ON λ-RINGS AND TOPOLOGICAL REALIZATION DONALD YAU Received 19 July 2005; Accepted 12 April 2006

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ON λ -RINGS AND TOPOLOGICAL REALIZATION

DONALD YAU

Received 19 July 2005; Accepted 12 April 2006

It is shown that most possibly truncated power series rings admit uncountably many filteredλ-ring structures. The question of how many of these filteredλ-ring structures are topologically realizable by theK-theory of torsion-free spaces is also considered for truncated polynomial rings.

1. Introduction

Aλ-ring is, roughly speaking, a commutative ringRwith unit together with operations λⁱ,i≥0, 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 ringR(G) of a groupGis aλ-ring, whereλⁱis induced by the map that sends a representation to itsith exterior power. Another example of aλ-ring is the complexK-theory of a topological spaceX. Here,λⁱarises from the map that sends a complex vector bundleη overXto theith exterior power ofη. In the algebra side, the universal Witt ring W(R) of a commutative ringRis aλ-ring.

The purpose of this paper is to consider the following two interrelated questions:

(i) classify theλ-ring structures over power series and truncated polynomial rings;

(ii) which ones and how many of theseλ-rings are realizable as (i.e., isomorphic to) theK-theory of a topological space?

The first question is purely algebraic, with no topology involved. One can think of the second question as aK-theoretic analogue of the classical Steenrod question, which asks for a classification of polynomial rings (over the field ofpelements and has an action by the modpSteenrod algebra) that can be realized as the singular modpcohomology of a topological space.

In addition to being aλ-ring, theK-theory of a space is filtered, makingK(X) a fil- teredλ-ring. Precisely, by a filteredλ-ring we mean a filtered ring (R,{R=I⁰⊃I¹⊃ ···}) in whichRis aλ-ring and the filtration idealsIⁿare all closed under theλ-operationsλⁱ (i >0). It is, therefore, more natural for us to consider filteredλ-ring structures over

Hindawi Publishing Corporation

International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 91267, Pages1–21

DOI10.1155/IJMMS/2006/91267

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filtered rings. Moreover, we will restrict to torsion-free spaces, that is, spaces whose inte- gral cohomology is Z-torsion-free. The reason for this is that one has more control over theK-theory of such spaces than non-torsion-free spaces.

A discussion of our main results follows. Proofs are mostly given in later sections. Our first result shows that there is a huge diversity of filteredλ-ring structures over most power series and truncated polynomial rings.

Theorem 1.1. Letx1,...,x_nbe algebraically independent variables with the same (arbitrary but fixed) filtrationd >0, and letr1,...,rn be integers≥2, possibly∞. Then the possibly truncated power series filtered ring Z[[x1,...,xn]]/(x^r1¹,...,xn^rⁿ) admits uncountably many isomorphism classes of filteredλ-ring structures.

Herex_i^∞ is by definition equal to 0. In particular, this theorem covers both finitely generated power series rings and truncated polynomial rings. The caseri= ∞(1≤i≤n) is proved in [11]. In this case, there are uncountably many isomorphism classes that are topologically realizable, namely, by the spaces in the localization genus of (BS³)^×ⁿ. The remaining cases are proved by directly constructing uncountably many filteredλ-rings.

In general, there is no complete classification of all of the isomorphism classes of filtered λ-ring structures. However, such a classification can be obtained for small truncated poly- nomial rings, in which case we can also give some answers to the second question above.

This will be considered below after a brief discussion of the Adams operations.

The Adams operations. The results below are all described in terms of the Adams oper- ations. We will use a result of Wilkerson [9] on recovering theλ-ring structure from the Adams operations. More precisely, Wilkerson’s theorem says that ifRis a Z-torsion-free ring which comes equipped with ring endomorphismsψ^k (k≥1) satisfying the conditions, (1)ψ¹=Id andψ^kψ^l=ψ^kl=ψ^lψ^k, and (2)ψ^p(r)≡r^p(modpR) for each prime pandr∈R, thenRadmits a uniqueλ-ring structure with theψ^kas the Adams operations. The obvious filtered analogue of Wilkerson’s theorem is also true for the filtered rings considered inTheorem 1.1. Therefore, for these filtered rings, in order to describe a filteredλ-ring structure, it suﬃces to describe the Adams operations.

When there are more than oneλ-ring in sight, we will sometimes writeψ_Rⁿto denote the Adams operationψⁿinR.

Truncated polynomial rings. Consider the filtered truncated polynomial rings (Z[x]/(xⁿ),

|x| =d) and (Z[x]/(xⁿ), |x| =d), withxin filtration exactlyd >0 andd>0, respectively. Let Λd denote the set of isomorphism classes of filtered λ-ring structures over (Z[x]/(xⁿ), |x| =d). DefineΛdsimilarly. Then it is obvious thatΛdandΛdare in one- to-one correspondence, and there is no reason to distinguish between them. Indeed, for R∈Λd one can associate to itR∈Λd such thatψ_R^k(x)=ψ_R^k(x) as polynomials for all k, and this construction gives the desired bijection betweenΛdandΛd. Therefore, using the above bijections, we will identify the setsΛd ford=1, 2,..., and writeΛ(Z[x]/(xⁿ)) for the identified set. Each isomorphism class of filteredλ-ring structures on Z[x]/(xⁿ) with|x| ∈ {1, 2,...}is considered an element inΛ(Z[x]/(xⁿ)).

When there is no danger of confusion, we will sometimes not distinguish between a filteredλ-ring structure and its isomorphism class.

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We start with the simplest casen=2, that is, the dual number ring Z[x]/(x²).

Theorem 1.2 ([10, Corollary 4.1.2]). There is a bijection betweenΛ(Z[x]/(x²)) and the set of sequences (b_p) indexed by the primes in which the componentb_pis divisible byp. The filteredλ-ring structure corresponding to (bp) has the Adams operationsψ^p(x)=bpx.

Moreover, such a filteredλ-ring is isomorphic to theK-theory of a torsion-free space if and only if there exists an integerk≥1 such thatb_p=p^kfor allp.

Therefore, in this case, exactly countably infinitely many (among the uncountably many) isomorphism classes are topologically realizable by torsion-free spaces. Indeed, theK-theory of the even-dimensional sphereS²^krealizes the filteredλ-ring withbp=p^k for all p. On the other hand, ifX is a torsion-free space withK(X)=Z[x]/(x²), then ψ^p(x)=p^kxfor allpifxlies in filtration exactly 2k[2, Corollary 5.2].

In what follows we will use the notationθp(n) to denote the largest integer for which p^θ^p⁽ⁿ⁾dividesn, wherepis any prime. By convention we setθp(0)= −∞.

To study the casen=3, we need to consider the following conditions for a sequence (b_p) of integers indexed by the primes.

(A)b2 =0,b_p≡0 (modp) for all primes p, andb_p(b_p−1)≡0 (mod 2^θ²^(b²⁾) for all odd primesp.

(B) Let (bp) be as in (A). Consider a prime p >2 (if any) for which bp =0 and θ_p(b_p)=min{θ_p(b_q(b_q−1)) :b_q =0}.

We will call them conditions (A) and (B), respectively. Notice that in (B),θp(bp)= θp(bp(bp−1)), sincepdoes not divide (bp−1). Moreover, there are at most finitely many such primes, since each suchpdividesb2(b2−1), which is nonzero.

The following result gives a complete classification forΛ(Z[x]/(x³)), the set of isomorphism classes of filteredλ-ring structures over the filtered truncated polynomial ring Z[x]/(x³). As before we will describe aλ-ring in terms of its Adams operations.

Theorem 1.3. LetRbe a filteredλ-ring structure on Z[x]/(x³). ThenRis isomorphic to one of the following filteredλ-rings.

(1)S((c_p))= {ψ^p(x)=c_px²:pprime}withc2≡1 (mod 2) andc_p≡0 (modp) forp >

2. Moreover, any such sequence (cp) gives rise to an element ofΛ(Z[x]/(x³)), and two such filteredλ-rings,S((cp)) andS((c_p)), are isomorphic if and only if (cp)= ±(c_p).

(2)S((bp),k)= {ψ^p(x)=bpx+cpx²:pprime}with (bp) satisfying condition (A) above and thecphaving the following form. Letp1,...,pnbe the list of all odd primes satis- fying condition (B) above. Then there exists an odd integerksuch that, writingGfor gcd(bp(bp−1) : all primesp),

(i) 1≤k≤G/2;

(ii)k≡0 (modp1···p_n);

(iii)c_p=kb_p(b_p−1)/Gfor all primesp.

Moreover, any such pair ((bp),k) gives rise to an element ofΛ(Z[x]/(x³)). Two such filteredλ-rings,S((b_p),k) andS((b_p),k), are isomorphic if and only ifb_p=b_p for all primespandk=k. NoS((c_p)) is isomorphic to anyS((b_p),k).

Notice that in this theorem, there are uncountably many isomorphism classes in each of cases (1) and (2). Also note that if there is no prime psatisfying condition (B), then p1···pnis the empty product (i.e., 1), andk≡0 (modp1···pn) is an empty condition.

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In case (2), if (bp) satisfies condition (A), then it follows that there are exactly G

4p1···pn

=

gcdbp

bp−1: allp 4p1···pn

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isomorphism classes of filteredλ-ring structures over Z[x]/(x³) with the property that ψ^p(x)≡bpx(modx²) for all primesp. Heresdenotes the smallest integer that is greater than or equal tos. Applying formula (1.1) to the cases (bp)=(p^r), wherer∈ {1, 2, 4}, we see that there are a total of 64 elements inΛ(Z[x]/(x³)) satisfyingψ^p(x)=p^rx(modx²) for all primesp. In fact, there is a unique such element whenr=1 (k=1), three such elements whenr=2 (k∈ {1, 3, 5}), and sixty such elements whenr=4 (k∈ {1, 3,..., 119}).

This simple consequence of Theorem 1.3 leads to the following upper bound for the number of isomorphism classes of filteredλ-ring structures over Z[x]/(x³) that are topologically realizable by torsion-free spaces.

Corollary 1.4. Let X be a torsion-free space whoseK-theory filtered ring is Z[x]/(x³).

Then, using the notation ofTheorem 1.3,K(X) is isomorphic as a filteredλ-ring toS((p^r),k) for somer∈ {1, 2, 4}and some k. In particular, at most 64 of the uncountably many iso- morphism classes of filteredλ-ring structures on Z[x]/(x³) can be topologically realized by torsion-free spaces.

Indeed, ifXis a torsion-free space whoseK-theory filtered ring is Z[x]/(x³), then by Adams’ result on Hopf invariant 1 [1], the generatorxmust have filtration exactly 2, 4, or 8. When the filtration ofxis equal to 2r, one hasψ^p(x)≡p^rx(modx²) for all primesp.

Therefore, byTheorem 1.3,K(X) must be isomorphic toS((p^r),k) for somer∈ {1, 2, 4} and somek. The discussion preceding this corollary then implies that there are exactly 64 such isomorphism classes of filteredλ-rings.

It should be remarked that at least 3 of these 64 isomorphism classes are actually realized by spaces, namely, the projective 2-spacesFP², wherePdenotes the complex numbers, the quaternions, or the Cayley octonions. These spaces correspond tor=1, 2, and 4, respectively. Further work remains to be done to determine whether any of the other 61 filteredλ-rings inCorollary 1.4are topologically realizable.

Before moving on to the casen=4, we would like to present another topological ap- plication ofTheorem 1.3, which involves the notion of Mislin genus. LetXbe a nilpotent space of finite type (i.e., its homotopy groups are all finitely generated and π_n(X) is a nilpotentπ1(X)-module for eachn≥2). The Mislin genus ofX, denoted by Genus(X), is the set of homotopy types of nilpotent finite type spacesY such that thep-localizations ofXandY are homotopy equivalent for all primesp. Let H denote the quaternions. It is known that Genus(HP^∞) is an uncountable set [8]. Moreover, these uncountably many homotopically distinct spaces have isomorphicK-theory filtered rings [11] but pairwise nonisomorphicK-theory filteredλ-rings [7]. In other words, K-theory filtered λ-ring classifies the Mislin genus of HP^∞. It is also known that Genus(HP²) has exactly 4 elements [6]. We will now show that the genus of HP²behaves very diﬀerently from that of HP^∞as far asK-theory filteredλ-rings are concerned.

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Corollary 1.5. K-theory filteredλ-ring does not classify the Mislin genus of HP². In other words, there exist homotopically distinct spacesX andY in Genus(HP²) whose K-theory filteredλ-rings are isomorphic.

Indeed, an argument similar to the one in [11] shows that the 4 homotopically distinct spaces in the Mislin genus of HP²all have Z[x]/(x³) as theirK-theory filtered ring, with x in filtration exactly 4. Therefore, in each one of these λ-rings, we haveψ^p(x)≡p²x (modx²) for all primesp. The corollary now follows, since byTheorem 1.3, there are only 3 isomorphism classes of filteredλ-ring structures on Z[x]/(x³) of the form S((p²),k) becausekmust be 1, 3, or 5. It is still an open question as to whetherK-theory filtered λ-ring classifies the genus of HPⁿfor 2< n <∞.

We now move on to the case n=4, that is, the filtered truncated polynomial ring Z[x]/(x⁴). A complete classification theorem along the lines of Theorems1.2 and 1.3 has not yet been achieved forn≥4. However, some sort of classification is possible if one imposes certain conditions on the linear coeﬃcients of the Adams operations that usually appear in theK-theory of spaces.

Theorem 1.6. LetRbe an element ofΛ(Z[x]/(x⁴)).

(1) Ifψ_R^p(x)≡px(modx²) for all primesp, thenRis isomorphic to the filteredλ-ring structure withψ^p(x)=(1 +x)^p−1 for all primesp.

(2) Ifψ_R^p(x)≡p²x(modx²) for all primesp, thenRis isomorphic to one of the following 60 mutually nonisomorphic filteredλ-ring structures on Z[x]/(x⁴):

Sk,d2

=

ψ^p(x)=p²x+kp²p²−1

12 x²+dpx³

, (1.2)

wherek∈ {1, 5},d2∈ {0, 2, 4,..., 58}, and dp= p²p⁴−1d2

60 +k²p²p²−1p²−4

360 (1.3)

for odd primesp.

(3) In general, ifψ_R^p(x)≡bpx(modx²) withbp =0 for all primesp, then there are only finitely many isomorphism classes of filteredλ-ring structuresSover Z[x]/(x⁴) such thatψ_S^p(x)≡bpx(modx²) for all primesp.

(4) If (in the notation of the previous statement)b2=0, thenRis of the formS((c_p), (dp))= {ψ^p(x)=cpx²+dpx³}. Any such collection of polynomials gives rise to a filteredλ-ring structure, provided thatψ^p(x)≡x^p (modp) for all primes p. Two such filteredλ-rings,S((c_p), (d_p)) andS(( ¯c_p), ( ¯d_p)), are isomorphic if and only if (i) (cp)= ±( ¯cp), and (ii) there exists an integerαsuch that ¯dp=dp+ 2cpαfor all primesp.

The first two statements of this theorem immediately leads to the following upper bound for the number of isomorphism classes of filteredλ-ring structures on Z[x]/(x⁴) that can be topologically realized by torsion-free spaces.

Corollary 1.7. Let X be a torsion-free space whoseK-theory filtered ring is Z[x]/(x⁴).

Then, as a filteredλ-ring,K(X) is isomorphic to one of the filteredλ-rings described in parts

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(1) and (2) inTheorem 1.6. In particular, at most 61 of the uncountably many isomorphism classes of filteredλ-ring structures on Z[x]/(x⁴) can be realized as theK-theory of a torsion- free space.

Indeed, ifXis such a space, then the filtration ofxmust be exactly 2 or 4 [4, Corollary 4L.10], and therefore the linear coeﬃcient ofψ^p(x) for any prime pmust be p(if the filtration ofxis 2) orp²(if the filtration ofxis 4). So the result follows immediately from Theorem 1.6.

It should be noted that at least 2 of the 61 isomorphism classes in the first two statements ofTheorem 1.6are topologically realizable, namely, by the projective 3-spacesFP³, whereFis either the complex numbers or the quaternions. TheK-theory of the former space is case (1) inTheorem 1.6, while the latter space hasS(1, 0) as itsK-theory. It is still an open question as to whether any of the other 59 isomorphism classes are topologically realizable.

What happens whenn >4, as far as the two questions stated in the beginning of this introduction is concerned, is only partially understood. We will discuss several conjectures and some partial results in this general setting.

Concerning the problem of topological realizations, we believe that the finiteness phe- nomenon in Corollaries1.4and1.7should not be isolated examples.

Conjecture 1.8. Letnbe any integer≥3. Then, among the uncountably many isomorphism classes of filtered λ-ring structures over the filtered truncated polynomial ring Z[x]/(xⁿ), only finitely many of them can be realized as theK-theory of torsion-free spaces.

As mentioned above, the casesn=3 and 4 are known to be true. Just like the way Corollaries1.4and1.7are proved, one way to approach this conjecture is to considerλ- rings with given linear coeﬃcients in its Adams operations. More precisely, we oﬀer the following conjecture.

Conjecture 1.9. Letnbe any integer≥3 and let{b_p∈pZ :pprime}be nonzero integers.

Then there exist only finitely many isomorphism classes of filteredλ-ring structures on Z[x]/(xⁿ) with the property thatψ^p(x)≡bpx(modx²) for all primesp.

In fact,Conjecture 1.8forn≥4 would follow from the cases, (b_p=p) and (b_p=p²), ofConjecture 1.9. Then=3 case ofConjecture 1.9is contained inTheorem 1.3. More- over, (1.1) gives the exact number of isomorphism classes in terms of thebp. One plausi- ble way to prove this conjecture is to consider extensions ofλ-ring structures one degree at a time.

Conjecture 1.10. Letnbe any integer≥3 and letRandSbe isomorphic filteredλ-ring structures on Z[x]/(xⁿ). Then there exists an isomorphismσ :R−→^∼⁼ S with the following property. IfR is a filteredλ-ring structure on Z[x]/(xⁿ⁺¹) such thatψ_R^p(x)≡ψ_R^p(x) (modxⁿ) for all primesp, then

S^def= ψ_S^p(x)=

σ⁻¹◦ψ_R^p◦σ(x)∈Z[x]

xⁿ⁺¹:pprimes

(1.4) is also a filteredλ-ring structure on Z[x]/(xⁿ⁺¹).

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Hereσ(x) is considered a polynomial in both Z[x]/(xⁿ) and Z[x]/(xⁿ⁺¹), andσ⁻¹(x) is the (compositional) inverse ofσ(x) in Z[x]/(xⁿ⁺¹). In the definition ofψ_S^p(x), the symbol

◦means composition of polynomials.

Observe that for any isomorphismσ:R→S, one has that (ψ_S^p◦ψ_S^q)(x)=(ψ_S^q◦ψ_S^p)(x) for all primes p and q. Thus, to prove Conjecture 1.10, one only needs to show that ψ_S^p(x)≡x^p(modp) for all primesp. Furthermore, ifConjecture 1.10is true, thenσ induces an isomorphismR^∼=S. Denote by ΛR the set of isomorphism classes of filtered λ-ring structuresRover Z[x]/(xⁿ⁺¹) such thatψ_R^p(x)≡ψ_R^p(x) (modxⁿ) for all primesp, and defineΛSsimilarly withSreplacingR. Sinceψ_S^p(x)≡ψ_S^p(x) (modxⁿ) for all primes pand any isomorphismτ:R→S, it follows thatConjecture 1.10implies thatσinduces an embeddingΛRΛS. In particular,Conjecture 1.9would follow fromConjecture 1.10 and the following finiteness result.

Theorem 1.11. Letnbe any integer≥3, and letap,ibe integers forpprimes and 1≤i≤ n−2 witha_p,1 =0 for every p. Then the number of isomorphism classes of filteredλ-ring structuresRon Z[x]/(xⁿ) satisfying

ψ_R^p(x)≡a_p,1x+···+a_p,n−2xⁿ⁻²modxⁿ⁻¹ (1.5) for all primespis at most min{|aⁿ_p⁻,1¹−a_p,1|:pprimes}.

Notice that ifConjecture 1.10is true forn≤Nfor someN, then, usingTheorem 1.11, one infers thatConjecture 1.9is true forn≤N+ 1, which in turn impliesConjecture 1.8 forn≤N+ 1. We summarize this in the following diagram:

(Conjecture 1.10)_n≤N+Theorem 1.11=⇒(Conjecture 1.9)_n≤N+1

=⇒(Conjecture 1.8)_n_≤_N+1. (1.6) In view of these implications, even partial results aboutConjecture 1.10would be of in- terest.

It was mentioned above that in order to proveConjecture 1.10, one only needs to prove the congruence identity,ψ_S^p(x)≡x^p(modp), for all primesp. In fact, this only needs to be proved forp < n, since the following result takes care of the rest.

Theorem 1.12. Let the assumptions and notations be the same as in the statement of Conjecture 1.10. Ifσ:R−→^∼⁼ Sis any filteredλ-ring isomorphism, thenψ_S^p(x)≡x^p (modp) for all primesp≥n.

Using this result, one can show thatConjecture 1.10is true for some small values ofn.

More precisely, we have the following consequence ofTheorem 1.12.

Corollary 1.13. Conjecture 1.10is true forn=3, 4, and 5. Therefore, Conjectures1.8and 1.9are true forn=3, 4, 5, and 6.

This corollary will be proved by directly verifying the congruence identity aboutψ_S^p(x) forp < n. The arguments for the three cases are essentially the same, and it does not seem

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to go through forn=6 (see the discussion after the proof of this corollary). A more sophisticated argument seems to be needed to proveConjecture 1.10in its full generality.

Organization. The rest of this paper is organized as follows. The following section gives a brief account of the basics ofλ-rings and the Adams operations, ending with the proof of Theorem 1.1. Proofs of Theorems1.3and1.6are in the two sections after the following section. The results concerning the three conjectures, namely, Theorems1.11and1.12 andCorollary 1.13, are proved in the final section.

2. Basics ofλ-rings

The reader may refer to the references [3,5] for more in-depth discussion of basic properties ofλ-rings. We should point out that what we call aλ-ring here is referred to as a

“special”λ-ring in [3]. All rings considered in this paper are assumed to be commutative, associative, and have a multiplicative unit.

2.1.λ-rings. By aλ-ring, we mean a commutative ringRthat is equipped with functions

λⁱ:R−→R (i≥0), (2.1)

calledλ-operations. These operations are required to satisfy the following conditions. For any integersi,j≥0 and elementsrandsinR, one has

(i)λ⁰(r)=1;

(ii)λ¹(r)=r;

(iii)λⁱ(1)=0 fori >1;

(iv)λⁱ(r+s)=_i

k=0λ^k(r)λⁱ⁻^k(s);

(v)λⁱ(rs)=Pi(λ¹(r),...,λⁱ(r);λ¹(s),...,λⁱ(s));

(vi)λⁱ(λ^j(r))=Pi,j(λ¹(r),...,λ^{i j}(r)).

TheP_iandP_i,jare certain universal polynomials with integer coeﬃcients, and they are defined using the elementary symmetric polynomials as follows. Given the variablesξ1,...,ξi

andη1,...,ηi, lets1,...,siandσ1,...,σi, respectively, be the elementary symmetric functions of theξ’s and theη’s. Then the polynomialP_iis defined by requiring that the ex- pressionPi(s1,...,si;σ1,...,σi) is the coeﬃcient oftⁱin the finite product

i m,n=1

1 +ξ_mη_nt. (2.2)

The polynomialPi,jis defined by requiring that the expressionPi,j(s1,...,si j) is the coef- ficient oftⁱin the finite product

l1<···<lj

1 +ξ_l1···ξ_ljt. (2.3)

Aλ-ring map is a ring map which commutes with all theλ-operations.

By a filtered ring, we mean a commutative ringRtogether with a decreasing sequence of ideals

R=I⁰⊇I¹⊇I²⊇ ···. (2.4)

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A filtered ring mapf :R→Sis a ring map that preserves the filtrations, that is, f(I_Rⁿ)⊆I_Sⁿ for alln.

A filteredλ-ring is a λ-ringR which is also a filtered ring in which each idealIⁿ is closed underλⁱ fori≥1. Suppose thatRandSare two filteredλ-rings. Then a filtered λ-ring map f :R→Sis aλ-ring map that also preserves the filtration ideals.

2.2. The Adams operations. There are some very useful operations inside aλ-ringR, the so-called Adams operations:

ψⁿ:R−→R (n≥1). (2.5)

They are defined by the Newton formula:

ψⁿ(r)−λ¹(r)ψⁿ⁻¹(r) +···+ (−1)ⁿ⁻¹λⁿ⁻¹(r)ψ¹(r) + (−1)ⁿnλⁿ(r)=0. (2.6) Alternatively, one can also define them using the closed formula:

ψ^k=Q_kλ¹,...,λ^k. (2.7) HereQkis the integral polynomial with the property that

Qk

σ1,...,σk

=x^k1+···+x^k_k, (2.8) where theσ_iare the elementary symmetric polynomials of thex’s. The Adams operations have the following properties.

(i) All theψⁿareλ-ring maps onR, and they preserve the filtration ideals ifRis a filteredλ-ring.

(ii)ψ¹=Id andψ^mψⁿ=ψ^mn=ψⁿψ^m.

(iii)ψ^p(r)≡r^p(modpR) for each primepand elementrinR.

Aλ-ring map f is compatible with the Adams operations, in the sense that f ψⁿ=ψⁿf for alln.

The following simple observation will be used many times later in this paper. Suppose thatRandSareλ-rings withSZ-torsion-free and that f :R→Sis a ring map satisfying f ψ^p=ψ^pf for all primesp. Thenf is aλ-ring map. Indeed, it is clear thatf is compatible with allψⁿby (ii) above. The Newton formula and the Z-torsion-freeness ofSthen imply that f is compatible with theλⁿas well.

As discussed in the introduction, Wilkerson’s theorem [9] says that ifRis a Z-torsion- free ring equipped with ring endomorphismsψⁿ(n≥1) satisfying conditions (ii) and (iii) above, then there exists a uniqueλ-ring structure onRwhose Adams operations are exactly the givenψⁿ. In particular, over the possibly truncated power series filtered ring Z[[x1,...,xn]]/(x^r1¹,...,x^rnⁿ) as in Theorem 1.1, a filteredλ-ring structure is specified by power seriesψ^p(xi) without constant terms,pprimes and 1≤i≤n, such that

ψ^pψ^qx_i=ψ^qψ^px_i, (2.9) ψ^pxi

≡x_i^p(modp) (2.10)

for all suchpandi.

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2.3. Proof ofTheorem 1.1

Proof. Denote byRthe possibly truncated power series filtered ring Z[[x1,...,xn]]/(x^r1¹, ...,x^rnⁿ) as in the statement ofTheorem 1.1.

As was mentioned in the introduction, the caser_i= ∞for alli is proved in [11]. It remains to consider the cases when at least oneriis finite.

Assume that at least oneriis finite. LetNbe the maximum of thoserjthat are finite.

For each primep≥Nand each indexjfor whichr_j<∞, choose an arbitrary positive in- tegerbp,j∈pZ such thatbp,j≥rjandbp,j =p. There are uncountably many such choices, sinceN <∞and there are countably infinitely many choices forbp,jfor eachp≥N. Con- sider the following power series inR:

ψ^pxi

=

⎧⎨

⎩

1 +xi_bp,i

−1 ifp≥N,ri<∞, 1 +xi_p

−1 otherwise. (2.11)

Herepruns through the primes andi=1, 2,...,n. The collection of power series{ψ^p(x_i) : 1≤i≤n}extends uniquely to a filtered ring endomorphismψ^pofR.

We first claim that these endomorphismsψ^p,pprimes, are the Adams operations of a filteredλ-ring structureSonR. SinceRis Z-torsion-free, by Wilkerson’s theorem [9], it suﬃces to show that

ψ^pψ^q=ψ^qψ^p (2.12)

and that

ψ^p(r)≡r^p(modpR) (2.13)

for all primes p andqand elements r∈R. Both of these conditions are verified easily using (2.11). Equation (2.12) is true because it is true when applied to eachxiand that thexiare algebra generators ofR. Equation (2.13) is true, since it is true for eachr=xi.

Now suppose that ¯Sis another filteredλ-ring structure onRconstructed in the same way with the integers{b¯p,j}. (Here again p runs through the primes ≥N and j runs through the indices for whichrj<∞.) So in ¯S,ψ^p(xi) looks just like it is in (2.11), except thatbp,iis replaced by ¯bp,i. Suppose, in addition, that there is a primeq≥Nsuch that

bq,j

=b¯q,j

(2.14)

as sets. We claim thatSand ¯Sare not isomorphic as filteredλ-rings.

To see this, suppose to the contrary that there exists a filteredλ-ring isomorphism

σ:S−→S.¯ (2.15)

Let jbe one of those indices for whichr_jis finite. Then, modulo filtration 2d, one has σx_j≡a1x1+···+a_nx_n (2.16)

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for somea1,...,an∈Z, not all of which are equal to 0. Ifri= ∞, we set ¯bq,i=q. Equating the linear coeﬃcients on both sides of the equation

σψ^qx_j=ψ^qσx_j, (2.17)

one infers that

b_q,j·

a_ix_i=

a_ib¯_q,ix_i. (2.18) Ifa_i =0 (and such ana_imust exist), then

bq,j=b¯q,i (2.19)

for somei. In particular, it follows that{bq,j}is contained in{b¯q,j}. Therefore the two sets are equal by symmetry. This is a contradiction.

This finishes the proof ofTheorem 1.1.

3. Proof ofTheorem 1.3

First we need to consider when a collection of polynomials can be the Adams operations of a filteredλ-ring structure on Z[x]/(x³). We will continue to describeλ-rings in terms of their Adams operations.

Lemma 3.1. A collection of polynomials,{ψ^p(x)=bpx+cpx²:pprime}, in Z[x]/(x³) ex- tends to (the Adams operations of) a filteredλ-ring structure if and only if the following three statements are satisfied:

(1)bp≡0 (modp) for all primesp;

(2)c2≡1 (mod 2) andcp≡0 (modp) for all primesp >2;

(3) (b²_q−b_q)c_p=(b²_p−b_p)c_qfor all primespandq.

Now suppose that these conditions are satisfied. Ifb2 =0, then (b²_p−bp)≡0 (mod 2^θ²⁽^b²⁾) for all odd primesp. Ifb2=0, thenb_p=0 for all odd primesp.

Proof. The polynomialsψ^p(x) in the statement above extend to a filteredλ-ring struc- ture on Z[x]/(x³) if and only if (2.9) and (2.10) are satisfied. Expandingψ^p(ψ^q(x)), one obtains

ψ^pψ^q(x)=bp

bqx+cqx²+cp

bqx+cqx²²

=

b_pb_qx+b_pc_q+b²_qc_px². (3.1) Using symmetry and equating the coeﬃcients ofx², it follows that (2.9) in this case is equivalent to

b²_q−bq cp=

b²_p−bp

cq. (3.2)

It is clear that (2.10) is equivalent to conditions (2) and (3) together, since x^p=0 for p >2.

Now assume that statements (1), (2), and (3) are satisfied. Ifb2 =0, then the right- hand side of (3.2) whenp=2 is congruent to 0 modulo 2^θ²⁽^b²⁾, and therefore so is the

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left-hand side. The assertion now follows, sincec2is odd. Ifb2=0, then the right-hand side of (3.2) when p=2 is equal to 0, and sob²_q=bq for all odd primesq, sincec2 =0.

Butbq =1, sobq=0.

Next we need to know when two filteredλ-ring structures over Z[x]/(x³) are isomorphic.

Lemma 3.2. LetS= {ψ^p(x)=b_px+c_px²}and ¯S= {ψ^p(x)=b¯_px+ ¯c_px²}(where p runs through the primes) be two filteredλ-ring structures over Z[x]/(x³). ThenSand ¯Sare iso- morphic filteredλ-rings if and only if the following two conditions are satisfied simultane- ously:

(1)bp=b¯pfor all primesp;

(2)(a) ifb2=b¯2=0, then there existsu∈ {±1}such thatc_p=uc¯_pfor all primesp;

(b) ifb2=b¯2 =0, then there existu∈ {±1}anda∈Z such that ab2

b2−1=c2−u¯c2. (3.3)

Proof. Suppose thatSand ¯Sare isomorphic, and letσ:S→S¯be a filteredλ-ring isomorphism. Then

σ(x)=ux+ax² (3.4)

for someu∈ {±1}and integera. Applying the mapσψ^p,pis any prime, to the generator x, one obtains

σψ^p(x)=ubpx+abp+cp

x². (3.5)

Similarly, one has

ψ^pσ(x)=ub¯_px+ab¯²_p+uc¯_px². (3.6) Recall from the previous lemma thatb2=0 impliesbp=0 for all odd primes p. There- fore, the “only if ” part now follows by equating the coeﬃcients in the equationσψ^p(x)= ψ^pσ(x).

Conversely, suppose that conditions (1) and (2)(a) in the statement of the lemma hold.

Then clearly the mapσ:S→S¯given on the generator byσ(x)=uxis the desired isomorphism.

Now suppose that conditions (1) and (2)(b) hold. The polynomialσ(x)=ux+ax² extends uniquely to a filtered ring automorphism on Z[x]/x³. The calculation in the first paragraph of this proof shows that ifb_p=0 for a certain primep, thenσψ^p(x)=ψ^pσ(x).

Ifbp =0, then (3.2) in the proof ofLemma 3.1implies that a=c2−uc¯2

b²2−b2 =c_p−uc¯_p

b²_p−b_p . (3.7)

Therefore, the argument in the first paragraph once again shows thatσψ^p(x)=ψ^pσ(x),

andσ:S→S¯is the desired isomorphism.

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Proof ofTheorem 1.3. It follows immediately from Lemmas3.1 and3.2that, in the notation of the statement ofTheorem 1.3, theS((cp)) are all filtered λ-ring structures on Z[x]/(x³) and that two of them are isomorphic if and only if the stated conditions hold.

Similar remarks apply to theS((bp),k). Also noS((cp)) is isomorphic to anS((bp),k). It remains only to show that any filteredλ-ring structureRon Z[x]/(x³) is isomorphic to one of them.

Writeψ^p(x)=bpx+cpx² for the Adams operations inR. Ifb2=0, then so isbp for each odd prime p. Then (cp) satisfies condition (2) inLemma 3.1, andRis isomorphic (in fact, equal) toS((c_p)).

Suppose thatb2 =0. Then byLemma 3.1the sequence (bp) satisfies condition (A).

First consider the case whenbp=0 for all odd primes p. In this case, condition (3) in Lemma 3.1implies that c_p=0 for all odd primes p. Let rdenote the remainder ofc2

modulob2(b2−1). Sinceris also an odd integer, there exists a unique odd integerkin the range 1≤k≤b2(b2−1)/2 that is congruent (modb2(b2−1)) to eitherror−r. By Lemma 3.2,Ris isomorphic toS((b_p),k).

Finally, consider the case whenb2 =0 and there is at least one odd prime pfor which bp =0. If

c2=qb2

b2−1+r (3.8)

for some integersqandrwith 0≤r < b2(b2−1), thenrmust be an odd integer, sinceb2

is even andc2is odd. Define

c¯2=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

r if 1≤r≤b2

b2−1

2 ,

b2

b2−1−r ifr >b2

b2−1

2 ,

c¯q=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

c_q−qb_qb_q−1 if 1≤r≤b2

b2−1

2 ,

(1 +q)b_qb_q−1−c_q ifr >b2

b2−1 2

(3.9)

forq >2. The three conditions (1)–(3) inLemma 3.1are all easily verified for the poly- nomialsS= {ψ^p(x)=b_px+ ¯c_px²}, and soSis a filtered λ-ring structure on Z[x]/(x³).

Observe that 1≤c¯2≤b2(b2−1)/2. Moreover, byLemma 3.2,Sis isomorphic toR. Equa- tion (3.2) implies that, ifpis a prime for whichbp =0, then

c¯2≡0

mod b2

b2−1 gcdb2

b2−1,b_pb_p−1

. (3.10)

Therefore, we have c¯2≡0

mod lcm

b2

b2−1 gcdb2

b2−1,bp

bp−1: all primesp

. (3.11)