ON ~-RING STRUCTURES OVER Z[[x]]



                            DONALD YAU



       Abstract. It is shown that the ~-ring structure over the power series
       ring Z[[x]] given by the K-theory of CP1 is uniquely determined by the
       following condition: _p(x)   px (mod x2) for each prime p, where _p
       is the Adams operation. Applications to algebraic topology and formal
       group laws are given.


                 1. Statement of the main result


  The  main  corollary  in  [11]  establishes  the  existence  of  uncountably
many mutually non-isomorphic ~-ring structures over the power series ring
Z[[x1, . .,.xn]] for any positive integer n. Although this is a purely algebraic
statement, its proof is not. In fact, it is proved by combining three algebraic
topological theorems from [8, 9, 11] about spaces in the localization genus of
classifying spaces of compact Lie groups and their K-theories. In particular,
the proof does not construct the ~-ring structures; it merely shows their ex-
istence. This prompts the following question: For a fixed n, how can these
uncountably many non-isomorphic ~-ring structures over Z[[x1, . .,.xn]] be
constructed algebraically?

  In this note we contribute to this question in the case n = 1 by ruling out
certain potential candidates for such ~-ring structures. To be more precise,
recall that a ~-ring R has certain mutually-commuting endomorphisms _n
(n   1), called Adams operations, such that _1 is the identity. Moreover,
for any prime p and any element r in R, the congruence relation

                       _p(r)     rp  (mod  pR)

holds.  The commutativity of the Adams operations implies that they are
determined uniquely by the _p for primes p.

  Specializing to the case where R is a ~-ring structure over the power series
ring Z[[x]], the abovePcongruence relation implies that the coefficients of the
power series _p(x) =   1i=0ap,xiiare all divisible by p, except for ap,p. In
algebraic topology, when a space has a power series ring as its K-theory, the
generators xiare usually in strictly positive filtrations and, since the Adams
operations preserve filtrations, the constant term ap, 0is 0. A case in point
is the infinite-dimensional complex projective space CP1 , which has Z[[x]]
as its K-theory ring with x in filtration exactly 2. Its Adams operations are
given by
               _p(x)  =  1 - (1 - x)p    px    (mod  x2)

for any prime p.
____________
   Date: June 8, 2004.
                                  1
2                            DONALD YAU


  It is, therefore, very tempting to ask the following variant of the question
stated at the end of the first paragraph above:

       Given the linear polynomials fp(x)  =  px,  where p runs
       through the primes, is it possible to extend them to a ~-
       ring structure over Z[[x]] with Adams operations satisfying
       _p(x)   fp(x) (mod  x2)?

There is, of course, at least one such ~-ring, which is given by the K-theory
of CP1 .  The main result of this note is that, up to ~-ring isomorphism,
this is the only one. We now record it formally in the following theorem.


Main Theorem. Let R be a ~-ring whose underlying ring is the power se-
ries ring Z[[x]]. Suppose that the Adams operations of R satisfy the property
that _p(x)   px (mod  x2) for any prime p.  Then R is isomorphic as a
~-ring to the K-theory of CP1 .


  It should be pointed out that the assumption on the linear coefficients of
the Adams operations are necessary in order that the conclusion be true.
In fact, the uncountably many non-isomorphic ~-ring structures over Z[[x]]
discovered in [11] all have the property that _p(x)   p2x (mod  x2). They
arise as the K-theories of the spaces in the localization genus of the classi-
fying space BSU (2); see [10] for a classification of these spaces. Therefore,
these ~-rings cannot be distinguished or identified simply by considering the
linear terms in the Adams operations, in stark contrast to the situation in
the theorem above.

  We now apply the Main Theorem to obtain a result about the K-theory
of spaces. Let X be a space. Say that it is even and torsionfree if its integral
cohomology is concentrated in even dimensions and is Z-torsionfree.


Corollary A. Let X be a space that is even and torsionfree. Suppose that
the K-theory ring of X is the power series ring Z[[x]] with x in filtration
exactly 2. Then K(X) is isomorphic as a ~-ring to K(CP1 ).


  Indeed, a classical result of Adams [1] implies that such a space has the
property that _p(x) = px+ terms of higher filtrations. The Corollary then
follows directly from the Main Theorem.

  Next we would like to discuss another consequence of the Main Theorem
that has to do with (one-dimensional, commutative) formal group laws. The
reader is referred to [4] for background information about formal group laws.

  Recall that the multiplicative group law is defined by

                Gm  =  1 - (1 - x)(1 - y) = x + y - xy,

and it is the formal group law associated to complex K-theory.  For an
integer n, the n-series of Gm  is the polynomial

                      [n]Gm (x)  =  1 - (1 - x)n,
                   ON ~-RING STRUCTURES OVER Z[[x]]                   3


which coincides with the Adams operation _n(x) in the K-theory of CP 1 .
Since we would like to construct ~-ring structures over Z[[x]] algebraically,
it is natural to ask the following question:

       Are there any formal group laws over the integers, other
       than Gm , whose n-series, n   1, form the Adams operations
       _n(x) of a ~-ring structure over Z[[x]]?

Since there are uncountably many formal group laws over the integers, a
positive answer to this question seems plausible.  However, this is not the
case, meaning that Gm  is the only formal group law whose n-series are the
Adams operations of a ~-ring structure over Z[[x]].


Corollary B. Let F be a (one-dimensional, commutative) formal group law
over Z. The following statements are equivalent:

   (1) F is strictly isomorphic to Gm .
   (2) There exists a ~-ring structure R over Z[[x]] with the property that,
       for n   1, _n(x) = [n]F (x), the n-series of F .
   (3) There exists a ~-ring structure R over Z[[x]] with the property that,
       for n   1, _n(x) = [n]F (x). Moreover, R is isomorphic, as a ~-ring,
       to K(CP  1).


  Indeed, it is clear that (1) implies (3) and that (3) implies (2).  The
n-series of a formal group law must satisfy

                       [n](x)     nx  (mod  x2).

Therefore, if R is as in (2), then it follows from the Main Theorem that
it is isomorphic as a ~-ring to K(CP  1). Moreover, the proof of the Main
Theorem shows that, when R is as in (2), log-1F(logGm (x)) is an integral
power series, where logF(x) is the logarithm of F . This implies that F and
Gm  are strictly isomorphic.

  This finishes the presentation of the results in this note.  In the next
section, we review some basics about ~-rings and Adams operations.  The
proof of the Main Theorem is given in the last section.


                             2. ~-rings


  The purpose of this section is to review some basics about ~-rings, which
is essential to understanding this article.  Some standard references for ~-
rings are the article by Atiyah and Tall [2] and the lecture notes by Knutson
[5].

  A ~-ring is a commutative ring R with a multiplicative identity equipped
with functions
                        ~i:R  !  R  (i   0),

called ~-operations, which 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
4                            DONALD YAU


     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 integral polynomials defined as follows.   Consider
the variables ,1, . .,.,i and j1, . .,.ji.  Denote by s1, . .,.si and oe1, . .,*
 *.oei,
respectively, the elementary symmetric functions of the ,'s and the j's.
The  polynomial  Pi is  defined  by  the  requirement  that  the  expression
Pi(s1, . .,.si; oe1, . .,.oei) be the coefficient of ti in the finite product

                           Yi
                                (1 + ,m jnt).
                          m,n=1

Similarly, if s1, . .,.sijare the elementary symmetric functions of ,1, . .,.,i*
 *j,
then the polynomial Pi,jis defined by the requirement that the expression
Pi,j(s1, . .,.sij) be the coefficient of ti in the finite product
                          Y
                               (1 + ,l1. .,.ljt).
                        l1<...<lj

  Given two ~-rings R and S, a ~-ring map f :R ! S is a ring map from R
to S which is compatible with the ~-operations in the sense that f~i= ~if
(i   0).

  A ~-ring has the so-called Adams operations

                         _n :R ! R  (n   1)

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.

These operations satisfy 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.


                  3.Proof of the Main Theorem


  Now we begin the proof of the Main Theorem in the previous section.
We need a couple of lemmas, whose proofs will be given at the end of this
section.


Lemma 3.1. Let f(x) and g(x) be power series over a field F .  Suppose
that f(x)   ffx   g(x) (mod  x2) for some element ff 2 F which is neither
0 nor a root of unity. Then for every element c 2 F , there exists a unique
power series h(x) over F satisfying the following conditions:

     o h(0) = 0,
     o h0(0) = c, where "0" means formal derivative, and
     o h(g(x)) = f(h(x)).
                   ON ~-RING STRUCTURES OVER Z[[x]]                   5


  Let R be a ~-ring as in the statement of the Main Theorem with Adams
operations satisfying the property that _q(x)   qx (mod  x2) for each prime
q.  Let p be an arbitraryPbut fixed prime.  Using Lemma 3.1 we obtain a
power series f(x) =   1i=1aixi with rational coefficients such that a1 = 1
and that

(3.1.1)                  pf(x)  =  f(_p(x)).

Note that f is dependent on the prime p (and R), and we will write fp for
f if it is necessary to mention p.

  In what follows, Z(p)denotes the ring of p-local integers, that is, the
subring of the rationals consisting of the elements m_nwith m and n integers

such that p does not divide n. For an integer r, the notation _1_prZ(p)denotes

the set of rational numbers of the form ff_prwhere ff 2 Z(p).

  We will need the following lemma about the p-integrality of the coefficients
of f. For an integer n, we define `p(n) to be the largest integer for which

p`p(n)divides n, so that n is the product of p`p(n)over its prime factors. In
particular, if p does not divide n, then `p(n) = 0.

                       P 1
Lemma 3.2. With f(x) =   i=1 aixi defined as above, weihave that forj

each n   1, an 2 __1__p`p(n)Z(p)and, if p divides n,  an - 1_pan=p2 Z(p).

                                            P 1
  Now we define another power series g(x) =   i=1cixi over the rationals
by the equation

                                        1    p
(3.2.1)                g(x)  def=f(x) - __f(x ).
                                        p

It is easy to see that the coefficients of g are given by
                     (
                      an           ifp does not dividen,
              cn  =        1
                      an - _pan=p  ifp dividesn.

In particular, it follows that c1 = 1 and that, by Lemma 3.2, cn 2 Z(p)for
n   2. So g(x) is actually a power series over the p-local integers. We will
write gp for g if it is necessary to mention the dependency of g on the prime
p.

  Let S be an arbitrary ~-ring structure over Z[[x]] with Adams operations
~_qsatisfying the property that ~_q(x)   qx (mod  x2) for each prime q. To

prove the Main Theorem, we need to show that R and S are isomorphic as
~-rings. There are similarly defined power series fp and gp associated with
S; we denote them by f~pand ~gpto avoid confusion with those associated
with R. By the Functional Equation Lemma from the theory of formal group
laws [4, 2.2(ii)], it follows that the power series 'p(x) defined by


                        'p(x)  def=f-1p(f~p(x)),

where f-1p(x) denotes the compositional inverse of fp(x), actually has co-
efficients in Z(p).  (In the notation of [4, 2.2], we are using the Functional
Equation Lemma in the case in which A = Z(p), K = Q, oe = Id, q = p,

s1 = p-1, sn = 0 for n   2, and A = pZ(p).)
6                            DONALD YAU


  Observe that
                        _p(x)  =  f-1p(pfp(x)).

This implies that

(3.2.2)  _p('p(x))  =  f-1p(pfp('p(x)))  =  f-1p(pf~p(x))  =  'p(_~p(x)).

Now if q is an arbitrary prime, then both _q('p(x)) and 'p(_~q(x)) are
power series over the rationals with constant terms 0 and linear terms qx.
Moreover, if h denotes either one of these two power series, then using eq.
(3.2.2)and the commutativity of Adams operations we infer that

(3.2.3)                 _p(h(x))  =  h(_~p(x)).

But since h(0) = 0 and h0(0) = q, eq. (3.2.3)and the uniqueness statement
of Lemma 3.1 implies that

(3.2.4)                _q('p(x))  =  'p(_~q(x)).

Combining eq. (3.2.2)(for the prime q) and eq. (3.2.4), using the uniqueness
part of Lemma 3.1 once again, we observe that

                           'p(x)  =  'q(x)

for arbitrary primes p and q. Denote the common value of the various 'p(x)
by '(x).  Since 'p(x) has coefficients in Z(p), we conclude that '(x) is a
power series with integer coefficients.

  The integral power series '(x) extends to a ring automorphism ' of Z[[x]]
with the property that

                        _p('(x))  =  '(_~p(x))

for each prime p. It follows that

                             _r'  =  '_~r

as endomorphisms for each positive integer r. But since every ~-operation
~r is a polynomial in the Adams operations _k (k   r), it follows that

                             ~r'  =  '~~r

as functions on Z[[x]] for each r   1, where ~~rdenotes the rth ~-operation
for S. Therefore, ' gives a ~-ring isomorphism from R to S, thereby proving
the Main Theorem.

  It remains to prove Lemma 3.1 and Lemma 3.2.

                                 P 1                   P 1
Proof of Lemma 3.1. Write f(x) =   i=1Paixi and g(x) =   i=1 bixi with
a1 = b1 = ff.  We will define h(x) =   1i=1cixi inductively, starting with
c1 = ff.

  Let n > 1 be an integer. Suppose that the coefficients cihave been defined
for i < n such that

(3.2.5)            h(g(x))     f(h(x))    (mod  xn)
                   ON ~-RING STRUCTURES OVER Z[[x]]                   7


and that the ci are unique. If cn exists, then we must have
            Xn   ` 1X     'j     1X   `Xn      ' i
(3.2.6)        cj     bixi          ai     cjxj       (mod  xn+1).
            j=1    i=1           i=1    j=1

Equating the coefficients of xn on both sides of eq. (3.2.6), we see that

(3.2.7)                  cnffn + t  =  cnff + s,

where t and s are rational polynomial expressions in, respectively, b1, . .,.bn,
c1, . .,.cn-1 and a2, . .,.an, c1, . .,.cn-1.  Solving for cn in eq. (3.2.7), we
conclude that
                                    s - t
(3.2.8)                     cn  =  _______,
                                   ffn - ff

which is a well-defined element in the field F , since n > 1 and ff is neither
0 nor a root of unity.  Therefore, if we define cn by the expression in eq.
(3.2.8), then eq. (3.2.5)holds modulo xn+1 instead of xn, and cn is unique
with respect to this property.

  The lemma can now be finished by an induction.

Remark 3.2.9. Lemma 3.1 is a generalization of a similar result of Lubin
[6, Proposition 1.1].  The proof above is rather standard and follows the
same pattern as Lubin's.



                                                   P 1
Proof of Lemma 3.2. In this proof we write _p(x) =   i=1bixi 2 Z[[x]]. In
particular, we have that b1 = p and that for j > 1,
                           (
                            0  (mod  p) if j 6= p,
                     bj
                            1  (mod  p) if j = p.


  We now prove the Lemma by induction on n, with the initial case a1 = 1
being trivial.

  Let, then, n > 1 be an integer and suppose that the Lemma has been
proved for integers less than n. Observe that by equating the coefficients of
xn on both sides of eq. (3.1.1), we obtain the equation

                                          n-1X
(3.2.10)              pan  =  bn + pnan +    ßnl,
                                          l=2

where ßnlhas the form

(3.2.11)        ßnl =  allpl-1bn-l+1 + alhnl(b1, . .,.bn-l).

Here the term hnlis given by
                                          X
(3.2.12)         hnl(b1, . .,.bn-l)  =           bi1. .b.il,
                                        i1+...+il=n

where in the sum the ij satisfy 1   ij   n - l.  We are writing ßnlin this
form because it is convenient for the arguments below.

  Notice that the first term in ßnl, allpl-1bn-l+1 (2   l   n-1), is always in
pZ(p)regardless of whether p divides n or not. In fact, if p does not divide
8                            DONALD YAU


l, then by induction hypothesis al 2 Z(p)and l - 1   1. If p does divide l,

then by induction hypothesis al2 __1__p`p(l)Z(p), but lpl-1 is divisible by pl *
 *and

l > `p(l).

  To prove the p-integrality statement about an, let's first consider the
situation when p does not divide n. For 2   l   n - 1, we can then rewrite
hnlas

                      l-2X    X        `l'
(3.2.13)       hnl =                       bspbi1. .b.il-s,
                      s=0i1+...+il-s=n-pss

in which none of the ij is equal to p. In particular, hnlis divisible by p`p(l)*
 *+1,

since it is divisible by lpl-s unless s = 0, in which case it is divisible by p*
 *l.
Together with the induction hypothesis on al, this implies that the second
term in ßnl, alhnl, is always in pZ(p).  Combining this with the previous

paragraph and eq. (3.2.10), we infer that anp(1 - pn-1) is in pZ(p), and
hence an is in Z(p).

  Consider now the case when p divides n.  The special case n = p needs
to be treated separately, but the argument is very similar to the one below,
and so we will omit it. We are now assuming, in addition, that n > p. Just
like we did above, we need to analyze hnl. If l 6= n=p, then hnlcan be written
in the form (3.2.13), and an argument similar to the one in the previous
paragraph shows that alhnllies in pZ(p).

  Consider the case l = n_p. We need the following result about hnl.


Lemma 3.3. hnn_p  1 (mod  p`p(n)Z)


Proof.Similar to the case when p does not divide n, we can rewrite hnn=pas

                      n_  l-2X     X        `n_'
(3.3.1)      hnn=p=  bpp+                    p   bspbi1. .b.in_p-s,
                          s=0i1+...+in_p-s=n-pss


in which none of the ij is equal to p. Since bp   1 (mod  p), we have that

                       `p(n_p)
                     bpp        1  (mod  p`p(n)Z),

and thus                n_

                       bpp    1  (mod  p`p(n)Z).

Therefore, it suffices to show that each term inside the double summation
is divisible by p`p(n).
                                                          n_
  To see this, consider first when s 6= 0. In this case,  ps bspbi1. .b.in_p-sis
                   n_-s
divisible by n_p. pp   , and hence by n since s   n_p- 2.  In particular, it is
                                                                 n_
divisible by p`p(n). If s = 0, then bi1. .b.in_pis divisible by pp , and hence *
 *by

p`p(n)as well.

  This finishes the proof of the Lemma.
                   ON ~-RING STRUCTURES OVER Z[[x]]                   9


  We can now finish the proof of Lemma 3.2.  As discussed above, the
number                           n  n_            X

                 ff  def=bn + an_p_pp -1bn-n_+1 +    ßnl
                                 p         p      l6=n_
                                                     p

is in pZ(p). Combining this with eq. (3.2.10)and the induction hypothesis
on an_p, we can infer that


                     anp(1 - pn-1)  =  ff + an_phnn_p


lies in ___1___p`p(n)-1Z(p), and hence an 2 __1__p`p(n)Z(p).


  Finally, both anp`p(n)and an_pp`p(n)-1lie in Z(p)(the latter by induction

hypothesis). We compute

anp`p(n)- an_pp`p(n)-1   anp`p(n)(1 - pn-1) - p`p(n)-1an_phnn_p  (mod  p`p(n)Z(*
 *p))


                     =  p`p(n)-1ff

                        0    (mod  p`p(n)Z(p)).

The first congruence is a consequence of Lemma 3.3.  This finishes the in-
duction step, and the proof of the Lemma is complete.


  The proof of the main theorem is now complete.


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[11]D. Yau, On adic genus and lambda-rings, Trans. Amer. Math. Soc., accepted f*
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  E-mail address: dyau@math.uiuc.edu


  Department of Mathematics, University of Illinois at Urbana-Champaign,
1409 W. Green Street, Urbana, IL 61801 USA