ALGEBRA OVER THE STEENROD ALGEBRA,
LAMBDA-RING, AND KUHN'S REALIZATION
CONJECTURE
DONALD YAU
Abstract. In this paper we study the relationships between operations
in K-theory and ordinary mod p cohomology. In particular, conditions
are given under which the mod p associated graded ring of a filtered
~-ring is an unstable algebra over the Steenrod algebra. This result
partially extends to the algebraic setting a topological result of Atiyah
about operations on K-theory and mod p cohomology for torsionfree
spaces. It is also shown that any polynomial algebra that is an algebra
over the Steenrod algebra can be realized as the mod p associated graded
of a filtered ~-ring. Another observation is that Atiyah's result gives *
*rise
to a K-theoretic analogue of Kuhn's Realization Conjecture concerning
the size of spaces in cohomology.
1. Introduction
A filtered ring is a ring R which comes equipped with a multiplicative
decreasing filtration {In } of ideals: R = I0 I1 I2 . ...A ~-ring is a ring
R equipped with functions ~i:R ! R (i 0), called ~-operations, satisfying
certain properties similar to those satisfied by exterior power operations.
(What we refer to as a ~-ring is what Atiyah and Tall call a "special" ~-
ring.) A filtered ~-ring is a filtered ring R which is also a ~-ring for which
the filtration ideals are all closed under the operations ~i for i > 0. Adams
operations in a ~-ring are denoted, as usual, by _n.
Let X be a torsionfree space; that is, a space which has no torsion in
integral cohomology. Then its integral cohomology ring H*(X; Z) can be
identified with the associated graded ring of its K-theory:
(1) Gr *K(X) ~=H*(X; Z)
The filtration on K(X) arises from a skeletal filtration on X: Letting Xn
denote the nth skeleton of X, the ith filtration ideal of K(X) is the kernel
Ii(X) = ker(K(X) ! K(Xi-1)) of the restriction map. With this filtration
the K-theory of a space is a filtered ring. Actually, the relationship between
K-theory and ordinary cohomology goes deeper. A well-known result of
Atiyah [1, Proposition 5.6 and Theorem 6.5] says that for a torsionfree space
X, its K-theory with ~-operations determine its mod p cohomology (for any
prime p) as an algebra over the mod p Steenrod algebra A:
____________
Date: August 24, 2002.
1
2 DONALD YAU
Theorem 1.1 (Atiyah). Let p be a prime and let X be a torsionfree space.
If ff 2 K(X) lies in filtration 2q, then there exist elements ffi 2 K(X)
(i = 0, 1, . .,.q) in filtration 2q + 2i(p - 1) such that
Xq
(2) _p(ff) = pq-iffi where ffq = ffp ifq > 0.
i=0
This yields well-defined functions
Pip:(Gr *K(X)) Fp ! (Gr *+2i(p-1)K(X)) Fp,
sending ~ff(the image of ff in the mod p associated graded) to ~ffi. With the
identification of eq. (1)mod p, these functions Pipare precisely the Steenrod
operations (with Pi2= Sq2iwhen p = 2).
Here and for the rest of this note, tensor product is taken over the ring
of integers Z, unless otherwise stated. The field of p elements is denoted by
Fp.
This result of Atiyah is a very effective tool when studying K-theory.
Here are a few examples. (1) Using the fact that Adams operations and
Chern character determine each other for a torsionfree space, Atiyah [1,
x7] used Theorem 1.1 to reprove a result of Adams about p-integrality of
Chern character for torsionfree spaces. (2) Theorem 1.1 is a key ingredient
in the proof by Notbohm and Smith [7] of the theorem that K-theory ~-
ring applied to the classifying space detects those fake Lie groups of type
G (a fixed connected compact Lie group) admitting a maximal torus. (3)
The author used Theorem 1.1 in [12] to relate Adams operations and Rector
invariants [8], classification invariants for the genus of BSU(2), and then to
give a classification of spaces in the genus of BSU(2) that are detectable by
CP1 .
It is customary to think of the mod p cohomology of a space as an object
in either the category of A-modules or the category of A-algebras, where A
denotes the mod p Steenrod algebra . We take the latter because we want to
consider the ring structure as well. Similarly, since every K-theory operation
is in a unique way a polynomial in the ~-operations, one can think of the
K-theory of a space as an object in the category of filtered ~-rings. From
this perspective, it is natural to ask if Atiyah's Theorem 1.1 is actually a
purely algebraic fact about filtered ~-rings and A-algebras. In other words,
we ask the question:
__*
When is the mod p associated graded, Rp := Gr *R Z Fp, of
a filtered ~-ring R an algebra over the mod p Steenrod algebra
with Steenrod operations induced by Adams operations on R?
The first few results of this paper give an answer to this question.
Before we describe our results, let us first discuss another reason as to
why it is interesting to consider this question. The author is interested in
obtaining K-theoretic refinements of results about topological realizations of
A-algebras. As mentioned above, for nice spaces X, its mod p cohomology
ALGEBRA OVER THE STEENROD ALGEBRA 3
can be obtained from its K-theory through the process of taking mod p
associated graded. It seems plausible and natural to "split" the realization
problems of A-algebras into two separate problems:
(1) An algebraic realization problem about existence and uniqueness of
filtered ~-rings that give rise to a given unstable A-algebra via the
process of mod p associated graded.
(2) A topological realization problem about existence and uniqueness of
spaces with K-theory a given filtered ~-ring.
The following diagram is a schematic presentation of this program.
(Spaces)N
K(-)nnnnn NNH*(-;Fp)NNN
nnn NNNN
vvnnnn N''
(Filtered ~-rings)__Gr*(-)_F_______//_(A-algebras)
p
The first step in such a program concerning the algebraic realization prob-
lem is to find out how one can pass from filtered ~-rings to A-algebras, in
a way compatible_with Atiyah's Theorem. Of course, the mod p associated
graded R *pis an_Fp-algebra. So the main questions are (i) how one obtains
operations on R *pfrom the operations on R, and (ii) whether such opera-
tions (if exist) behave like Steenrod operations. In Atiyah's Theorem 1.1,
the Steenrod operations Piparise from the Adams operation _p via eq. (2).
We encapsulate this in the following definition.
Definition 1.2 (Atiyah formula). Let r be an element in a filtered ~-ring R
in filtration 2q and let p be a prime . We say that r satisfies Atiyah formula
at p if there exist elements ri2 R (i = 0, 1, . .,.q) in filtrations 2q +2i(p-1)
such that
Xq
(3) _p(r) = pq-iri where rq = rp ifq > 0.
i=0
We say that R satisfies Atiyah formula at p if every element in R satisfies
Atiyah formula at p.
We call eq. (3) an Atiyah formula for r, usually leaving the prime p
implicit.
For example, Atiyah's Theorem 1.1 tells us that if X is a torsionfree space
with I2n+1(X) = I2n+2(X) for any n (for instance, if X has cells only in
even dimensions), then every element in K(X) satisfies Atiyah formula at
any prime.
We now make some remarks about this definition. Note that an element
r 2 R can be considered to lie in different filtrations, since if r lies in
filtration n 1 then it also lies in filtration n - 1. Thus, the condition that
R satisfy Atiyah formula at p means that every element in R, regardless of
what filtration (say, 2q) it is considered to be in, has an Atiyah formula at
p for that filtration. When we say that "r satisfies Atiyah formula at p",
4 DONALD YAU
what we mean is that whenever q 0 and r lies in filtration 2q, r has an
Atiyah formula at p when it is considered to be in filtration 2q. Also, if r
lies in filtration 2q with q > 0, then an Atiyah formula for r also yields an
Atiyah formula when r is considered to be in filtration 2(q - 1), since we can
rewrite eq. (3)as
_p(r) = pq-1(pr0) + . .+.p2(prq-3) + p(prq-2 + rq-1) + rp.
In particular, when R is Hausdorff in the topology induced by the filtration
(that is, \n 1 In = (0)), R satisfies Atiyah formula at p provided that every
non-zero element r in R has an Atiyah formula at p when it is considered to
be in its äm ximal" filtration.
As an algebraic analogue of Atiyah's Theorem 1.1, our first result shows
that Atiyah formula implies the existence of operations on the mod p asso-
ciated graded algebra. We will use the terminology evenly filtered ~-ring to
denote a filtered ~-ring R = (R, {In }) for which I2n+1 = I2n+2 for every n.
Theorem 1.3. Let p be any prime and let R = (R, {In }) be an evenly
filtered ~-ring which satisfies Atiyah formula at p. Then there exist well-
defined operations
__* __*+2i(p-1)
(4) Pip:Rp ! Rp (i 0)
on the mod p associated graded of R defined as follows. Given any element
__r2 __R2q
__ p lift it to any element r 2 R in filtration exactlyP2q whose image
in R 2qpis __r, write down any Atiyah formula _p(r) = qi=0pq-iri for r (in
filtration 2q) as in eq. (3), and then take
(__ __2(q+i(p-1))
ri 2 Rp if0 i q
Pip(__r) =
0 ifi > q.
Proofs will be given in x4
__2q
Since given any element __r2 R p there always exists a lift to an element
r 2 I2q\I2q+2, the point of the above theorem is that, despite the ambiguity
in the different choices of lifts r and the possibly different ways of expressi*
*ng
_p(r) (for each lift r) in Atiyah formula, the elements __riare well-defined in
the mod p associated graded.
To answer the question posed above, we need to know whether the op-
erations in eq. (4) behave like Steenrod operations. The next result shows
that, as a formal consequence of Atiyah formula, they at least satisfy the
Cartan formula, the additivity, öt p square", and ü nstable" conditions.
Theorem 1.4. Let the notations and hypotheses be the same as in Theorem
1.3. Then the operations Pipin eq. (4)satisfy the following properties.
(1) Each Pipis additive.
__2q __2pq
(2) If q > 0 then Pqp:Rp ! Rp is the pth power map.
__2q __
(3) If __r2 Rp then Pipr= 0 for every_i > q.
P __ *
* _
(4) If __rand _sare two elements in R*p, then Pip(__r_s) = l+k=i(Plpr) (Pk*
*ps).
ALGEBRA OVER THE STEENROD ALGEBRA 5
Two additional_properties are still needed in order that the mod p asso-
ciated graded R *pbe an A-algebra, namely, P0p= Idand the Adem relation.
One might first suspect that these two properties are also consequences of
Atiyah formula. This, however, is not true. Examples can be constructed
easily to show that these two properties are not necessarily satisfied even in
the presence of Atiyah formula.
Example 1.5 (Atiyah formula does not imply P0p= Id). For any prime p,
there exists an evenly filtered ~-ring R which satisfies Atiyah formula at p
but whose operation P0pis not equal to Id. The underlying ring of R is the
ring Z["] ("2 = 0) of dual numbers with the ä- dic filtration, where " lies in
filtration precisely 4.
Example 1.6 (Atiyah formula does not imply the Adem relation). For any
prime p > 2, there exists an evenly filtered ~-ring R which satisfies Atiyah
formula at p but whose operations Pipdo not satisfy the Adem relation. The
underlying ring of R is the filtered polynomial ring Z(p)[x] with the x-adic
filtration, where x lies in filtration precisely 2(p - 1) and Z(p)is the ring of
integers localized at p.
Examples 1.5 and 1.6 tell us that in order to make the mod p associated_
graded of R into an A-algebra, we should add extra assumptions so that R *p
satisfy the Adem relation and P0p= Id. Since the Adem relation is about
composition of certain Steenrod operations, we need to assume something
about _p applied to elements appearing in Atiyah formula. The following
result should now come as no surprise. (We will use the notation [m=n] to
denote the integer part of m=n.)
Theorem 1.7. Let p be a prime and let the notations and hypotheses_be
the same as in Theorem 1.3. Then the mod p associated graded R *pwith the
operations Pip(Sq 2i= Pi2if p = 2) in eq. (4)is an unstable algebra over the
mod p Steenrod algebra, provided that the following two additional conditions
hold:
(1) P0p= Id.
(2) For each element r 2 R in filtration 2q, there exist Atiyah formulas
Xq q+i(p-1)X
(5) _p(r) = pq-iri _p(ri) = pq+i(p-1)-jri,j
i=0 j=0
such_that whenever i, j > 0 and i < pj, the following equality holds
in R 2(q+(i+j)(p-1))p:
hii
_p ` '
__r X i+t (p - 1)(j - t) - 1__
j,i= (-1) rt,i+j-t if p > 2
(6) t=0 i - pt
[i_2]` '
__r X 2j - 2t - 1 __
j,i= rt,i+j-t if p = 2.
t=0 2i - 4t
6 DONALD YAU
It is worth pointing out that, in view of Examples 1.5 and 1.6, Theorem
1.7 is a best possible result in the sense that the conclusion will no longer
hold if either one of the two stated conditions is removed.
Having given conditions under which the mod p associated graded of a
filtered ~-ring is an A-algebra, we now turn to the realization question:
Which A-algebras can be realized as the mod p associated
graded of a filtered ~-ring via Atiyah formula?
While we do not know whether every A-algebra can be realized, we do
have the following result showing that polynomial algebras are realizable.
Theorem 1.8. Let p be any prime and let H* be an unstable A-algebra of
the form
H* = Fp [{xff}ff2S]
where S is an indexing set and the xffare algebraically independent variables
in even, positive dimensions. Then there exists an evenly filtered ~-ring R
satisfying Atiyah formula at p such that the following statements hold.
(1) The underlying filtered ring of R is the power series ring
Z(p)[[{Xff}ff2S]] where the Xffare algebraically independent variables
and_in which Xfflies in filtration equal to exactly the degree of xff.
(2) R_*pwith the operations Pipin eq. (4)is an unstable A-algebra.
(3) R *pis isomorphic to H* as unstable A-algebras.
We remark that in this result, the p-adic integers could also have been
used in place of Z(p)as the coefficients of R.
Our last result is a K-theoretic analogue of a conjecture of N. Kuhn. In [5]
Kuhn made an interesting conjecture, the Realization Conjecture, about the
size of the mod p cohomology of topological spaces: The mod p cohomology
of a space should be either finite as a set or infinitely generated as a module
over the mod p Steenrod algebra. Kuhn verified this conjecture in the case
when the Bockstein is zero in sufficiently high degrees [5, Theorem 0.1].
Using reduction steps in Kuhn's paper [5], the Realization Conjecture was
proved recently by L. Schwartz [9].
One naturally wonders if there are analogous results concerning the size
of spaces in other cohomology theories. Using Atiyah's Theorem 1.1 and
Kuhn's original result, we will see that there is such an analogue for K-
theory. To generalize the result of Kuhn and Schwartz, we first introduce a
K-theoretic notion which corresponds to a module over the Steenrod algebra.
Filtered _p-module. Let p be a prime. We define a filtered _p-module to be
an ordered pair ((M, {In}), _p) (or simply (M, _p) or even just M) consisting
of a filtered abelian group (M, {In}) and a distinguished endomorphism _p.
For example, the K-theory of a space X is a filtered _p-module with the
usual filtration and the Adams operation _p; this is the only way in which
we make K(X) into a filtered _p-module. We say that an element ff in a
ALGEBRA OVER THE STEENROD ALGEBRA 7
filtered _p-module M in filtration 2q satisfies Atiyah formula if there exists
elements ffi (i = 0, . .,.q) in filtration 2q + 2i(p - 1) such that
Xq
_p(ff) = pq-iffi.
i=0
Such an expression is referred to as an Atiyah formula for ff. The filtered
_p-module is said to satisfy Atiyah formula if every element in it satisfies
Atiyah formula. For example, the filtered _p-module K(X) satisfies Atiyah
formula (at least when X is torsionfree).
Now we can ask what a K-theoretic analogue of a finitely generated A-
module is. The A-linear multiples of an element in an A-module are the
finite sums of iterated Steenrod operations acting on that element. Since
Atiyah's result above tells us that the Steenrod operations on H*(X; Fp)
come from Atiyah formula decomposition (eq. (2)) of _p, a K-theoretic
analogue of A-linear multiples should involve iterated applications of _p on
Atiyah formula. We arrive at the following K-theoretic finiteness condition,
which corresponds to H*(X; Fp) being a finitely generated A-module.
_p-finitely generated. Let (M, _p) be a filtered _p-module. We say that
it is _p-finitely generated by the elements m1, . .,.mn in M if the following
condition is true: There exist Atiyah formulas
Xq1
_pm1 = pq1-j1m(1,j1)
j1=0
..
.
qnX
_pmn = pqn-j1m(n,j1)
j1=0
q1+j1(p-1)X
(7) _pm(1,j1) = pq1+j1(p-1)-j2m(1,j1,j2)(0 j1 q1)
j2=0
..
.
qn+j1(p-1)X
_pm(n,j1) = pqn+j1(p-1)-j2m(n,j1,j2)(0 j1 qn)
j2=0
..
.
etc. etc. such that M is generated as an abelian group by the elements
m(i,j1,...,jr)(1 i n, r 0). The filtered _p-module M is said to be
_p-finitely generated if there exists a finite set of elements m1, . .,.mn in M
with the above property.
Having a K-theoretic analogue of a finitely generated A-module, we are
now ready for the promised generalization of Kuhn's Realization Conjecture.
Theorem 1.9. Let X be a torsionfree space of finite type whose integral
cohomology is concentrated in even dimensions. If there exists a prime p
8 DONALD YAU
for which the filtered _p-module K(X) is _p-finitely generated, then the
underlying abelian group of K(X) must be finitely generated.
As in the case of modules over the Steenrod algebra, purely algebraic
counterexamples are easily constructed. For example, let p be an arbitrary
prime, and consider the abelian group A = 1n=0Z with xpn in filtration
2pn and the endomorphism _p sending xpn to xpn+1. It is readily checked
that this filtered _p-module is _p-finitely generated by {x}, and yet it is not
finitely generated as an abelian group. Thus, Theorem 1.9 says that many
algebraically allowed filtered _p-modules cannot be realized as the K-theory
of spaces.
This finishes the presentation of the results of this paper. The rest of this
paper is organized as follows. In x2 some basics of ~-rings and algebras over
the Steenrod algebra are recalled. Section 3 contains an observation about
Atiyah formula for a sum of elements. This will be used in x4, in which
proofs of the theorems and examples above are given in the order in which
they were presented.
2. ~-rings and algebras over the Steenrod algebra
The purpose of this section is to recall the definitions and basic properties
of a ~-ring and of an (unstable) algebra over the Steenrod algebra. All rings
considered in this paper are commutative with unit. The reader can consult
[2, 4] for more information on ~-rings.
2.1. ~-rings and Adams operations. A ~-ring is a commutative ring R
with unit equipped with functions ~i: R ! R (i 0) such that for any
elements r and s in R, the following conditions hold:
o ~0(r) = 1.
o ~1(r) = r.
o ~n(1) = 0 forPevery n > 1.
o ~n(r + s) = ni=0~i(r)~n-i(s).
o ~n(rs) = Pn(~1(r), . .,.~n(r); ~1(s), . .,.~n(s)).
o ~m (~n(r)) = Pm,n(~1(r), . .,.~mn (r)).
The last three statements are required to hold for every n and m 0. Here
the Pn and Pm,n are certain universal polynomials with integer coefficients
(see Atiyah and Tall [2] or Knutson [4] for detail). The functions ~i are
called ~-operations. A ~-ring map f :R ! S between two ~-rings is a
ring homomorphism f :R ! S which is compatible with the ~-operations,
f~i= ~if for each i.
A filtered ~-ring is a filtered ring R = (R, {In }) for which the filtration
ideals In are all closed under the operations ~i for i > 0. A filtered ~-ring
map is a ~-ring map which is also a filtered ring map (that is, it preserves
the filtrations).
ALGEBRA OVER THE STEENROD ALGEBRA 9
Given a ~-ring R, there are Adams operations _n :R ! R (n 1) defined
by the Newton formula
(8) _n(r) - ~1(r)_n-1 (r) + . .+.(-1)n-1~n-1(r)_1(r) + (-1)nn~n(r) = 0.
The Adams operations satisfy the following properties:
o _1 = Id.
o All the _n are ~-ring maps.
o _m _n = _mn for any n, m 1.
o _p(r) rp (mod pR) for every prime p and every element r in R.
It follows from the Newton formula eq. (8) that any ~-ring map also com-
mutes with the Adams operations. If R is a filtered ~-ring, then the Adams
operations are filtered ~-ring maps. Also note that any Adams operation
can be computed from the operations _p, p prime.
For a ~-ring R, one might wonder whether or not the Adams operations
actually determine the ~-ring structure. According to a result of Wilkerson
[11, Prop. 1.2] this is, in fact, the case provided the ring R is torsionfree a*
*s a
Z-module. We now recall this result, since we will use it several times later
on in this paper.
Theorem 2.1 (Wilkerson). Let R be a torsionfree ring (as a Z-module)
equipped with ring homomorphisms _n :R ! R for n 1 satisfying the
properties:
(1) _1 = Id and _m _n = _mn for every m and n.
(2) _p(r) rp (mod pR) for every prime p and every element r in R.
Then there is a unique ~-ring structure over R with the given _n as Adams
operations.
2.2. Unstable algebras over the Steenrod algebra. Here we briefly
recall the definition of an (unstable) algebra over the Steenrod algebra. The
reader can consult the books [3, 10] for more information on this subject.
The field of p elements is denoted by Fp.
Let p be a prime. Denote by A the mod p Steenrod algebra. It is the
graded associative Fp-algebra generated by the Bockstein fi in degree 1 and
the Steenrod operations Piin degree 2i(p-1) (resp. Sqiwhen p = 2) (i 0).
They are subject to the conditions P0 = Id (resp. Sq0 = Id when p = 2),
fi2 = 0 and the Adem relation. A module over A is assumed to be Z-graded.
An A-module M is called an A-algebra if both of the following conditions
hold:
o The Steenrod operations satisfy the Cartan formula on products,
X
Pn(mm0) = Pi(m)Pj(m0)
i+j=n
for any n 0 and elements m, m02 M (similarly when p = 2).
o Pi(m) = mp (resp. Sqi(m) = m2 when p = 2) if 2i (resp. i when
p = 2) is equal to |m|, the degree of m.
10 DONALD YAU
An unstable A-algebra is an A-algebra M which satisfies the unstable
condition: Pi(m) = 0 if 2i > |m| (resp. Sqi(m) = 0 if i > |m| when p = 2).
3.An observation about Atiyah formula
The purpose of this section is to record an observation about Atiyah
formula on sums of elements. This will be used a few times in the next
section.
Proposition 3.1. Let p be any prime and let R = (R, {In }) be an evenly
filtered ~-ring. Suppose that r and s are elements in R with r 2 I2n \ I2n+2
and s 2 I2m \ I2m+2 for some integers n < m. If both r and s satisfy Atiyah
formula at p, then so does r + s.
Proof.Write t = r +s and note that t lies in I2n \I2n+2. The proof is easy if
n = 0, soPwe assume from now onPthat n > 0. Write down Atiyah formulas
_p(r) = ni=0pn-iri, _p(s) = mi=0pm-i si for r and s, respectively. Define
the following elements
s0= pm-n s0 + . . .+ psm-n-1 + sm-n
(r + s)p - rp - sp
c = ___________________
p
8
>>>r0 + s0 ifi = 0
><
ri+ si if1 i n - 2
ti=
>>>rn-1 + sn-1 - c ifi = n - 1
>:
r + s ifi = n.
Then we have that
(9)
Xn Xm
_p(t) = pn-iri + pm-i si
i=0 i=0
n-2X
= pn(r0 + s0) + pn-i(ri+ si) + p(rn-1 + sn-1 - c) + tp
i=1
Xn
= pn-iti.
i=0
It is now easy to check that eq. (9) is an Atiyah formula for t = r + s (in
filtration 2n). Therefore, by the remarks after Definition 1.2, eq. (9) also
yields an Atiyah formula for t when it is considered to be in any filtration
2n.
This finishes the proof of the proposition.
The previous proposition admits the following variant involving not the
sum of two elements but an infinite sum.
ALGEBRA OVER THE STEENROD ALGEBRA 11
Proposition 3.2. Let p and R be as in Proposition 3.1. Assume in addition
that R is complete Hausdorff in the topology induced by the given filtration
on R. Suppose that {ri} is a sequence of elements in R with ri2 I2ni\I2ni+2
and n1 0. In this case we have that rq = rp = r0q, and in particular __rq= __*
*r0q.
Now if m is an integer, 0 m q - 1, then in the quotient R=I2q+2m(p-1)+2
one computes
_p(r)_ m
= rm + prm-1 + . .+.p r0
pq-m
= r0m+ pr0m-1+ . .+.pm r00
Thus, the images of rm and r0m in the associated graded Gr 2q+2m(p-1)R
can only differ by an element that is divisible by p. Therefore, they must
coincide once we reduce modulo p. This proves the first assertion.
12 DONALD YAU
For the second assertion, the case q = 0 is again easy, so we assume that
q > 0. First write down Atiyah formulas for h and f:
q-1X q+n-1X
_p(h) = pq-ihi + hp, _p(f) = pq+n-ifi + fp
i=0 i=0
with hi in filtration 2(q + i(p - 1)) and fi in filtration 2(q + n + i(p - 1)).
Define elements si in R as follows:
8
>>>: j=q-1
sp if i = q
Also, define an element fl in R by the equation
rp + php + fp = sp + pfl = (r + ph + f)p + pfl.
Now one calculates
_p(s) = _p(r) + p_p(h) + _p(f)
q-1X q-1X q+n-1X
= pq-iri + p pq-ihi + pq+n-ifi + rp + php + fp
i=0 i=0 i=0
Xq
= pq-isi.
i=0
It is not hard to see that_the_elements si satisfy the required properties. For
instance, _sq-1= __rq-1in R 2q+2(q-1)(p-1)pbecause fl lies in filtration at lea*
*st
2pq, fj (for j q - 1) lies in filtration at least 2(q + n + (q - 1)(p - 1)), *
*and
phq-1 is p-divisible. This proves the second assertion.
This finishes the proof of Theorem 1.3.
Proof of Theorem 1.4.The first three statements are immediate from the
definitions of the Ppieq. (4)and that of Atiyah formula eq. (3).
Now we consider the last statement. Let __rand _sbe in degrees 2m and 2n,
respectively. Without loss of generality we may assume that m n. The
case when both m and n are equal to 0 is immediate. We will denote by r
and s (arbitrary) lifts of __rand _s, respectively, to R in filtrations precise*
*ly
2m and 2n.
Let us now consider the case when m = 0 and n > 0. We write down
Atiyah formulas:
_p(r) = r0 = rp + pr0
Xn
_p(s) = pns0 + . . .+ psn-1 + sp = pn-isi.
i=0
ALGEBRA OVER THE STEENROD ALGEBRA 13
Here r0 and r0 are some elements in R. Therefore, using the fact that the
Adams operation _p is multiplicative, we have that
_p(rs) = r0(pns0 + . . .+ psn-1 + sp)
= pnr0s0 + . . .+ pr0sn-1 + r0sp
n-2X
= pn-ir0si + p(r0sn-1 + spr0) + (rs)p
i=0
Since Ppi(__r) = __r0if i = 0 and is 0 if i > 0, and since spr0 lies in filtrat*
*ion at
least 2np, the last statement of the theorem when m = 0 and n > 0 follows.
Finally, we consider the case when both m and n are positive. The Atiyah
formula for s is as above, but that for r looks like
mX
_p(r) = pm r0 + . . .+ prm-1 + rp = pm-i ri.
i=0
Therefore, we have that
m+nX X
_p(rs) = pm+n-i ci where ci = rlsk.
i=0 l+k=i
The case when m, n > 0 for the last statement of the theorem follows.
This finishes the proof of the last statement of the theorem.
Proof of Example 1.5.Fix a prime p and let R be the filtered ring Z["]
("2 = 0) of dual numbers with the ä- dic filtration, where " lies in filtration
precisely 4. Let k be any integer and define the filtered ring endomorphisms
_q (q prime) on R by specifying
(
0 ifq 6= p
_q(") =
p2k" ifq = p.
Then it follows from Wilkerson's Theorem 2.1 that there is a unique filtered
~-ring structure on R with these Adams operations. Using Proposition 3.1 it
is easy to check that R satisfies Atiyah formula at the prime p with "0 = k",
and so Pp0(_") = k_"which is equal to _"if and only if k 1 (mod p). In other
words, Pp0= Idif and only if k 1 (mod p).
__*
It is worth pointing out that the Adem relation is satisfied in R p, since
only Pp0can be non-zero.
Proof of Example 1.6.Fix a prime p > 2 and let Z(p)denote the ring of
integers localized at p. Let R be the filtered polynomial ring Z(p)[x] with
the x-adic filtration, where x lies in filtration precisely 2(p - 1). Define
filtered ring endomorphisms _q (q prime) on R by specifying
(
0 ifq 6= p
_q(x) = P p
-pp-2x2 + i=1pp-ixi ifq = p.
Then they satisfy the following properties:
14 DONALD YAU
o _u_v = _v_u for any primes u and v.
o If q 6= p then q is invertible in R, and so it is trivially true that
_q(f) fq (mod qR) for any element f 2 R. It is also clear that
_p(f) fp (mod pR), since it holds for f = x and every element
ff 2 Z(p)satisfies ffp ff (mod pZ(p)).
Therefore, by Wilkerson's Theorem 2.1, there is a unique filtered ~-ring
structure on R with these _q as Adams operations. Moreover, it follows from
the argument in the next-to-the-last paragraph of the proof of Theorem 1.4
(Cartan formula) and Proposition 3.1 that R satisfies Atiyah formula at p.
__2(p-1)
Now the operation Ppi(0 i p - 1) takes __x2 Rp to
(__
xi+1 ifi 6= 1
Ppi(__x) =
0 ifi = 1.
In particular, we have that
Pp1Pp1(__x) = Pp1(0) = 0,
which is not equal to 2Pp2(__x) = 2__x3, since p > 2. In other words, Pp1Pp16=
2Pp2.
In summary, R is a filtered_~-ring that satisfies Atiyah formula at p, but
the operations Ppion R*pdo not satisfy the Adem relation (Pp1Pp1= 2Pp2).
Proof of Theorem 1.7.Since we are dealing with a fixed prime, we will omit
the subscript p.
In view of Theorems 1.3 and 1.4 and the hypothesis P 0= Id, we only
need demonstrate the Adem relation. With the notations as in eq. (5), we
know that P iP j(__r) = __rj,ifor any i and j. Therefore, the Adem relation is
satisfied by the hypothesis eq. (6).
Proof of Theorem 1.8.We will give the proof only when S is a finite set; the
proof of the general case requires only a slight modification of the argument
below but is more tedious.
So we have H* = Fp[x1, . .,.xn] for some n 1. Let Xi (1 i n) be
independent variables and define the evenly filtered power series ring
R = Z(p)[[X1, . .,.Xn]]
with Xi in filtration exactly the degree of xi, say, 2di. Then it is clear that
there is an isomorphism of graded Fp-algebras
__* __ __ ~= *
(10) oe :Rp = Fp[X 1, . .,.X n] -! H
__ __ __2di
with oe sending X ito xi, where X iis the image of Xi in R p .
To define Adams operations on R, we first look at the Steenrod operations
applied to the xi. For every i (1 i n) and j (1 j di- 1), there
ALGEBRA OVER THE STEENROD ALGEBRA 15
exists an n-variable polynomial fi,j= fi,j(y1, . .,.yn) with coefficients in Fp
such that
P j(xi) = fi,j(x1, . .,.xn).
Moreover, if yk has weight 2dk, then fi,jis homogeneous of weight 2di +
2j(p - 1). We can lift fi,jto a polynomial over Z by replacing each non-
zero coefficient in it by an integral lift; denote such a lift by Fi,j. Then Fi*
*,j
is also a homogeneous polynomial over Z (and hence over Z(p)) of weight
2di+ 2j(p - 1).
We now define Adams operations on R. Define filtered ring endomor-
phisms _q (q prime) on R by specifying
(
0 ifq 6= p
_q(Xi) =
pdiXi,0+ . .+.pXi,di-1+ Xi,di ifq = p,
in which the Xi,jare defined as
8
>>>:
Xpi ifj = di.
These filtered ring maps have the following properties:
o _u_v = _v_u for any primes u and v.
o _q(r) rq (mod qR) for any prime q and element r in R. This is
clear if q 6= p, since in this case q is invertible in R. This is true
for q = p because it holds for r = Xi and every element ff in Z(p)
satisfies ffp ff (mod pZ(p)).
Since R is Z-torsionfree, Wilkerson's Theorem 2.1 now implies that there is
a unique filtered ~-ring structure on R with these _q as Adams operations.
Since Xi,jlies in filtration 2di+2j(p-1), each Xisatisfies Atiyah formula
at p. Combined with the fact that any non-zero element ff in Fp satisfies
ffp = ff, the argument for the last statement of Theorem 1.4 now shows that
any monomial in R satisfies Atiyah formula at p. It then follows immediately
from Proposition 3.1 that R satisfies Atiyah formula_at_p as well. Therefore,
by Theorem 1.3 there are operations P i= Ppi:R*p! R*+2i(p-1)p(with P2i=
Sq2i). We will omit the subscript p. These operations have the following
properties:
__ __
o P 0= Id, since P 0(X i) = X ifor each i.
o For each i and j with 1 i n, 1 j di- 1, one has that
__ __ __ __2di+2j(p-1)
P j(X i) = fi,j(X 1, . .,.X n) 2 Rp .
We will make use of the following algebra to show that oe is actually an
A-algebra isomorphism. Let
__ 1 2
A p = Fp[Q , Q , . .].
16 DONALD YAU
be the graded Fp-algebra freely generated_by_the Qk (k 1) in degree
2k(p - 1). Then H* is naturally a graded A p-algebra with
Qkxi def=P k(xi).
__* __
Similarly, one can regard R pas a graded A p-algebra with Qk acting as P k.
__
We now claim that oe as in eq. (10) is an A p-algebra isomorphism. To
prove this claim it suffices to show that
oeQk = Qkoe
for every integer k, for which_it is enough to demonstrate that the equality
holds when applied to each X i. But we have that
__ __ __
oeQk(X i) = oe(fi,k(X 1, . .,.X n))
__ __
= fi,k(oeX 1, . .,.oeX n)
= fi,k(x1, . .,.xn)
= Qkxi
__
= Qkoe(X i).
__
So oe is an A p-algebra isomorphism._
Now consider the ideal J in A pgenerated by the elements
X[_ip] ` '
(p - 1)(j - t) - 1 i+j-t t
QiQj - (-1)i+t Q Q ifp > 2,
t=0 i - pt
X[_i2]` '
2j - 2t - 1 i+j-t t
QiQj - Q Q ifp = 2
t=0 2i - 4t
in which i, j > 0_and_i < pj. Since H* is actually an A-algebra, when
considered_as an A p-algebra_it is annihilated by J. Therefore, since oe is an
Ap-algebra isomorphism, R *pis also annihilated by J.
__*
But we already know that the operations P ion Rp satisfy the properties in
Theorem 1.4 with P0_= Id. Together with the previous paragraph, therefore,
we conclude that R*pwith the operations P iis, in fact, an unstable A-algebra
and that oe is an isomorphism of unstable A-algebras.
This finishes the proof of the theorem.
Proof of Theorem 1.9.We begin with three reductions.
Reduction step 1. To show that K(X) is a finitely generated abelian
group, it suffices to show that its associated graded Gr* K(X) = H*(X; Z)
is such. To see this, first note that K(X) with the topology induced by
the filtration {In = ker(K(X) ! K(Xn-1))} (Xn-1 the n - 1 skeleton of
X) is Hausdorff; that is, the intersection \nIn is 0. Indeed, an element ff
in \nIn is represented by a map ff: X ! BU whose restriction to each
skeleton Xn-1 is nullhomotopic, i.e. ff is a phantom map from X to BU.
But since Hn (X; Q) and ßn+1BU Q cannot be simultaneously nonzero for
ALGEBRA OVER THE STEENROD ALGEBRA 17
any integer n, there can be no essential phantom maps from X to BU (see
[6]). Therefore, ff must be 0 and so \nIn = 0; that is, K(X) is Hausdorff.
Now if Gr* K(X) = H*(X; Z) is a finitely generated abelian group, then
there exists an integer N > 0 such that H*(X; Z) = H