. The variable x is given degree 1 here.
EXTENSIONS OF FILTERED ~-RINGS 5
Proposition 2.1 (Corollary 4.1.2 in [5]). Let J be a set of primes.
Then there is a one-to-one correspondence between the set of isomorphismQ
classesQof filtered ~-ring structures over Z(J)[x]=x2 and the set p62JZ(J)x
p2JpZ(J). This correspondence associates (bp) to the filtered ~-ring struc-
ture with Adams operations, _p(x) = bpx.
In particular, Z(J)[x]=x2 admits uncountably many isomorphism classes
of filtered ~-ring structures.
Therefore, we can and will specify a filtered ~-ring structure over
Z(J)[x]=x2 by a sequence of J-local integers {bp}, one for each prime p.
The J-local integer bp is the linear coefficient in _p(x).
Before we go any further, we would like to mention a simple but interesting
fact about the abundance of filtered ~-ring structures over truncated power
series rings. One can think of it as a partial generalization of the above
proposition.
Theorem 2.2. Let x1, . .,.xk be algebraically independent variables of
degree 1 and let J be a set of primes. Let n1, . .,.nk be elements in
{2, 3, . .,.1} with at least one nj < 1. Then the truncated power series
ring R = Z(J)[[x1, . .,.xk]]=(xn11, . .,.xnkk) admits uncountably many pair-
wise non-isomorphic filtered ~-ring structures.
In the statement above, we take x1i to mean 0. So if ni = 1, then no
positive power of xi is equal to 0 in R.
Proof.Let N be the maximal of those nj that are finite. For each prime
p N and each index j for which nj < 1, choose an arbitrary positive
integer bp,j2 pZ. Consider the following power series in R:
(
(1 + xi)bp,i- 1 ifp N and ni< 1,
(2.1) _p(xi) =
(1 + xi)p - 1 otherwise.
Here p runs through the primes and i = 1, 2, . .,.k. Each one of these power
series extends to a filtered ring endomorphism of R.
We first claim that these power series are the Adams operations (applied
to the xi) of a filtered ~-ring structure S on R. Since R is Z-torsionfree, by
Wilkerson's Theorem [4], it suffices to show that
(2.2) _p_q = _q_p
and that
(2.3) _p(r) rp (mod pR)
for all primes p and q and elements r 2 R. Both of these are verified easily
using (2.1). Equation (2.2)is true because it is true when applied to each
6 DONALD YAU
xi and that the xi are algebra generators of R. Equation (2.3)is true, since
it is true for r = xi.
Now suppose that ~Sis another filtered ~-ring structure on R constructed
in the same way with the integers {~bp,j2 pZ}. (Here again p runs through
the primes N and j runs through the indices for which nj < 1.) So in
~S, _p(xi) looks just like it is in (2.1), except that bp,iis replaced by ~bp,i.
Suppose, in addition, that there is a prime q N such that
{bq,j} [ {q} 6= {~bq,j} [ {q}
as sets. We claim that S and ~Sare not isomorphic as filtered ~-rings.
To see this, suppose to the contrary that there exists a filtered ~-ring
isomorphism
oe :S ! ~S.
Let j be one of those indices for which nj is finite. Then
oe(xj) a1x1 + . .+.akxk (mod degree 2)
for some a1, . .,.ak 2 Z(J), not all of which are equal to 0. Equating the
linear coefficients on both sides of the equation
oe_q(xj) = _qoe(xj),
one infers that
bq,j = q or ~bq,i
for some i. In particular, it follows that
{bq,j} [ {q} {~bq,j} [ {q},
and therefore the two sets are equal by symmetry. This is a contradiction.
This finishes the proof of the theorem.
3. The dual number ring
Let R be a filtered ~-ring structure over the dual number ring Z(J)[x]=x2.
A filtered ~-ring extension of R to Z(J)[x]=x3 is a filtered ~-ring structure
S over Z(J)[x]=x3 such that S=(x3) is identical to R as a ~-ring. In other
words, when applied to x, the Adams operations in R are exactly those of
S truncated by x3. Sometimes we will just say that S is an extension of R.
First we would like to have a usable criterion that guarantees that an
extension of R exists.
Theorem 3.1. Let J be a set of primes and let R be a filtered ~-ring struc-
ture over Z(J)[x]=x2 corresponding to the sequence (bp). Then R admits a
filtered ~-ring extension to Z(J)[x]=x3 if, and only if, for all odd primes q,
bq b2q (mod 2`2(b2)Z(J)).
EXTENSIONS OF FILTERED ~-RINGS 7
Proof.We will do the proof for Z, which corresponds to the case when J is
the set of all primes. The case when J is not the set of all primes can be
proved with essentially the same argument.
The existence of an extension of R is equivalent to the existence of integers
cp, p primes, such that the polynomials _p(x) = bpx+cpx2 in Z[x]=x3 satisfy
(3.1) _p(_q(x)) = _q(_p(x))
and
(3.2) _p(x) xp (mod pR)
for all primes p and q. The equation (3.2)is equivalent to c2 1 (mod 2)
when p = 2 and cq 0 (mod q) when q is an odd prime. In Z[x]=x3 one
computes
_p(_q(x)) = bp(bqx + cqx2) + cp(bqx + cqx2)2
= (bpbq)x + (bpcq + b2qcp)x2.
Using symmetry and equating the coefficients of x2, it follows that (3.1)is
equivalent to
(3.3) (b2q- bq)cp = (b2p- bp)cq.
Let's consider first the case b2 = 0. If an extension exists, then since
c2 6= 0, (3.3) implies that bq = 0 for all odd primes q because bq, being
divisible by q, is not equal to 1. This is exactly the condition stated in
the theorem when b2 = 0. Conversely, suppose that the stated condition is
satisfied, meaning that bq = 0 for all odd primes q. Then (3.3)is a vacuous
statement, and therefore one obtains an extension of R by simply choosing
integers cp with c2 1 (mod 2) and cq 0 (mod q) when q is an odd prime.
Now let's consider the case b2 6= 0. Consider (3.3)with p = 2. If an
extension of R exists, then since c2 is not divisible by 2, it follows that
2`2(b2)divides (b2q- bq). Conversely, suppose that (b2q- bq) is divisible by
2`2(b2). We take 8
>:bp(bp_-_1)_ ifp > 2.
2`2(b2)
Then (3.3)is clearly satisfied. Moreover, c2 is an odd integer, since both
b2=2`2(b2)and (b2- 1) are. Likewise, for an odd prime p, (b2p- bp) is divisible
by p (since p divides bp) and therefore so is (b2p-bp)=2`2(b2). This shows that
R admits an extension.
One consequence of this theorem is that there must be an extension when
2 is invertible, since in this case the congruence relation in the theorem is
automatically satisfied.
8 DONALD YAU
Corollary 3.2. If J is a set of primes that does not contain 2, then ev-
ery filtered ~-ring structure over Z(J)[x]=x2 can be extended to one over
Z(J)[x]=x3.
Another consequence of the theorem is that integrally, when b2 = 0, the
moduli set of isomorphism classes of filtered ~-ring extensions is uncount-
able.
Example 3.3 (`2(b2) = -1). In the proof of Theorem 3.1, we saw that
when b2 = 0 (and therefore bq = 0 for all odd primes q as well), an extension
of R to a filtered ~-ring structure over Z[x]=x3 is given by a sequence of inte-
gers (cp) satisfying c2 1 (mod 2) and cq 0 (mod q) if q > 2. If (cp) and
(~cp) correspond to two such extensions, then a filtered ~-ring isomorphism
between them is given by a polynomial
oe(x) = fflx + ax2
for some ffl 2 { 1} and some integer a. It is compatible with the Adams
operations in the sense that oe_p = _poe, which is equivalent to
cp = ffl~cp
for all primes p. Therefore, the set of isomorphism classes of filtered ~-ring
structures over Z[x]=x3 extending R is parametrized by
` Y '
(1 + 2Z) x qZ ={ 1}
q
in which q runs through the odd primes.
In the integral case, since b2 is an even integer, we have either `2(b2) 1
or = -1. In contrast to Example 3.3, there are also many filtered ~-ring
structures over the dual number ring for which the moduli set is empty.
Example 3.4 (`2(b2) 2). Over the dual number ring Z[x]=x2, there exist
uncountably many isomorphism classes of filtered ~-ring structures that do
not admit any filtered ~-ring extension to Z[x]=x3. Indeed, let b2 be any
even integer with `2(b2) 2. For each odd prime q, let lq be any positive
integer and set
bq = 2qlq.
Denote by R the filtered ~-ring structure over Z[x]=x2 corresponding to (bp).
Then 2`2(b2)does not divide bq for any odd prime q. Since (bq- 1) is an odd
integer, it is not divisible by 2`2(b2)either. Therefore, Theorem 3.1 implies
that R does not admit any filtered ~-ring extension to Z[x]=x3.
EXTENSIONS OF FILTERED ~-RINGS 9
Observe that the product b2(b2-1) is always divisible by 2 in Z(J). In par-
ticular, the congruence condition in Theorem 3.1 is automatically satisfied
if `2(b2) = 1, and so the moduli set is non-empty.
Corollary 3.5. Let R be a filtered ~-ring structure over Z(J)[x]=x2 corre-
sponding to a sequence (bp) with `2(b2) = 1. Then R admits a filtered ~-ring
extension to Z(J)[x]=x3.
Of course, this result is only interesting if 2 2 J, since otherwise we
already know from Corollary 3.2 that an extension exists.
We will see below that over Z, R admits a unique extension if b2 = 2.
Integrally the condition `2(b2) = 1 is not enough to guarantee the existence
of a unique extension. However, over the integers localized at 2, `2(b2) = 1
does guarantee that the extension is unique.
In order to further understand uniqueness of extensions and the moduli
set of extensions, we would like to know when two extensions are isomorphic.
Let R be a filtered ~-ring structure over Z(J)[x]=x2 corresponding to a
sequence (bp), where J is a set of primes containing 2. Assume that b2 6= 0.
Suppose that S and ~Sare two filtered ~-ring extensions of R to Z(J)[x]=x3
corresponding to, respectively, (cp) and (~cp). In other words, the Adams
operations of S is given by
_p(x) = bpx + cpx2,
and similarly for ~S. Denote the set of invertible elements in Z(J)by Zx(J).
Theorem 3.6. With the assumptions and notations as above, the following
statements are equivalent:
(1) S and ~Sare isomorphic filtered ~-rings.
(2) There exist u 2 Zx(J)and a 2 Z(J)such that
abp(1 - bp) = u~cp- u2cp
for all primes p.
(3) There exist u 2 Zx(J)and a 2 Z(J)such that
ab2(1 - b2) = u~c2- u2c2.
If one of these equivalent conditions is satisfied, then a filtered ~-ring iso-
morphism
oe :S ! ~S
is given by
oe(x) = ux + ax2,
extended linearly and multiplicatively to all of Z[x]=x2.
10 DONALD YAU
Proof.(1) ) (2). Let oe :S ! ~Sbe a filtered ~-ring isomorphism. Then
(3.4) oe(x) = ux + ax2
for some J-local unit u and J-local integer a. Applying the map oe_p to the
generator x, one obtains
oe_p(x) = bp(ux + ax2) + cp(ux + ax2)2
= ubpx + (abp + u2cp)x2.
Similarly, one has
_poe(x) = ubpx + (ab2p+ u~cp)x2.
Condition (2) now follows by equating the coefficients of x2.
(2) ) (3). This is trivial.
(3) ) (1). Since 2 is not invertible in Z(J), it follows that c2 cannot be
0. Therefore, (3.3)implies that cp = 0 if and only if bp 2 {0, 1}. For those
bp 62 {0, 1}, we infer from (3.3)again that
u~c2- u2c2 u~cp- u2cp
a = __________ = __________.
b2 - b22 bp - b2p
The argument for the proof of (1) ) (2) now implies that the filtered ring
isomorphism oe :S ! ~Sdefined by (3.4)satisfies oe_p = _poe for all primes
p, since they agree on x. Therefore, oe is a filtered ~-ring isomorphism.
We will discuss several consequences of this theorem.
Suppose that we are working over the 2-local integers and assume that
`2(b2) = 1. Then
~c2 c2 (mod 2Z(2)),
since both c2 and ~c2are congruent to 1 modulo 2Z(2). Moreover, (1 - b2) is
a 2-local unit, and therefore so is b2_2(1 - b2). It follows that
~c2-_c2_ ~c2- c2 2
= _______. _________
b2 - b22 2 b2(1 - b2)
lies in Z(2). This shows that the third condition in Theorem 3.6 is satisfied.
Corollary 3.7. Any filtered ~-ring structure over Z(2)[x]=x2 with `2(b2) = 1
admits, up to isomorphism, a unique filtered ~-ring extension to Z(2)[x]=x3.
This corollary is not true over the integers, as the following example shows.
Example 3.8. Consider the filtered ~-ring structure R over Z[x]=x2 with
(
-2x ifp = 2
_p(x) =
0 ifp > 2.
We claim that R admits, up to isomorphism, exactly two filtered ~-ring
extensions to Z[x]=x3.
EXTENSIONS OF FILTERED ~-RINGS 11
To see this, first note that, since bq = 0 for odd primes q, in any extension
corresponding to, say, {cp}, cq must be equal to 0 for all odd primes q.
In other words, in any extension of R, the Adams operation _q for q odd
sends x to 0. Therefore, an extension of R determines and is determined by
a choice of an odd integer c2. Consider the two extensions S and ~Swith
Adams operations
_2(x) = -2x + x2
and
_2(x) = -2x + 3x2,
respectively. We will show that S and ~Sare non-isomorphic and any exten-
sion of R is isomorphic to either S or ~S.
In order to see that they are not isomorphic, observe that b2(1-b2) = -6,
which divides neither 2 nor 4. Therefore, the third condition in Theorem
3.6 implies that S and ~Scannot be isomorphic as filtered ~-rings.
Now let T be any extension of R with
_2(x) = -2x + cx2.
Since c is an odd integer, it follows that
c 1, 3, or5 (mod 6).
In particular, for some choice of ffl 2 { 1}, either (fflc-1) or (c-3) is divis*
*ible
by 6. Therefore, by the third condition in Theorem 3.6, T is isomorphic as
a filtered ~-ring to either S or ~S.
We can achieve uniqueness of an extension over the integers if we insist
that b2 = 2. Indeed, in this case, c2 is an odd integer, and so ~c2- c2 is
divisible by 2. In particular, the third condition in Theorem 3.6 is satisfied
with u = 1 and a = (c2- ~c2)=2. Since Corollary 3.5 tells us that in this case
there is at least one extension, we infer that the moduli set contains exactly
one point.
Corollary 3.9. Any filtered ~-ring structure over Z[x]=x2 with _2(x) = 2x
admits, up to isomorphism, a unique filtered ~-ring extension to Z[x]=x3.
We saw in Example 3.3 that the filtered ~-ring structure over Z[x]=x2
with b2 = 0 admits uncountably many isomorphism classes of filtered ~-ring
extensions to Z[x]=x3. It is natural to ask if there are other filtered ~-ring
structures over the dual number ring that admit uncountably many, or at
least countably infinitely many, extensions to Z[x]=x3 up to isomorphism.
The answer, as we will see shortly, is negative. Employing Theorem 3.6
once again, we will prove a slightly more general result here and obtain the
integral result as an immediate consequence.
12 DONALD YAU
Corollary 3.10. Let J be a set of primes containing 2 and let R be a filtered
~-ring structure over Z(J)[x]=x2 with _2(x) = b2x. Assume that b2 62 {0, 1}
and that all the prime factors of the numerator of b2(b2 - 1), written in
lowest terms, lie in J. Then R admits at most finitely many filtered ~-ring
extensions to Z(J)[x]=x3.
Proof.By the third condition in Theorem 3.6, it suffices to show that in
Z(J), there are only finitely many cosets modulo b2(b2 - 1)Z(J). This will
follow from two elementary facts about the integers.
First, if r is an element in Z(J), write it in lowest terms as r = k=l. Then
there is a canonical isomorphism of groups
~=
Z(J)=rZ(J) -! Z(J)=kZ(J).
Indeed, since l is a J-local unit, it follows that the two subsets, rZ(J)and
kZ(J), of Z(J)are equal. The isomorphism is given by multiplication by l.
Second, if k 6= 0 is an integer, all of whose prime factors lie in J, then the
natural inclusion
i: Z ,! Z(J)
induces an isomorphism of groups
~=
i*: Z=kZ -! Z(J)=kZ(J).
To show that i* is surjective, it is enough to show that for every prime p
that does not lie in J, there exists an integer m such that 1_p-m lies in kZ(J).
Since p and k are relatively prime, there exist integers m and h such that
1 = mp + hk.
This implies that
1_ h
- m = k . __,
p p
which of course lies in kZ(J). The argument for the injectivity of i* is equally
easy.
Now write b2(b2- 1) in lowest terms as k=l. Then there are isomorphisms
Z(J)=b2(b2 - 1)Z(J) ~= Z(J)=kZ(J) ~= Z=kZ.
Therefore, there are only finitely many cosets in Z(J)modulo b2(b2- 1)Z(J).
As mentioned above, this implies that there are at most finitely many filtered
~-ring extensions of R to Z(J)[x]=x3.
Over the integers, b2 is an even integer, so it cannot be equal to 1. In
particular, the hypothesis in the previous corollary is satisfied, provided that
b2 6= 0.
EXTENSIONS OF FILTERED ~-RINGS 13
Corollary 3.11. Let R be a filtered ~-ring structure over Z[x]=x2 with
_2(x) = b2x, b2 6= 0. Then R admits at most finitely many filtered ~-ring
extensions to Z[x]=x3.
One might wonder if there is some uniform upper bound for the number of
elements in the moduli set of isomorphism classes of filtered ~-ring extensions
of Z[x]=x2. The example below illustrates that there is no such upper bound.
This means that by allowing different filtered ~-ring structures on Z[x]=x2,
one can obtain (finitely) arbitrarily large moduli sets of isomorphism classes
of filtered ~-ring extensions to Z[x]=x3.
Example 3.12. Let l be a positive odd integer and let nl be l(2l - 1).
Consider the filtered ~-ring structure R on Z[x]=x2 with Adams operations
(
2lx ifp = 2
_p(x) =
0 ifp > 2.
We claim that R admits exactly (nl+ 1)=2 isomorphism classes of filtered
~-ring extensions to Z[x]=x3.
The argument here is very similar to that of Example 3.8. Consider
the following filtered ~-ring extensions of R: For each odd integer i 2
{1, 3, . .,.nl}, let Si be the filtered ~-ring structure on Z[x]=x3 with Adams
operations (
2lx + ix2 ifp = 2
_p(x) =
0 ifp > 2.
Then each Si is an extension of R. We will show that (i) they are pair-
wise non-isomorphic filtered ~-rings and that (ii) each extension of R is
isomorphic to some Si. To see (i), notice that for two distinct odd integers
i, j 2 {1, 3, . .,.nl}, we have
i 6 j (mod 2l(2l - 1)).
The third condition in Theorem 3.6 now tells us that Si and Sj cannot be
isomorphic as filtered ~-rings.
To prove (ii), one observes that if S is a filtered ~-ring extension of R to
Z[x]=x3, then in S,
_2(x) = 2lx + cx2
for some odd integer c. Since 2l(2l - 1) is an even integer, it follows that
there exists an element i 2 {1, 3, . .,.nl} such that either i or -i is congrue*
*nt
to c modulo 2l(2l - 1). Using Theorem 3.6 once again, we conclude that S
is isomorphic to Si.
14 DONALD YAU
References
[1]M. F. Atiyah and D. O. Tall, Group representations, ~-rings and *
*the J-
homomorphism, Topology 8 (1969), 253-297.
[2]F. J. B. J. Clauwens, Commuting polynomials and ~-ring structures on Z[x], *
*J. Pure
Appl. Algebra 95 (1994) 261-269.
[3]D. Knutson, ~-rings and the representation theory of the symmetric group, L*
*ecture
Notes in Math. 308, Springer-Verlag, Berlin-New York, 1973.
[4]C. Wilkerson, Lambda-rings, binomial domains, and vector bundles over CP(1),
Comm. Algebra 10 (1982), 311-328.
[5]D. Yau, Moduli space of filtered lambda-ring structures over a filtered rin*
*g, Int. J.
Math. Math. Sci., accepted for publication.
[6]D. Yau, On adic genus and lambda-rings, Trans. Amer. Math. Soc., accepted f*
*or
publication.
E-mail address: dyau@math.uiuc.edu
Department of Mathematics, University of Illinois at Urbana-Champaign,
1409 W. Green Street, Urbana, IL 61801 USA