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