Operations in connective K-theory
aa r X i v : . [ m a t h . K T ] J un OPERATIONS IN CONNECTIVE K -THEORY ALEXANDER MERKURJEV AND ALEXANDER VISHIK
Abstract.
In this article we classify additive operations in connective K-theory withvarious torsion-free coefficients. We discover that the answer for the integral case re-quires understanding of the b Z one. Moreover, although integral additive operations aretopologically generated by Adams operations, these are not reduced to infinite linearcombinations of the latter ones. We describe a topological basis for stable operationsand relate it to a basis of stable operations in graded K-theory. We classify multiplicativeoperations in both theories and show that homogeneous additive stable operations with b Z -coefficients are topologically generated by stable multiplicative operations. This is nottrue for integral operations. Introduction
Let k be a field of characteristic 0. An oriented cohomology theory A ∗ over k is afunctor from the category Sm opk of smooth quasi-projective varieties over k to the categoryof Z -graded commutative rings equipped with a push-forward structure and satisfyingcertain axioms. In this article, we study the, so-called, small theories . For these, theappropriate choice is [11, Definition 2.1] which employs a strong form of the localisationaxiom and is some breed of the axioms of Panin-Smirnov [8, 9] and that of Levine-Morel[7, Definition 1.1.2]. In particular, every oriented cohomology theory A ∗ admits a theoryof Chern classes c An of vector bundles. Among such theories there is the universal one- the algebraic cobordism of Levine-Morel Ω ∗ [7]. We will work with the free theories,i.e. theories obtained from Ω ∗ by change of coefficients. These are exactly the theories of rational type for which the results of [11] apply.Examples of free oriented cohomology theories are: • Chow theory CH ∗ that assigns to a smooth variety X over k the Chow ring CH ∗ ( X ); • Graded K -theory K ∗ gr taking X to the Laurent polynomial ring K ( X )[ t, t − ] (gradedby the powers of the Bott element t of degree −
1) over the Grothendieck ring K ( X ); • Connective K -theory taking a smooth variety X to the ring CK ∗ ( X ) of X (see [3] and[5]).The connective K -theory is the “smallest” oriented cohomology theory “living” aboveChow theory and graded K -theory: there are natural graded morphismsCK ∗ ( X ) y y rrrrrrrrrr % % ❑❑❑❑❑❑❑❑❑❑ CH ∗ ( X ) K ∗ gr ( X ) The first author has been supported by the NSF grant DMS that yield graded isomorphismsCK ∗ ( X ) /t CK ∗ +1 ( X ) ∼ → CH ∗ ( X ) and CK ∗ ( X )[ t − ] ∼ → K ∗ gr ( X ) . Moreover, multiplication CK n +1 ( X ) t −→ CK n ( X ) by the Bott element t ∈ CK − ( k ) is anisomorphism if n <
0. The map CK ( X ) → K gr ( X ) = K ( X ) is also an isomorphism, sowe can identify CK n ( X ) with K ( X ) for all n n > n ( X ) t n −→ CK ( X ) = K ( X ) is the subgroup K ( n )0 ( X ) ⊂ K ( X ) generated by the classes of coherent O X -modules with codimension of support atleast n . Note that the map t n may not be injective in general if n > A ∗ and B ∗ be two oriented cohomology theories. An additive operation G : A ∗ → B ∗ is a morphism between functors A ∗ and B ∗ considered as contravariant functors from Sm k to the category of abelian groups. Examples of additive operations are Adamsoperations in algebraic K -theory and Steenrod operations in the Chow groups modulo aprime integer.If A ∗ is an oriented cohomology theory and R is a commutative ring, we write A nR ( X )for A n ( X ) ⊗ Z R and OP n,mR ( A ) for the R -module of R -linear operations A nR → A mR .It is proved in [11, § A ∗ admits the Adams operations Ψ Am ∈ OP n,nR ( A ) for all n and m . The operation Ψ Am in OP , R ( A )satisfies Ψ Am ( c A ( L )) = c A ( L ⊗ m )for a line bundle L . Moreover, there is an R -linear mapAd n : R [[ x ]] → OP n,nR ( A )taking the power series (1 − x ) m to the Adams operation Ψ Am for all m ∈ Z .In general, the map Ad n is neither injective not surjective. But it is shown in [11, § n is an isomorphism if A ∗ is the graded K -theory, thus, OP n,nR ( K gr ) ≃ R [[ x ]] . Since the power series (1 − x ) m generate R [[ x ]] as topological group in the x -adic topol-ogy, we can say that the R -module OP n,nR ( K gr ) is topologically generated by the Adamsoperations in the graded K -theory. Moreover, since multiplication by the Bott element isan isomorphism in K ∗ gr , we have OP n,mR ( K gr ) = R [[ x ]] · t n − m .In the present paper we study the groups OP n,mR := OP n,mR (CK) of operations in theconnective K -theory over R . We write for simplicity OP n,m for OP n,m Z .The groups CK n ( X ) for n K ( X ), hence translating the aboveresult on the operations in graded K -theory, we see that Ad n : R [[ x ]] → OP n,nR is anisomorphism for n is trivial on CK nR for n >
1, i.e. Ad n (1) = 0, so we considerthe restriction Ad ′ n : xR [[ x ]] → OP n,nR of the map Ad n . The R -module CK R ( X ) is acanonical direct summand of CK R ( X ) = K ( X ) R with the complement R ·
1. This leadsto a ring isomorphism OP , R ≃ R × OP , R . Moreover, the map Ad ′ : xR [[ x ]] → OP , R isan isomorphism. PERATIONS IN CONNECTIVE K -THEORY 3 The structure of the groups OP n,nR with n > R . The homomorphisms Ad ′ n : xR [[ x ]] → OP n,nR for n > OP n,nR is very simple over the ring of profinite integers b Z = lim( Z /n Z ): Theorem.
The map Ad ′ n : xR [[ x ]] → OP n,n b Z is an isomorphism if n > . In particular,the b Z -module OP n,n b Z is topologically generated by the Adams operations. Over Z the map Ad ′ n is not surjective if n > Theorem.
The group OP n,n of integral operations is isomorphic canonically to a subgroupof OP n,n b Z . Moreover, there is an exact sequence → x Z [[ x ]] Ad ′ n −−→ OP n,n → ( b Z / Z ) n − → if n > . Thus, the group b Z also shows up in the computation of OP n,n over Z . For example, OP , as a subgroup of OP , b Z = x b Z [[ x ]] is generated by x Z [[ x ]] and the power series P i> c − c i i x i for all c ∈ b Z and integers c i such that c − c i is divisible by i for all i >
0, i.e., c i in Z represents congruence class of c modulo i .We prove that the rings OP n,n and OP n,n b Z are commutative. Moreover, the rings OP n,n are “almost” integral domains: the only zero divisors are the multiples of Ψ ± Ψ − .An operation G : A ∗ → B ∗ is called multiplicative if G is a morphism of functors Sm k → Rings . Examples are twisted
Adams operations Ψ cb defined as follows. Let b ∈ b Z and c ∈ b Z × . Then the operation Ψ cb is homogeneous and equal to c − n · Ψ bc on CK n b Z , whereΨ bc is the (generalized) Adams operation with the power series (1 − x ) bc . We classify allmultiplicative operations on CK ∗ b Z in Section 5.The notion of “stability” in topology can be considered in algebraic setting as follows(see [11, § SmOp be a category whose objects are pairs (
X, U ), where X ∈ Sm k and U is an open subvariety of X . Any theory A ∗ extends from Sm k to SmOp by therule: A ∗ (( X, U )) := Ker( A ∗ ( X ) → A ∗ ( U )) . and every additive operation A ∗ → B ∗ on Sm k extends uniquely to an operation on SmOp . There is an identification σ AT : A ∗ (( X, U )) ∼ = −→ A ∗ +1 (Σ T ( X, U )) , where Σ T ( X, U ) := (
X, U ) ∧ ( P , P \ G : A ∗ → B ∗ we define its desuspension as the uniqueoperation Σ − G : A ∗ → B ∗ such that G ◦ σ AT = σ BT ◦ Σ − G. A stable additive operation G : A ∗ → B ∗ is the collection { G ( n ) | n > } of operations A ∗ → B ∗ such that G ( n ) = Σ − G ( n +1) . A. MERKURJEV AND A. VISHIK
In Section 6 we classify stable operations in connective K -theory over b Z . We prove thatunder the identification OP n,n b Z = (cid:26) b Z [[ x ]] , if n x b Z [[ x ]] , if n > − ( G ) = (cid:26) Φ( G ) , if n G ) − Φ( G )(0) , if n > G ∈ OP n,n b Z and Φ( G ) = ( x −
1) d G d x . Thus, the desuspension map Σ − yields atower of injective maps b Z [[ x ]] = OP , b Z ← ֓ OP , b Z ← ֓ . . . ← ֓ OP n,n b Z ← ֓ . . . . The group of homogeneous degree 0 stable operations CK ∗ b Z → CK ∗ b Z is canonically isomor-phic to the group S := ∩ n Im(Φ n ) ⊂ b Z [[ x ]] . We identify this group in Section 6. In particular we prove that S is the closure in the x -adic topology of b Z [[ x ]] of the set of all (finite) b Z -linear combinations of the Adamspower series A r for r ∈ b Z × . The b Z -module S and its integral version S appear to be ofan uncountable rank. We describe a topological basis for them.We call a multiplicative operation G stable if the constant sequence ( G, G, G, . . . ) isstable. We prove that stable multiplicative operations CK ∗ b Z → CK ∗ b Z are exactly operationsΨ c , for c ∈ b Z × . Thus, we obtain: Theorem.
Homogeneous degree stable additive operations on CK ∗ b Z are topologicallygenerated by the stable multiplicative operations there. Similarly, stable multiplicative operations on CK ∗ are Ψ ± . This time though, theydon’t generate the group of stable additive operations which is of uncountable rank.Recall that operations in (graded) K -theory were determined in [11, § K gr . We describea basis of the group of stable K gr -operations and relate it to the basis of stable CK-operations. The ring of stable operations is dual to the Hopf algebra of co-operationsdefined over Z and therefore has a structure of (topological) Hopf algebra. The Hopfalgebra of co-operations coincides with K ( K ) in topology and has been studied in [1],[2], [4], [6] and [10].The main tool used in our proofs is the general result of the second author [11, Theorem6.2] that asserts, when applied to the connective K -theory, that an operation G ∈ OP n,mR for n > G l ∈ R [[ x , . . . , x l ]] for all l > n satisfying certain conditions. In particular, G l divisible by x · . . . · x l and − G l +1 = ∂ ( G l ),the partial derivative of G l (see Definition 2.1) for all l > n , i.e., all power series G l aredetermined by G n . We show that if R is torsion free, then G n can be integrated over K = R ⊗ Q : there is a unique power series H ∈ xK [[ x ]] such that G n = ∂ n − ( H ). Thus,the operation G is determined by a power series H in one variable over K such that ∂ n − ( H ) ∈ R [[ x , . . . , x n ]]. PERATIONS IN CONNECTIVE K -THEORY 5 The article is organized as follows. In Section 2 we prove general results which willpermit us to integrate the multivariate symmetric power series and reduce the classifica-tion of operations to the description of power series in one variable with certain integralityproperties. These properties are then studied and the respective power series are classifiedin Section 3. In Section 4 we apply the obtained results in combination with [11, Theorem6.2] to produce a description of additive operations in CK with integral and b Z -coefficients.We describe the ring structure on the set of homogeneous operations. The description ofoperations in K gr comes as an easy by-product. In the latter case, we also describe thedual bi-algebra of co-operations. Multiplicative operations in CK and K gr are studied inSection 5. Finally, Section 6 is devoted to the computation of stable operations.2. Symmetric power series
Partial derivatives.
Let F ( x, y ) be a (commutative) formal group law over a com-mutative ring R . We write x ∗ y := F ( x, y ).Let G ( x , . . . , x n ) ∈ R [[ x , . . . , x n ]] be a power series in n > Definition 2.1.
The partial derivative of G (with respect to F ) is the power series( ∂G )( x , x , . . . , x n +1 ) = G ( x ∗ x , x , . . . , x n +1 ) − G ( x , x , . . . , x n +1 ) − G ( x , x , . . . , x n +1 ) + G (0 , x , . . . , x n +1 ) ∈ R [[ x , . . . , x n +1 ]] . Note that the partial derivative is always taken with respect to the first variable (inthis case x ) in the list of variables. Write ∂ m for the iterated partial derivative. We alsoset ( ∂ G )( x , . . . , x n ) = G ( x , . . . , x n ) − G (0 , x , . . . , x n ).For a subset I ⊂ [1 , m + 1] := { , . . . , m + 1 } write x I for the ∗ -sum of all x i with i ∈ I .In particular, x ∅ = 0. Then( ∂ m G )( x , . . . x m + n ) = P ( − | I | G ( x I , x m +2 , . . . , x m + n ) ∈ R [[ x , x , . . . , x m + n ]] , where the sum is taken over all 2 m +1 subsets I ⊂ [1 , m + 1]. In particular, ∂ m G issymmetric with respect to the first m + 1 variables. Observation 2.2. If G ∈ R [[ x , . . . , x n ]] is so that ∂G is symmetric power series, then ∂ m G is symmetric for all m > ∂G is symmetric, ∂ m G = ∂ m − ( ∂G ) is symmetric with respect to the last n variables. But ∂ m G is symmetric with respect to the first m + 1 variables, hence it issymmetric. Notation 2.3.
For any commutative Q -algebra K writelg ( x ) := log(1 − x ) = − P i > x i i ∈ K [[ x ]]and for any n >
0, lg n ( x ) := 1 n ! (cid:0) lg ( x ) (cid:1) n ∈ K [[ x ]] . In particular, lg ( x ) = 1. A. MERKURJEV AND A. VISHIK
For the rest of this section ∗ denotes the multiplicative formal group law, i.e., x ∗ y = x + y − xy .The power series lg ( x ) belongs to the kernel of ∂ . Moreover, we have the followingstatement. Proposition 2.4.
For any commutative Q -algebra K and any n > , the kernel of ∂ n − : K [[ x ]] → K [[ x , . . . , x n ]] is equal to P r We change the variables: y i = lg ( x i ) = log(1 − x i ), where x = x . The multiplica-tive group law ∗ translates to the additive one. In the new variables the partial derivativeis homogeneous and lowers the degree in y by 1. Therefore, the kernel of ∂ n is spannedby 1 , y , . . . , y n − . (cid:3) The following formula is very useful. Proposition 2.5. Let K be a Q -algebra, G ∈ K [[ x ]] and n a positive integer. Then ( ∂ n G )( x , x , . . . x n +1 ) = ∞ P k =1 k ! ∂ n − (cid:16) (1 − x ) k d k G d x k (cid:17) ( x , x , . . . , x n ) · x kn +1 . Proof. Note that both sides don’t contain monomials x α := x α x α · · · x α n +1 n +1 if at least one α i is zero. We prove that for every multi-index α with α i > i , the x α - coefficientsof both sides are equal. Set k = α n +1 .The x α -coefficient of the left hand side is the same as the x α -coefficient of G ( x ∗ x ∗ · · ·∗ x n +1 ). To determine this coefficient, we differentiate (in the standard way) k times the series G ( x ∗ x ∗· · ·∗ x n +1 ) by x n +1 , plug in x n +1 = 0 and divide by k !. Notethat dd x n +1 ( x ∗ x ∗· · ·∗ x n +1 ) = (1 − x )(1 − x ) · · · (1 − x n ) . It follows that the x α -coefficient in the left hand side is equal to the x α x α · · · x α n n -coefficient of 1 k ! (1 − x ) k (1 − x ) k · · · (1 − x n ) k d k G d x k ( x ∗ x ∗· · ·∗ x n ) . On the other hand, note that the x α -coefficient of the right hand side is equal to the x α x α · · · x α n n -coefficient of k ! ∂ n − (cid:16) (1 − x ) k d k G d x k (cid:17) ( x , x , . . . , x n ). This is the same as the x α x α · · · x α n n -coefficient of1 k ! (1 − x ∗ x ∗· · ·∗ x n ) k G ( k ) ( x ∗ x ∗· · ·∗ x n ) = 1 k ! (1 − x ) k (1 − x ) k · · · (1 − x n ) k d k G d x k ( x ∗ x ∗· · ·∗ x n ) . (cid:3) For a nonzero power series H ∈ R [[ x , . . . , x n ]] denote by v ( H ) the smallest degree ofmonomials in H . Set also v (0) = ∞ . Observation 2.6. Suppose that a commutative ring R is torsion free. A direct calculationshows that for positive integers n and m , we have v ( ∂ n − ( x m )) = m if m > n . It followsthat v ( ∂ n − ( G )) = v ( G ) for every G ∈ R [[ x ]] such that v ( G ) > n . PERATIONS IN CONNECTIVE K -THEORY 7 Integration of symmetric power series.Definition 2.7. A power series G ∈ R [[ x , . . . , x n ]] is called double-symmetric if G itselfand ∂G are both symmetric.In the following proposition we prove that double-symmetric power series can be sym-metrically integrated over any commutative Q -algebra. Proposition 2.8. Let K be a commutative Q -algebra and G ∈ K [[ x , . . . , x n ]] , n > , bea symmetric power series divisible by x · . . . · x n . The following are equivalent: (1) G is double-symmetric; (2) All derivatives ∂ m ( G ) , m > , are symmetric power series; (3) There is a power series L ∈ K [[ x ]] such that G = ∂ n − ( L ) ; (4) There is H ∈ K [[ x , . . . , x n − ]] such that ∂ ( H ) = G ; (5) There is a unique symmetric H ∈ K [[ x , . . . , x n − ]] , divisible by x · . . . · x n − , withzero coefficient at x · . . . · x n − and such that ∂ ( H ) = G .Proof. Note that (1) ⇔ (2) by Observation 2.2. We will prove the equivalence of allstatements by induction on n . The implication (3) ⇒ (2) is clear, (2) ⇒ (1) and (3) ⇒ (4)are trivial.(5) ⇒ (3) follows by induction applied to H .(1) or (4) ⇒ (5) Over a commutative Q -algebra every formal group law is isomorphicto the additive one. So we may assume that the group law is additive, i.e., the derivativeis defined by ( ∂G )( x, y, ¯ t ) = G ( x + y, ¯ t ) − G ( x, ¯ t ) − G ( y, ¯ t ) + G (0 , ¯ t ) . We first prove uniqueness. Indeed if ∂H = 0, then H is linear in x , and since H issymmetric and divisible by x · . . . · x n − , we must have H = 0.Case n = 2: The implication (4) ⇒ (5) is obvious. We prove (1) ⇒ (5). We mayassume that G is a homogeneous polynomial of degree d > 1. The symmetry of thederivative of G ( x, y ) results in the following cocycle condition: G ( x + y, z ) + G ( x, y ) = G ( x + z, y ) + G ( x, z ) . In particular, we have the following equalities: G ( x + y, x + y ) + G ( x, y ) = G (2 x + y, y ) + G ( x, x + y ) ,G (2 x + y, y ) + G (2 x, y ) = G (2 x, y ) + G ( y, y ) ,G ( x, x + y ) + G ( x, y ) = G (2 x, y ) + G ( x, x ) . It follows that ∂ ( G ( x, x ))( x, y ) = G ( x + y, x + y ) − G ( x, x ) − G ( y, y )= G (2 x, y ) − G ( x, y )= (2 d − G ( x, y )) , hence G ( x, y ) = ∂ ( H ), where H ( x ) = G ( x, x ) / (2 d − A. MERKURJEV AND A. VISHIK Case n = 3: Write G ( x, y, z ) = P i > G i ( x, y ) z i . Clearly, all G i ( x, y ) satisfy (1) or(4) and hence (5) by induction. Integrating each G i ( x, y ), we get a power series H = P i,j > a i,j x i y j in two variables such that ∂H = G .Note that we can change H by any series P i c i xy i without changing ∂H . This way, wecan make H = P i,j > a i,j x i y j with a i, = a ,i and a , = 0. We claim that H is symmetric.Indeed, from the symmetry of ∂H , we have: (cid:18) i + ki (cid:19) a i + k,j = (cid:18) j + kj (cid:19) a j + k,i , for any i, j, k > 1. This implies that i + l (cid:18) i + li (cid:19) a i + l − , = a l,i , and so, a i,l = a l,i , for any i, l > 2. This shows that H is symmetric. Observe that suchsymmetric integration is unique provided a , = 0.Case n > 3: Write G = P i > G i · x in with G i ∈ K [[ x , . . . , x n − ]]. The slices G i of G are double-symmetric. By the inductive assumption, these can be uniquely integratedto symmetric power series H i ∈ K [[ x , ..., x n − ]] as in (5). Putting these power seriestogether, we obtain H = P i > H i · x in − ∈ K [[ x , ..., x n − ]]such that ∂H = G . Write H = P i ,...,i n − a i ,...,i n − x i . . . x i n − n − . Modifying H by x . . . x n − L ( x n − ) for an appropriate power series L , we may assumethat a i, ,..., = a , ,...,i for all i .We claim that H is symmetric. The x i . . . x i n n -coefficient of G = ∂H is equal to (cid:18) i + i i (cid:19) a i + i ,i ,...,i n . Therefore, since G is symmetric, H is symmetric with respect to x , ..., x n − , if i > 1. Recall than H is also symmetric in x , . . . , x n − . Therefore, itsuffices to show that the coefficient a ,i ,...,i n − does not change if we interchange i n − with i k for some k = 2 , . . . , n − i , . . . , i n − but one are equal to 1. Then the statement follows fromthe equality a , ,...,i = a i, ,..., = a ,i,..., for all i . Otherwise, at least two indices, say i k = u and i l = v with k < l are greater than 1.If l < n − 1, set w = i n − . We have (here and below we indicate only the indices whichare permuted, hidden indices remain unchanged): a ,u,v,w = a v,u, ,w = a v,w, ,u = a ,w,v,u , so we interchanged i k and i n − . If l = n − 1, we can write a ,u,v = a u, ,v = a u,v, = a v,u, = a v, ,u = a ,v,u , i.e., we again interchanged i k and i n − . (cid:3) PERATIONS IN CONNECTIVE K -THEORY 9 The groups Q nR The formal group law is multiplicative in this section. Let R be a commutative ringand K = R ⊗ Z Q . We assume that R is torsion free (as abelian group), i.e., R can beidentified with a subring of K . Definition 3.1. For any integer n > 1, let us denote by Q nR the R -module of power series G in xK [[ x ]], for which ∂ n − ( G ) ∈ R [[ x , ..., x n ]]. For example, Q R = xR [[ x ]]. We also set Q nR = R [[ x ]] if n xR [[ x ]] and P Suppose R has no nontrivial Z -divisible elements. Then xR [[ x ]] ∩ (cid:16) P Observe that Φ(lg r ( x )) := ( x − · dd x (lg r ( x )) = lg r − ( x ) . Suppose P 0, i.e., nq r is a nonzero Z -divisible element in R , a contradiction. (cid:3) Definition 3.3. Let n and m be integers. If n > Q n,mR the submodule of Q nR consisting of all power series G such that v ( ∂ n − G ) > m . If n 0, set Q n,mR = x max (0 ,m ) · R [[ x ]].Since v ( ∂ n − G ) > n for every G ∈ Q nR with n > 0, we have Q n,mR = Q n,nR = Q nR if n > m . Note also that Q ,mR = x max (1 ,m ) · R [[ x ]].3.1. The groups Q n b Z . In this section we determine the structure of the modules Q n b Z overthe ring b Z = lim( Z /n Z ). We write b Q for b Z ⊗ Q . Note that b Q = b Z + Q and Z = b Z ∩ Q in b Q . Lemma 3.4. Let b , b , . . . , b m ∈ b Z be such that b i ≡ b j ( mod j ) for every i divisible by j . Then there is b ∈ Z such that b ≡ b i ( mod i ) for all i = 1 , . . . , m .Proof. Let p , p , . . . , p s be all primes at most m . For every k , let q k = p r k k be the largestpower of p k such that q k m . By Chinese Remainder Theorem, we can find b ∈ Z suchthat b ≡ b q k ( mod q k ) for all k . We claim that b works. Take any i m . We prove that b ≡ b i ( mod i ). Write i as the product i = Q q ′ k , where q ′ k is a power of p k . Clearly, q ′ k divides q k . We have b q ′ k ≡ b i ( mod q ′ k ) by assumption ,b q k ≡ b q ′ k ( mod q ′ k ) by assumption ,b ≡ b q k ( mod q k ) by construction . It follows that b ≡ b i ( mod q ′ k ) for all k , hence b ≡ b i ( mod i ). (cid:3) Let G ( x ) = P ∞ i =1 a i x i with a i ∈ b Q . Lemma 3.5. For positive integers j s , the x j y s -coefficient of ∂G is equal to j P i =0 ( − j − i (cid:18) s + is (cid:19)(cid:18) sj − i (cid:19) a s + i . Proof. We have 1 s ! d s G d x s = ∞ P i =0 (cid:18) s + is (cid:19) a s + i x i . The statement follows from Proposition 2.5. (cid:3) Set b i = ia i for all i > Corollary 3.6. If ∂G ∈ b Z [[ x, y ]] then b i − b ∈ b Z for all i > . In particular, if a ∈ b Z ,then all b i are in b Z .Proof. The xy j -coefficient of ∂G is equal to b j +1 − b j . (cid:3) Proposition 3.7. Let G ∈ Q b Z and let n > be an integer such that a i ∈ b Z for all i < n .Let p t < n be power of a prime integer p such that p t divides n . Then p t divides b n .Proof. Take j = p t and s = n − p t > p t . By Lemma 3.5, the x j y s -coefficient of ∂G is equalto j P i =0 ( − j − i (cid:18) s + is (cid:19)(cid:18) sj − i (cid:19) a s + i ∈ b Z . By assumption, all terms in the sum but the last one belong to b Z , hence so does the lastone: (cid:0) np t (cid:1) a n ∈ b Z . But (cid:0) np t (cid:1) a n = (cid:0) n − p t − (cid:1) b n /p t , hence (cid:0) n − p t − (cid:1) b n is divisible by p t . As (cid:0) n − p t − (cid:1) isprime to p , the coefficient b n is divisible by p t . (cid:3) Proposition 3.8. We have Q b Z = b Q · lg ( x ) ⊕ x b Z [[ x ]] . Proof. Let G ( x ) = P ∞ i =1 a i x i ∈ Q b Z and set as before b i = ia i . Adding a lg ( x ) to G ( x )we may assume that a = 0. By Corollary 3.6, b i ∈ b Z for all i .We claim that for every positive integers i < n such that i divides n we have b n ≡ b i modulo i . We prove this by induction on n . By Lemma 3.4 applied to m = n − 1, thereis b ∈ Z such that b ≡ b i modulo i for all i < n . Subtracting b lg ( x ) from G ( x ), we mayassume that b i is divisible by i for all i < n , or equivalently, a i ∈ b Z for all i < n . Weprove that b n is divisible by i .Case 1: n = p k is a power of a prime p . Then i = p t is a smaller power of p . ByProposition 3.7, i divides b n .Case 2: n is not power of a prime. Write n as a product of powers of distinct primes: n = q q · · · q s . By Proposition 3.7, q k divides b n for every k , hence n divides b n . Inparticular, i divides b n . The claim is proved.Let b ∈ b Z be such that b ≡ b n ( mod n ) for all n . We have G = b lg ( x ) + P n > b n − bn x n ∈ b Z · lg ( x ) + x b Z [[ x ]] . (cid:3) PERATIONS IN CONNECTIVE K -THEORY 11 Corollary 3.9. Let G ( x ) = ax + . . . ∈ Q b Z be a power series with a ∈ b Z . Then G ( x ) ∈ b Z · lg ( x ) + x b Z [[ x ]] . Lemma 3.10. Let H ( x, y ) = P i,j > a i,j x i y j ∈ b Q [[ x, y ]] be such power series that both ∂ -partial derivatives of H have coefficients in b Z and a i, as well as a ,i are in b Z , for all i .Then H ( x, y ) ∈ b Z [[ x, y ]] .Proof. Consider some j -th row of H : y j · P i > a i,j x i . We know that P i > a i,j x i ∈ Q b Z .By Corollary 3.9, P i > a i,j x i is equal to c j · lg ( x ) modulo x b Z [[ x ]] for some c j ∈ b Z . Hence c j i ≡ a i,j ( mod b Z ) for all i . Applying the same considerations to the i -th column, weobtain: c j i ≡ d i j ( mod b Z ) , for certain d i ∈ b Z . Let us show that all c i ’s (and d j ’s) are zeros. Indeed, we have: jc j ≡ id i ( mod ij ) . Hence, jc j is divisible by i , for any i and, hence c j = 0. This implies that a i,j ∈ b Z for any i, j . (cid:3) Lemma 3.11. Suppose, H ( x , ..., x n ) = P i ,...,i n > a i ,...,i n x i · . . . · x i n n ∈ b Q [[ x , . . . , x n ]] besuch a power series that all ∂ -partial derivatives of H with respect to all variables havecoefficients in b Z and a i ,...,i n ∈ b Z as long as all i j ’s but one are equal to . Then H hascoefficients in b Z .Proof. Induction on n . For n = 1 there is nothing to prove. For n = 2 this is Lemma 3.10.We can assume that n > 3. Suppose we know the statement for r < n . Note, that all thecells of H (where we set certain i j ’s to be 1) also satisfy the conditions of the Lemma.By our assumption, these have all coefficients in b Z . That is, a i ,...,i n ∈ b Z provided, atleast, one of i j ’s is 1. Consider a hyper-slice H i of H (we fix i ). Then H i satisfies theconditions of the Lemma (note that n > H i has coefficients in b Z and so does H . (cid:3) The following theorem is a generalization of Proposition 3.8. Theorem 3.12. For every n > , Q n b Z = ` The statement is clear if n 0. Now assume that n > 1. It follows from Lemma3.2 that ` 1. Since H is symmetric, by Lemma 3.11, allcoefficients of the power series H are in b Z . By the induction hypothesis, G ( x ) ∈ Q n = ` For every sequence a , we have ( x − · dd x ( a · lg r ( x )) = a · lg r − ( x ) . Proof. Write ( − r a · lg r ( x ) = P b i x i and ( − r − a · lg r − ( x ) = P c i x i . We need to provethat ( m + 1) b m +1 − mb m = c m for every m . We have( m + 1) b m +1 = P
For every c ∈ b Z and every integer r > there is a sequence ˜ c = ( c i ) i > ofintegers c i ∈ Z such that c i ≡ c ( mod i ) for all i and ( c − ˜ c ) · lg i ( x ) ∈ b Z [[ x ]] for all i = 1 , . . . , r , where c − ˜ c is the sequence ( c − c i ) i > . PERATIONS IN CONNECTIVE K -THEORY 13 Proof. Take any collection ˜ c = ( c i ) i > of integers. Note that for every i > k =1 , . . . , r , the x i + k − -coefficient of ˜ c · lg k ( x ) is a linear combination of c , . . . , c i with rationalcoefficients where the c i -coefficient is equal to ( − k / ( i ( i + 1) . . . ( i + k − c , c , . . . inductively to make all coefficients of the power series G k = ( c − ˜ c ) · lg k ( x )integral for all k = 1 , . . . , r . Let c be an integer congruent to c modulo r !, so the x k -coefficient of G k is integer for every k = 1 , . . . , r . Suppose we have modified c , . . . c n sothat the x j -coefficient of G k is integral for all k = 1 , . . . , r and j n + k − k = 1 , . . . , r , we will modify c n +1 to make integral the x n + k -coefficientof G k . Note that the integral x j -coefficients of G k for j n + k − k = 1 we don’t modify c n +1 : the power series G is already integral. k ⇒ k + 1: By Lemma 3.13, ( x − · d G k +1 d x = G k . Hence, if G k = P i > k b i x i and G k +1 = P i > k +1 a i x i , then a n + l +1 = − n + l +1 ( b k + . . . + b n + l )for all l .By induction, b k , . . . , b n + k are integral. Recall that these are linear combinations of the c ′ i ’s, where c ′ i = c − c i and c ′ n +1 appears only in b n + k . We modify c n +1 by adding to c n +1 the integer t ( n + 1)( n + 2) . . . ( n + k ) with some t ∈ Z . Note that b k , . . . , b n + k − remainunchanged and b n + k changes to b n + k + t , so it stays integral. Choose t to make a n + k +1 integral.Note that c ′ n +1 comes with coefficient ( − l / (( n + 1) . . . ( n + l )) in the x n + l -coefficientof G l . Since ( n + 1) . . . ( n + l ) divides ( n + 1) . . . ( n + k ) when l k , the x n + l -coefficientof G l remains integral for l k . (cid:3) Now we prove that the map ρ n : Q n → b Q n − is surjective. Since q · lg r ∈ Q n for all q ∈ Q and r = 1 , . . . , n − 1, we have Q n − ⊂ Im( ρ n ). It suffices to show that b Z n − ⊂ Im( ρ n ).Choose c r ∈ b Z for r = 1 , . . . , n − 1. By Lemma 3.14, there are sequences of integers ˜ c r such that ( c r − ˜ c r ) · lg r ( x ) ∈ b Z [[ x ]].As ˜ c r · lg r ( x ) = c r · lg r ( x ) − ( c r − ˜ c r ) · lg r ( x ) , we have ρ n (cid:0)P 1, the group Q n is generated by x Z [[ x ]] and the powerseries ( c − ˜ c ) · lg r ( x ) as in Lemma 3.14, where c ∈ b Z and r = 1 , . . . , n − Q n can be approximated by polynomials as follows: Lemma 3.16. For every m > and n , we have Q n ⊂ Z [ x ] m − + P 1. In view of (3.15), the group Q n modulo x Z [[ x ]] + P 1. Then the x m -truncation F of (˜ c − d ) · lg r ( x ) is contained in Z [ x ] m − and ˜ c · lg r ( x ) is congruent to F modulo Z · lg r ( x ) + x m Q [[ x ]]. (cid:3) Operations Let k be a field of characteristic 0 and write Sm k for the category of smooth quasi-projective varieties over k . An oriented cohomology theory A ∗ over k is a functor from Sm opk to the category of Z -graded commutative rings equipped with a push-forward struc-ture and satisfying certain axioms (see [11, Definition 2.1]). We write A ∗ ( X ) = ` n ∈ Z A n ( X )for a variety X in Sm k and let A ∗ ( k ) denote the coefficient ring A ∗ (Spec k ).Let A ∗ be an oriented cohomology theory. There is a (unique) associated formal grouplaw F A ( x, y ) = P i,j > a Ai,j x i y j = x + y + a , · xy + higher terms ∈ A ∗ ( k )[[ x, y ]]that computes the first Chern class of the tensor product of two line bundles L and L ′ (see, for example, [7, § § c A ( L ⊗ L ′ ) = F A ( c A ( L ) , c A ( L ′ )) . Example 4.1. The Chow theory CH ∗ takes a smooth variety X to the Chow ring CH ∗ ( X )of X . We have CH ∗ ( k ) = Z and F CH ( x, y ) = x + y is the additive group law. Example 4.2. (see [7, Example 1.15]) The graded K -theory K ∗ gr takes X to the Laurentpolynomial ring K ( X )[ t, t − ] (graded by the powers of the Bott element t of degree − K ( X ) of X . We have K ∗ gr ( k ) = Z [ t, t − ] and F K gr ( x, y ) = x + y − txy is the multiplicative group law. Example 4.3. (see [3] and [5]) The connective K -theory takes X to the ring CK ∗ ( X ) of X . We have CK ∗ ( k ) = Z [ t ] and F CK ( x, y ) = x + y − txy . PERATIONS IN CONNECTIVE K -THEORY 15 All cohomology theories in these examples are of rational type (see [11, § A ∗ is an oriented cohomology theory and R a commutative ring, the functor A ∗ R defined by A ∗ R ( X ) = A ∗ ( X ) ⊗ Z R is also an oriented cohomology theory with values inthe category of graded R -algebras. Definition 4.4. Let A ∗ and B ∗ be two oriented cohomology theories. An R -linear opera-tion G : A ∗ R → B ∗ R is a morphism between functors A ∗ R and B ∗ R considered as contravariantfunctors from Sm k to the category of R -modules (cf. [11, Definition 3.3]). Note that G may not respect the gradings on A ∗ R and B ∗ R .Let n, m ∈ Z . A morphism G : A nR → B mR between contravariant functors from Sm k tothe category of R -modules can be viewed as an R -linear operation via the obvious compo-sition A ∗ R →→ A nR → B mR ֒ → B ∗ R . All such operations form an R -module OP n,mR ( A ∗ , B ∗ ).The composition of operations yields an R -linear pairing OP n,mR ( A ∗ , B ∗ ) ⊗ R OP m,rR ( B ∗ , C ∗ ) → OP n,rR ( A ∗ , C ∗ ) . In particular, OP n,nR ( A ∗ ) := OP n,nR ( A ∗ , A ∗ ) has a structure of an R -algebra. Example 4.5. (see [3] and [5]) Multiplication by t yields an operation CK n +1 R → CK nR that is an isomorphism if n < 0. There are graded R -linear operationsCK ∗ R → CH ∗ R and CK ∗ R → ( K ∗ gr ) R . The sequence CK n +1 ( X ) t −→ CK n ( X ) → CH n ( X ) → n and X .If n > n ( X ) → K ngr ( X ) = K ( X ) t − n ≃ K ( X )is generated by the classes of coherent O X -modules with codimension of support at least n . If n Theorem 4.6. Let A ∗ be a cohomology theory of rational type and B ∗ be any orientedcohomology theory over k . Let R be a commutative ring. Then there is a -to- corre-spondence between the set OP n,mR ( A ∗ , B ∗ ) of R -linear operations G : A nR → B mR and theset consisting of the following data { G l , l ∈ Z > } : G l ∈ Hom R (cid:0) A n − l ( k ) ⊗ R, B ∗ ( k )[[ x , . . . , x n ]] ( m ) ⊗ R (cid:1) satisfying (1) G l ( α ) is a symmetric power series for all l and α ∈ A n − l ( k ) ⊗ R , (2) G l ( α ) is divisible by x · . . . · x l for all l and α , (3) G l ( α )( y + B z, x , . . . , x l ) = P i,j G i + j + l − ( α · a Ai,j )( y × i , z × j , x , . . . , x l ) , where a Ai,j are the coefficients of the formal group law of A ∗ and the sum y + B z is taken withrespect to the formal group law of B ∗ . Here B ∗ ( k )[[ x , . . . , x n ]] ( m ) is the subgroup in B ∗ ( k )[[ x , . . . , x n ]] consisting of all homo-geneous degree m power series (all the x i ’s have degree 1).The functions G l are determined by the operation G as follows (see [11, § L i for the pull-back of the canonical line bundle on P ∞ with respect to the i -th projection ( P ∞ ) l → P ∞ . Then(4.7) G l ( α ) (cid:0) c B ( L ) , . . . , c B ( L l ) (cid:1) = G (cid:0) α · c A ( L ) · . . . · c A ( L l ) (cid:1) , where c is the first Chern class. Remark 4.8. Theorem 4.6 was proved in [11, Theorem 6.2] in the case R = Z . Thegeneral case readily follows. Indeed, multiplication by an element r ∈ R yields operations r : A nR → A nR and r : B mR → B mR . An additive operation G : A nR → B mR is R -linear if andonly if G ◦ r = r ◦ G for all r ∈ R . The latter is equivalent to the equality G l ◦ r = r ◦ G l for all l , i.e., that all G l are R -linear. Example 4.9. (see [11, § A ∗ be a cohomology theory of rational type and m ∈ Z .Consider the power series [ m ]( x ) := x + A . . . + A x ∈ A ∗ ( k )[[ x ]] ( m times). The Adamsoperation Ψ Am ∈ OP ∗ , ∗ R is determined by ( G l ) l > , where G l is multiplication by the powerseries [ m ]( x ) · . . . · [ m ]( x l ). The Adams operations satisfy the relationsΨ Ak ◦ Ψ Am = Ψ Akm = Ψ Am ◦ Ψ Ak for all k and m .4.1. Operations in connective K -theory. We would like to determine the R -module OP n,mR of all R -linear operations G : CK nR → CK mR for any pair of integers n and m . ByTheorem 4.6, G is given by a collection of power series G l ( α ) ∈ R [ t ][[ x , . . . , x n ]] ( m ) , where α ∈ CK n − lR ( k ) and l > 0, satisfying conditions of the theorem. The group CK n − lR ( k ) istrivial if l < n and CK n − lR ( k ) = R · t l − n otherwise. (Recall that t has degree − G l ( α ) = 0 and in the latter case the power series G l ( α ) are uniquely determinedby G l ( t l − n ). We will simply write G l for G l ( t l − n ).If l > max(1 , n ), condition (3) in Theorem 4.6 reads as follows: G l ( x + y − txy, ¯ z ) = G l ( x, ¯ z ) + G l ( y, ¯ z ) − G l +1 ( x, y, ¯ z ) . In other words,(4.10) G l +1 = − ∂ t G l , where the derivative ∂ t is taken with respect to F CK ( x, y ) = x + y − txy . Thus, G l +1 isuniquely determined by G l .If n > 0, the operation G yields the double-symmetric power series G n ∈ R [ t ][[ x , . . . , x n ]] ( m ) that is divisible by x · . . . · x n . Conversely, if H ∈ R [ t ][[ x , . . . , x n ]] ( m ) is a double-symmetricpower series divisible by x · . . . · x n , then setting G n + i := ( − i ∂ it ( H ) for all i > 0, weget a sequence of power series that determines an R -linear operation G (see Observation2.2).If n G is determined by G ∈ R [ t ] m and power series G ∈ R [ t ][[ x ]] m that is uniquely determined by ( G ) | t =1 ∈ x max (1 ,m ) R [[ x ]]. If m > G = 0,otherwise G ∈ R · t − m and we can combine G and G together into the power series H = ( G − G ) | t =1 ∈ R [[ x ]].If L ∈ R [ t ][[ x , . . . , x n ]] ( m ) , then v ( L | t =1 ) > m . Conversely, for every J ∈ R [[ x , . . . , x n ]]with v ( J ) > m , there is a unique homogeneous power series L ∈ R [ t ][[ x , . . . , x n ]] ofdegree m such that L | t =1 = J . If L is double-symmetric and divisible by x · . . . · x n , then PERATIONS IN CONNECTIVE K -THEORY 17 so is L | t =1 (with respect to the derivative ∂ given by the formal group law x + y − xy )and conversely.We have proved the following statement. Proposition 4.11. Let R be a commutative ring and let n and m be two integers. An R -linear operation G : CK nR → CK mR is determined by (1) A power series H ∈ x max (0 ,m ) R [[ x ]] if n . In this case G = H (0) · t − m and G ∈ xR [ t ][[ x ]] ( m ) is a unique homogeneous power series such that H = ( G − G ) | t =1 and G l = ( − l − ∂ l − t ( G ) for l > , (2) A double-symmetric power series J ∈ R [[ x , . . . , x n ]] divisible by x · . . . · x n suchthat v ( J ) > m if n > . In this case G l = 0 for l = 0 , . . . , n − and G n ∈ R [ t ][[ x , . . . , x n ]] ( m ) is a unique homogeneous power series such that G n | t =1 = J and G l = ( − l − n ∂ l − nt ( G n ) for l > n . Let R be a commutative ring that is torsion free as abelian group. Define an R -modulehomomorphism λ n,m : Q n,mR → OP n,mR as follows. If n λ n,m ( H ) for H ∈ Q n,m = x max (0 ,m ) · R [[ x ]] is the operation givenby Proposition 4.11(1). If n > λ n,m ( H ) for H ∈ Q n,m is the operation given by thepolynomial J = ( − n ∂ n − ( H ) as in Proposition 4.11(2).The following theorem determines the R -module of operations OP n,mR in terms of themodules Q n,mR of power series in one variable. Theorem 4.12. Let R be a commutative ring that is torsion free as abelian group and K = R ⊗ Q . The homomorphisms λ n,m yield an R -linear isomorphisms between OP n,mR and the factor module of Q n,mR by the K -subspace spanned by lg i ( x ) , i = 1 , . . . , n − . Inparticular, OP n,mR ≃ x max (0 ,m ) · R [[ x ]] if n and OP ,mR ≃ x max (1 ,m ) · R [[ x ]] .Proof. The surjectivity of λ n,m follows from Propositions 2.8 and 4.11. The kernel of λ n,m is determined in Proposition 2.4. (cid:3) Corollary 4.13. The map λ n,m yields an isomorphism Q nR ∩ x max (0 ,n,m ) · K [[ x ]] ∼ → OP n,mR . Proof. The case m n follows from the theorem. Otherwise, by Observation 2.6, v ( ∂ n − x i ) = i for all i > n . (cid:3) Let n, m ∈ Z and i, j non-negative integers. We define an R -linear homomorphism Q n,mR → Q n + i,m − jR as follows. If n , m n + i > Q n,mR = R [[ x ]] → xR [[ x ]] ֒ → Q n + i,m − jR takes H to ∂ ( H ) = H − H (0). Otherwise, Q n,mR ⊂ Q n + i,m − jR , and the map we define isthe inclusion.Multiplication by t k yields an operation CK ∗ + kR → CK ∗ R and therefore, the homomor-phisms OP n,mR → OP n + i,m − jR for all i, j > Proposition 4.14. The diagram Q n,mRλ n,m (cid:15) (cid:15) / / Q n + i,m − jR λ n + i,m − j (cid:15) (cid:15) OP n,mR / / OP n + i,m − jR , is commutative.Proof. The case i = 0 follows directly from the definition. It remains to consider the case i = 1 and j = 0.Suppose first that n > 0. Let H ∈ Q n,mR ⊂ Q n + i,m − jR and G = λ n,m ( H ) ∈ OP n,mR . Inparticular, G n | t =1 = ( − n − ∂ n − ( H ). Denote by G ′ the image of G in OP n +1 ,mR . Write L i for the pull-back of the canonical line bundle on P ∞ with respect to the i -th projection( P ∞ ) n +1 → P ∞ . The power series G ′ n +1 is determined by the equality (see (4.7)) G ′ n +1 ( c ( L ) , . . . , c ( L n +1 )) = G ′ ( c ( L ) · . . . · c ( L n +1 ))= G ( tc ( L ) · . . . · c ( L n +1 ))= G n +1 ( t )( c ( L ) , . . . , c ( L n +1 ))= G n +1 ( c ( L ) , . . . , c ( L n +1 )) , hence G ′ n +1 = G n +1 . It follows from (4.10) that G ′ n +1 | t =1 = G n +1 | t =1 = − ( ∂ t G n ) | t =1 = − ∂ ( G n | t =1 ) = − ∂ (( − n ∂ n − ( H )) = ( − n +1 ∂ n ( H ) , and therefore, G ′ = λ n +1 ,m ( H ).If n < n = 0 and m > Q n,mR ⊂ Q n +1 ,mR and the statement followsimmediately from the definitions. It remains to consider the case n = 0 and m 0. Let H ∈ Q ,mR = R [[ x ]] and G = λ ,m ( H ) ∈ OP ,mR . In particular, H = ( G − G ) | t =1 . Denoteby G ′ the image of G in OP ,mR . A computation as above shows that G ′ = G . Hence G ′ | t =1 = G | t =1 = − ( H − H (0))Therefore, G ′ = λ ,m ( H − H (0)) and H − H (0) is the image of H in Q ,mR . (cid:3) Corollary 4.13 and Proposition 4.14 yield: Corollary 4.15. If m n then the map OP n,nR → OP n,mR is an isomorphism. In particular, there is a canonical ring homomorphism OP n,nR → OP n +1 ,nR ∼ → OP n +1 ,n +1 R . Example 4.16. Note that the identification OP , R = R [[ x ]] is not a ring isomorphism.The corresponding ring structure on R [[ x ]] will be described in Section 4.5. The naturalsurjective homomorphism R [[ x ]] = OP , R → OP , R = xR [[ x ]]takes a power series G ( x ) to G ( x ) − G (0). Its kernel is generated by 1. The complementaryoperation G ( x ) G (0) on CK = K is an idempotent that takes the class of a vector PERATIONS IN CONNECTIVE K -THEORY 19 bundle E to rank( E ) · 1, where 1 is the identity in K . In particular, we get a natural R -algebra isomorphism OP , R ≃ R × OP , R .4.2. Adams operations. Let R be a torsion free ring. We define the compositionAd n : R [[ x ]] → Q nR λ n,n −−→ OP n,nR , where the first map is the identity if n ∂ : R [[ x ]] → xR [[ x ]] and the inclusion of xR [[ x ]] into Q nR . The image of Ad n is denoted OP n,nR,cl and called the submodule of classical operations .If n 0, we have OP n,nR,cl = OP n,nR = R [[ x ]]. If n > R has no nontrivial Z -divisible elements (for example. R = Z or b Z ), the restriction of Ad n on xR [[ x ]] is injective and therefore, OP n,nR,cl ≃ xR [[ x ]].Let m be an integer. In the notation of the Example 4.9, [ m ]( x ) = (1 − (1 − tx ) m ) /t .In view of Proposition 4.11, the Adams operations Ψ m ∈ OP n,nR,cl are defined by(4.17) Ψ k = Ad n ((1 − x ) m ) . Since the power series (1 − x ) m generate R [[ x ]] as topological group in the x -adic topol-ogy, the group of classical operations OP n,nR,cl is topologically generated by the Adamsoperations.By Proposition 4.14, the operations Ψ k are compatible with the canonical homomor-phisms OP n,nR → OP n +1 ,n +1 R .For every k > k = P ki =0 ( − i (cid:18) ki (cid:19) Ψ i . Then Υ k = λ n,n ( x k ) if k > 0. Recall that Υ = 0 if n > 1. It follows that the R -module OP n,nR,cl consists of all linear combinations P k > α k · Υ k with α k ∈ R (cf. [11, Theorem 6.8]). If R has no nontrivial Z -divisible elements, the coefficients α k , (where k > n k > n > 1) are uniquely determined by the operation.4.3. Operations over b Z . In Section 3 we determined the modules Q nR over the ring R = b Z . Theorems 3.12 and 4.12 yield: Theorem 4.18. There are canonical isomorphisms OP n,n b Z = OP n,n b Z ,cl ≃ (cid:26) b Z [[ x ]] , if n ; x b Z [[ x ]] , if n > . In particular, the natural map OP n,n b Z → OP n +1 ,n +1 b Z is an isomorphism for all n > n and m , OP n,m b Z ≃ (cid:26) x max (0 ,m ) · b Z [[ x ]] , if n { G ∈ x b Z [[ x ]] | v ( ∂ n − ( G )) > m } , if n > Operations over Z . Now we turn to the case R = Z and for simplicity write OP n,m for OP n,m Z .Corollary 4.13 implies that the natural homomorphism OP n,m → OP n,m b Z is injective.In particular, we can identify OP n,n with a subgroup of OP n,n b Z = x b Z [[ x ]] for all n > 1, so we have a sequence of subgroups OP , ⊂ OP , ⊂ . . . ⊂ OP n,n ⊂ . . . ⊂ x b Z [[ x ]] . Recall (Theorem 4.12) that OP n,m ≃ x max (0 ,m ) · Z [[ x ]] if n OP n,m ≃ OP n,n if m n by Corollary 4.15.Let m > n > 1. By Theorem 4.12, we can identify OP n,m with the factor groupof Q n,m by the subgroup P n − r =1 Q · lg r ( x ). It follows that the map ρ n in (3.15) yields ahomomorphism OP n,m → ( b Q / Q ) n − = ( b Z / Z ) n − . By the proof of Lemma 3.16, this map is surjective. Its kernel is denoted OP n,mcl and calledthe subgroup of classical operations . In the case n = m this group coincides with the groupof classical operation defined earlier. In view of Corollary 4.13, OP n,mcl is identified withthe group (cid:0)` n − r =1 Q · lg r ( x ) + x Z [[ x ]] (cid:1) ∩ x m Q [[ x ]].We view the group x Z [ x ] m − of integral polynomials of degree at most m − Q -space x Q [[ x ]] / ( x m ). Denote by L n,m the intersection of x Z [ x ] m − with the imagein x Q [[ x ]] / ( x m ) of the space ` n − r =1 Q · lg r ( x ). Then L n,m is a subgroup of x Z [ x ] m − ofrank n − OP n,mcl = L n,m ⊕ x m Z [[ x ]] . If m = n > 1, the map of Q -spaces is an isomorphism and L n,n = x Z [ x ] n − . It followsthat OP n,ncl = x Z [[ x ]] . Recall that OP n,ncl = OP n,n = Z [[ x ]] if n OP n,m = OP n,n if m n .We summarize our results in the following statement. Theorem 4.19. The natural homomorphism OP n,m → OP n,m b Z is injective. For anyintegers m > n > there is an exact sequence → OP n,mcl → OP n,m → ( b Z / Z ) n − → , where OP n,mcl = L n,m ⊕ x m Z [[ x ]] . Moreover, OP n,ncl = x Z [[ x ]] . Remark 4.20. Similar arguments yield the following formula for m > n > OP n,m b Z = L n,m b Z ⊕ x m b Z [[ x ]] , where L n,m b Z = L n,m ⊗ b Z .4.5. Composition. The R -module homomorphism Ad n : R [[ x ]] → OP n,nR is not a ringhomomorphism. In this section we introduce a new product on R [[ x ]] so that Ad n becomesan R -algebra homomorphism.Let H, H ′ ∈ R [[ x ]], write H ′ = P i > a i x i and define the composition in H and H ′ bythe formula H ◦ H ′ = a · H (0) + P i > ( − i a i · ( ∂ i − H )( x × i ) . The composition ◦ is distributive in H and H ′ with respect to addition. (Note that theusual substitution of power series is only one-sided distributive.) The polynomial 1 − x PERATIONS IN CONNECTIVE K -THEORY 21 is the identity for the composition: (1 − x ) ◦ H = H = H ◦ (1 − x ) for all H . We view R [[ x ]] as an R -algebra with product given by the composition. Lemma 4.21. The maps Ad n : R [[ x ]] → OP n,nR are R -algebra homomorphisms.Proof. In view of Proposition 4.14 it suffices to consider the case n = 0. Let H, H ′ ∈ R [[ x ]] and write H ′ = P i > a i x i . If G , G , . . . ∈ R [ t ][[ x ]] is the sequence of power seriescorresponding to Ad ( H ) (see Proposition 4.11), then G = H (0) ∈ R , H = ( G − G ) | t =1 and G i = ( − i − ∂ i − t ( G ) for i > 1. Note that G ( t, x ) = − H ( tx ) + H (0).Write L for the canonical line bundle on P ∞ . By (4.7) and (4.10),Ad ( H )( c ( L ) i ) = G i ( c ( L ) × i ) = ( − i − ( ∂ i − t G )( c ( L ) × i ) = ( − i ( ∂ i − H )( tc ( L ) × i ) . Therefore, we have(Ad ( H ) ◦ Ad ( H ′ ))( c ( L )) = − Ad ( H )( P i > a i c ( L ) i )= − P i > a i (Ad ( H ))( c ( L ) i )= P i > ( − i − a i · ( ∂ i − H )( tc ( L ) × i ) . On the other hand, write H ◦ H ′ = ( G ′′ − G ′′ ) | t =1 , where G ′′ = a · H (0) and G ′′ = P i > ( − i − a i · ( ∂ i − H )( tx × i ) . It follow thatAd ( H ◦ H ′ )( c ( L )) = G ′′ ( c ( L )) = P i > ( − i − a i · ( ∂ i − H )( tc ( L ) × i ) = (Ad ( H ) ◦ Ad ( H ′ ))( c ( L )) . If r ∈ R = CK R ( k ), thenAd ( H ◦ H ′ )( r ) = G ′′ · r = a · H (0) · r = Ad ( H )( a · r ) = (Ad ( H ) ◦ Ad ( H ′ ))( r ) . Overall, Ad ( H ◦ H ′ ) = Ad ( H ) ◦ Ad ( H ′ ). (cid:3) The polynomials A m := (1 − x ) m satisfy Ad n ( A m ) = Ψ m in OP n,nR . It follows fromLemma 4.21 and Example 4.9 that A k ◦ A m = A km = A m ◦ A k for all k and m . Proposition 4.22. Let R be a commutative ring and K a Q -algebra. Then (1) The composition ◦ in R [[ x ]] is commutative. (2) The power series lg r ( x ) ∈ K [[ x ]] , r > , are orthogonal idempotents that partitionthe identity, that is, lg n ( x ) ◦ lg m ( x ) = δ n,m · lg n ( x ) and − x = P r > lg r ( x ) .Proof. (1) It follows from the definition that the power series x n ◦ G and G ◦ x n arecontained in x n R [[ x ]] for all n and G . Let H, G ∈ R [[ x ]]. Fix an integer n > H = H + H and G = G + G , where H and G are linear combinations of the Adamspolynomials A i and H , G ∈ x n R [[ x ]]. As H and G commute, the remark above yields H ◦ G − G ◦ H ∈ x n R [[ x ]]. Since this holds for all n , we have H ◦ G = G ◦ H . (2) The iterated derivative ∂ i (lg n ( x )) is zero if i > n and( ∂ n − lg n )( x , . . . , x n ) = n Q i =1 log(1 − x i ) . It follows that lg n ( x ) ◦ x m = 0 if m > n andlg n ( x ) ◦ x n = ( − n ( ∂ n − lg n )( x × n ) = ( − n (log(1 − x )) n = ( − n n ! lg n ( x ) . This calculation together with the first part of the proposition show that the power serieslg r ( x ) are orthogonal idempotents.Write 1 − x = P i > a i lg i ( x ) for a i ∈ K . Composing with lg n ( x ) we get a n = 1 for all n . (cid:3) Theorems 4.18 and 4.19 together with Proposition 4.22 yield the following corollary. Corollary 4.23. The rings OP n,n b Z and OP n,n are commutative. Let K be a Q -algebra. We view K [[ x ]] as a ring with respect to addition and compo-sition. Let G ∈ K [[ x ]] and write G = P i > a i lg i for (unique) a i ∈ K . It follows fromProposition 4.22 that the map(4.24) b : K [[ x ]] → K [0 , ∞ ) , taking G to the sequence ( a i ) i > is a ring isomorphism. It takes x n K [[ x ]] onto K [ n, ∞ ) forevery n . Example 4.25. The image of the polynomial A m ( x ) = (1 − x ) m is equal to (1 , m, m , . . . ).Indeed substituting y = log(1 − x ) in the equality e my = P i > m i y i i ! yields A m ( x ) = P i > m i lg i ( x ).4.6. Topology. In this section we introduce three topologies on b Z [[ x ]]. Proposition 4.26. Let G ∈ OP n,nR and m > n . The following conditions are equivalent: (1) G ∈ Im( OP n,mR → OP n,nR ) ; (2) G is zero on every smooth variety of dimension < m .Proof. (1) ⇒ (2) Since CK mR ( X ) = 0, for any variety X of dimension < m , the operation G is zero on X .(2) ⇒ (1) Let n > 1. By Proposition 4.11(2), the operation G is given by a double-symmetric power series H ( x , ..., x n ) ∈ R [[ x , ..., x n ]] ( n ) such that H = ( G n ) | t =1 . We needto prove that v ( H ) > m . We will show that any monomial x r = x r · . . . · x r n n of H with P i r i < m is zero.Consider X r := Q i P r i . This is a variety of dimension < m . Write x i for the first Chernclass in CK R ( X r ) of the pull-back of the canonical line bundle on P r i with respect to the i -th projection X r → P r i . By formula (4.7),0 = G ( x · . . . · x n ) = G n ( x , . . . , x n ) ∈ CK nR ( X r ) . By Projective Bundle Theorem,CK nR ( X r ) = R [[ x , . . . , x n ]] / ( x r +11 , . . . , x r n +1 n ) . Therefore the monomial x r of H is trivial.The case n (cid:3) PERATIONS IN CONNECTIVE K -THEORY 23 Corollary 4.27. Let d > be an integer and G ∈ OP n,n . Then there is a Z -linearcombination G ′ ∈ OP n,n of the Adams operations Ψ k with k = 0 , . . . , d such that G and G ′ agree on CK n ( X ) for all smooth varieties X of dimension d .Proof. By Lemma 3.16, applied to m = d + 1, there is a polynomial G ′ ∈ Z [ x ] of degreeat most d such that G − G ′ ∈ P We introduce three topologies on b Z [[ x ]]: • τ s is generated by the neighborhoods of zero U m consisting of power series divisibleby x m , for some m > 0, i.e., τ s is the x -adic topology. • τ w is generated by the neighborhoods of zero U m + V N , where V N consists of allpower series divisible by some N ∈ N . • τ o is generated by the neighborhoods of zero W m consisting of power series, wherethe respective operation acts trivially on varieties of dimension < m .Recall, that a topology ϕ is coarser than the topology ψ , denoted ϕ ψ , if any setopen with respect to ϕ is also open with respect to ψ . Proposition 4.29. τ w τ o τ s .Proof. Since v ( G ( x )) > m implies v ( ∂ n − G ( x )) > m and hence G ∈ Im( OP n,m b Z → OP n,n b Z )by Theorem 4.12. Therefore, it follows from Proposition 4.26 that τ o τ s .The topology τ w is generated by the neighborhoods of zero U N,m = ( N, x m ) ⊂ b Z [[ x ]],and τ o is generated by the neighborhoods of zero W k = { G ∈ b Z [[ x ]] | v ( ∂ n − ( G )) > k } byProposition 4.26. We need to show that for every N and m there is k with W k ⊂ U N,m .We have similar compact (Hausdorff) topology τ w on b Z [[ x , . . . , x n ]] so that the map ∂ n − is continuous in τ w . Note that the map ∂ n − : b Z [[ x ]] → b Z [[ x , . . . , x n ]] is injectiveand the induced map from b Z [[ x ]] to the image of ∂ n − is a homeomorphism (since theimage of every closed subset is closed as b Z [[ x ]] is compact and the target is Hausdorff).In particular, if G k ∈ b Z [[ x ]] is a sequence such that the sequence ∂ n − ( G k ) converges to0, then the sequence G k converges to 0 in b Z [[ x ]].Now we prove that for every N and m there is k with W k ⊂ U N,m . Assume on thecontrary that for every k we can find G k ∈ W k , but G k / ∈ U N,m . Then ∂ n − ( G k ) convergesto 0, but G k does not converge to 0 in b Z [[ x ]], a contradiction. (cid:3) Observation 4.30. 1) For n = 1, τ o = τ s ;2) For n > τ w = τ o = τ s . Proof. 1) This follows from Proposition 4.26, since n = 1.2) For n > W m contains, in particular, all power series P i a i x i ∈ b Z [[ x ]], where a = ia i , for all 0 < i < m , which is not contained in any U l , for l > 1. Thus, τ o = τ s .For m > n > W m /U m is a free b Z -module of rank ( n − U m + V N ) /U m is afree b Z -module of rank ( m − τ w = τ o . (cid:3) We view OP n,n and OP n,n b Z as the topological rings for the topologies τ w , τ o and τ s respectively via the inclusions OP n,n ֒ → OP n,n b Z ֒ → b Z [[ x ]].Note that the x -adic topology τ s can be defined on R [[ x ]] for every R .Consider the restriction b : R [[ x ]] → K [0 , ∞ ) of the map (4.24). We view K [0 , ∞ ) as atopological ring with the basis of neighborhoods of zero given by the ideals K [ n, ∞ ) for all n > 0, so that the map is continuous. Proposition 4.31. The image of the map b : R [[ x ]] → K [0 , ∞ ) is contained in R [0 , ∞ ) .Proof. By Example 4.25, the image of the Adams polynomial A m under the map (4.24) iscontained in R [0 , ∞ ) . But the set of all linear combinations of Adams polynomials is densein R [[ x ]] in the topology τ s . The statement follows since R [0 , ∞ ) is closed in K [0 , ∞ ) . (cid:3) Proposition 4.31 identifies the ring OP n,n b Z ⊂ b Z [[ x ]] with a subring of b Z [ n, ∞ ) and OP n,n with a subring of Z [ n, ∞ ) if n > 0. Indeed, if n > Q n b Z λ n,n −−→→ OP n,n b Z b −→ b Z [0 , ∞ ) → b Z [ n, ∞ ) is generated by lg r with 0 < r < n and all these logarithms are contained in the kernel of λ n,n .The ring OP n,n is not a domain: we have (Ψ + Ψ − )(Ψ − Ψ − ) = 0. Let e ± = (Ψ ± Ψ − ) ∈ OP n,n [ ] , so e + and e − are orthogonal idempotents and e + + e − = 1. There is an embedding OP n,n ֒ → OP n,n [ ] = OP n,n [ ] e + × OP n,n [ ] e − . Proposition 4.32. If n > the rings OP n,n (cid:2) (cid:3) e ± are domains.Proof. Recall that there is an injective ring homomorphism b : OP n,n ֒ → Z [1 , ∞ ) such that b (Ψ m ) = ( m, m , m , . . . ) for all m . In particular, b ( e + ) = (0 , , , , . . . ) and b ( e − ) = (1 , , , , . . . ) . Lemma 4.33. Let ( a , a , . . . ) ∈ Im( b ) . Then for any prime integer p , we have a i ≡ a j modulo p if i ≡ j modulo p − .Proof. It suffices to prove the statement for b (Ψ m ). We have a i − a j = m i − m j = m j ( m i − j − m is not divisible by p , then m i − j − p . (cid:3) Let G · H = 0 in OP n,n . Set ( a , a , . . . ) = b ( G ) and ( b , b , . . . ) = b ( H ). We have a i b i = 0 for all i . To prove the statement it suffices to show that if a i = 0 for some i , then b j = 0 for all j ≡ i modulo 2.Choose an odd prime p that does not divide a i . By Lemma 4.33, a j is not divisible by p for all j such that i ≡ j modulo p − 1. In particular a j = 0, hence b j = 0. Thus, wehave proved that b j = 0 for all j ≡ i modulo p − Lemma 4.34. There are infinitely many primes q such that gcd( q − , p − 1) = 2 . PERATIONS IN CONNECTIVE K -THEORY 25 Proof. Let c be the odd part of p − p − /c is a 2-power). By Dirichlet, thereare infinitely many primes q such that q ≡ q ≡ c . Clearly,gcd( q − , p − 1) = 2 for such q . (cid:3) Let j be such that j ≡ i modulo 2. We need to prove that b j = 0. Take any prime q as inLemma 4.34. There are positive integers k and m such that t := i +( p − k = j +( q − m .We have proved that b t = 0 since t ≡ i modulo p − 1. By Lemma 4.33, 0 = b t ≡ b j modulo q , i.e., b j is divisible by q . We have proved that b j is divisible by infinitely many primes q , hence b j = 0. (cid:3) Operations in graded K -theory. In this section we determine the R -module ofall R -linear operations G : K ngrR → K mgrR for any pair of integers n and m denoted by OP n,mR ( K ∗ gr ). Recall that K ngrR = K grR · t − n = CK R · t − n , hence by Theorem 4.12, OP n,mR ( K ∗ gr ) = OP , R ( K ∗ gr ) · t m − n = OP , R (CK ∗ ) · t m − n = R [[ x ]] · t m − n . Recall that product operation in the ring OP n,mR ( K ∗ gr ) = R [[ x ]] is the composition ◦ (seeSection 4.5). Moreover, R [[ x ]] is a (topological) bi-algebra over R with co-product definedby the rule (1 − x ) n → (1 − x ) n ⊗ (1 − x ) n for all n > Ψ ⊗ Ψ in thelanguage of operations.Let us describe the dual bi-algebra A (over Z ) of co-operations as follows. Let A be thesubring of the polynomial ring Q [ s ] consisting of all polynomials f such that f ( a ) ∈ Z forall a ∈ Z . In particular, Z [ s ] ⊂ A . The polynomials e n := 1 n ! ( − s )(1 − s ) . . . ( n − − s ) = ( − n (cid:18) sn (cid:19) ∈ A for all n > A as an abelian group. Consider a pairing A ⊗ R [[ x ]] → R, a ⊗ G 7→ h a, G i ∈ R, such that h e n , x m i = δ n,m . This pairing identifies R [[ x ]] with the dual co-algebra for A viathe isomorphism Hom Z ( A, R ) ∼ → R [[ x ]] , taking a homomorphism α : A → R to the power series P n > α ( e n ) x n . Lemma 4.35. For every polynomial f ∈ A , we have h f, (1 − x ) m i = f ( m ) .Proof. We may assume that f = e n for some n . Then h f, (1 − x ) m i = h e n , (1 − x ) m i = ( − n (cid:18) mn (cid:19) = e n ( m ) = f ( m ) . (cid:3) The lemma shows that a co-operation f evaluated at the Adams operation Ψ m is equalto f ( m ).It follows from Lemma 4.35 that h s n , (1 − x ) km i = ( km ) n = k n · m n = h s n , (1 − x ) k i · h s n , (1 − x ) m i . As the composition in R [[ x ]] satisfies (1 − x ) k ◦ (1 − x ) m = (1 − x ) km , the composition in R [[ x ]] is dual to the co-product of A taking s n to s n ⊗ s n in A ⊗ A .The equality h s i + j , (1 − x ) m i = m i + j = m i · m j = h s i , (1 − x ) m i · h s j , (1 − x ) m i shows that the product in A is dual to the co-product in R [[ x ]]. Thus, the bi-algebra R [[ x ]] of operations is dual to the bi-algebra A of co-operations. Remark 4.36. The polynomial ring Z [ s ] is a bi-algebra with respect to the co-product s → s ⊗ s . The dual bi-algebra over R is R [0 , ∞ ) . The dual of the embedding Z [ s ] → A isthe homomorphism b : R [[ x ]] → R [0 , ∞ ) defined in Proposition 4.31 since by Lemma 4.35, h s n , (1 − x ) m i = m n = h s n , b ((1 − x ) m ) i as b ((1 − x ) m ) = (1 , m, . . . , m n , . . . ).5. Multiplicative operations Definition 5.1. A multiplicative operation G : A ∗ → B ∗ is a morphism of functors from Sm k to the category of rings. That is, the ring structure is respected. (We don’t assumethat G is a graded ring homomorphism.)If A ∗ and B ∗ are cohomology theories over k , to any multiplicative operation G : A ∗ → B ∗ we can assign the morphism ( ϕ G , γ G ) : ( A ∗ ( k ) , F A ) → ( B ∗ ( k ) , F B ) of the respectiveformal group laws, where ϕ G : A ∗ ( k ) → B ∗ ( k ) is the restriction of G to Spec( k ) and γ G ( x ) ∈ xB ∗ ( k )[[ x ]] is defined by the condition: G ( c A ( O (1))) = γ G ( c B ( O (1))) ∈ B ∗ ( P ∞ ) = B ∗ ( k )[[ x ]] . (The power series γ G ( x ) /x is called the inverse Todd genus of G .) Theorem 5.2. ([11, Theorem 6.9]) Let A ∗ be a theory of rational type and B ∗ any orientedcohomology theory. Then the assignment G ( ϕ G , γ G ) is a bijection between the set ofmultiplicative operations G : A ∗ → B ∗ and the set of morphisms of formal group laws. Example 5.3. Let R be either Z , Z p or b Z and b ∈ R . The Adams operation Ψ b : CK ∗ R → CK ∗ R is homogeneous and multiplicative. The corresponding map ϕ is the identity and γ = − (1 − tx ) b t . If c ∈ R × , write Ψ cb for the homogeneous multiplicative twisted Adamsoperation with ϕ ( t ) = ct and γ = − (1 − tx ) bc ct (in particular, Ψ b = Ψ b ). It follows from theequality Ψ cb ( tx ) = Ψ cb ( t )Ψ cb ( x ) = ct · γ ( x ) = 1 − (1 − tx ) bc that on CK nR the operation Ψ cb isequal to c − n · Ψ bc . For any c ∈ R , let Ψ c be the homogeneous multiplicative operation with ϕ ( t ) = ct and γ = 0. This operation is zero in positive degrees and is equal to c n · rankon CK − nR = ( K ) R for n > ∗ R → CK ∗ R which is identity on CK R ,multiplication by t n : CK nR → CK R if n > nR → CK R (inverse to multiplication by t − n ) if n 0. This operation is not homogeneous and itsimage is CK R . Set e Ψ cb := Θ ◦ Ψ cb . This is a multiplicative operation with image in CK R .The corresponding function ϕ ( t ) = c and γ = − (1 − tx ) bc c .Introduced operations satisfy the following relations: Ψ = e Ψ andΨ cb ◦ Ψ ed = Ψ cebd ; Ψ cb ◦ e Ψ ed = e Ψ ecbd ; e Ψ cb ◦ Ψ ed = e Ψ cebd ; e Ψ cb ◦ e Ψ ed = e Ψ ecbd . Over Q every formal group law is isomorphic to the additive one. Hence, for everytheory C ∗ , we have isomorphisms of formal group laws.(id , exp C ) : ( C ∗ ⊗ Z Q , F C ) . . ( C ∗ ⊗ Z Q , F add ) : (id , log C ) n n PERATIONS IN CONNECTIVE K -THEORY 27 Suppose that the coefficient ring B ∗ ( k ) of the target theory has no torsion. Then thecomposition (id , exp B ) ◦ ( ϕ G , γ G ) ◦ (id , log A ) identifies the set of multiplicative operations A ∗ → B ∗ with a subset of morphisms of formal group laws ( A ∗ ⊗ Z Q , F add ) → ( B ∗ ⊗ Z Q , F add ). The latter morphism is defined by ( ψ, γ ), where, in our case, ψ = ϕ ⊗ Z Q , forsome ring homomorphism ϕ = ϕ G : A ∗ ( k ) → B ∗ ( k ) and γ ( x ) = b · x , for some b ∈ B ∗ ( k ).In other words, ( ϕ G , γ G ) = (id , log B ) ◦ ( ϕ G , γ ) ◦ (id , exp A ). Then γ G ( x ) = ϕ G (exp A )( b · log B ( x )) . Multiplicative operations in CK . For A ∗ = B ∗ = CK ∗ b Z we have: A = B = b Z [ t ], F A = F B = x + y − txy andlog CK ( x ) = log(1 − tx ) t , exp CK ( z ) = 1 − e zt t . Note that a ring homomorphism ϕ from b Z [ t ] to a ring T such that T n> nT = 0 isuniquely determined by ϕ ( t ) in T . Indeed, suppose that ϕ and ψ satisfy ϕ ( t ) = ψ ( t ). Forany f ∈ b Z [ t ] and n > f = g + nh for some g ∈ Z [ t ] and h ∈ b Z [ t ]. Then ϕ ( g ) = ψ ( g )and hence ϕ ( f ) − ψ ( f ) ∈ nT . Since this holds for all n > 0, we have ϕ ( f ) − ψ ( f ) = 0 forall f .Thus, the map ϕ G : b Z [ t ] → b Z [ t ] is determined by ϕ G ( t ) = c ( t ) ∈ b Z [ t ]. Let b = b ( t ) ∈ b Z [ t ]. Note that any choice of b ( t ) and c ( t ) gives a morphism of rational formal group lawsand so, a multiplicative operation G : CK ∗ b Z ⊗ Z Q → CK ∗ b Z ⊗ Z Q with γ G ( t, x ) = 1 − (1 − tx ) b ( t ) c ( t ) t c ( t ) = P n > ( − n − ( tx ) n (cid:0) b ( t ) c ( t ) t n (cid:1) c ( t ) , which lifts to an operation CK ∗ b Z → CK ∗ b Z if and only if the coefficients of our power seriesbelong to b Z . The coefficient at x n is(5.4) a n = ( − n − b ( t ) Q n − k =1 ( b ( t ) c ( t ) − kt ) n ! . Denote as b p ( t ) , c p ( t ) the Z p -components of our polynomials. If deg( b p ( t ) c p ( t )) > p , the leading term of our t -polynomial will be clearly non-integral (for some n ).Similarly, if for some p , the constant term of b p ( t ) c p ( t ) is non-zero, then the smallestterm of the p -component of our t -polynomial will be non-integral, for some n . Hence, thepolynomial b ( t ) c ( t ) is linear. Then, for a given prime p , either b p ( t ) = b p and c p ( t ) = c p t ,or b p ( t ) = b p t and c p ( t ) = c p , for some b p , c p ∈ Z p . Then the Z p - component of ourcoefficient is: ( a n ) p = ( − n − t m b p (cid:0) b p c p − n − (cid:1) n , where m = n − , or m = n. If b p = 0, then this will be integral for all n if and only if c p ∈ Z × p , while if b p = 0, then c p can be an arbitrary element from Z p . Let us denote the ( Z p -components of) operationswith m = n − c p b p , while the ones with m = n as e Ψ c p b p (see Example 5.3; we suppress p from notations). Here Ψ c p b p respects the grading on CK ∗ Z p , while e Ψ c p b p maps CK ∗ Z p to CK Z p .The pairs ( b p , c p ) run over the set ( Z p \ × Z × p ∪ { } × Z p and, in addition, Ψ = e Ψ . Thus, any multiplicative operation G on CK ∗ b Z = × p CK ∗ Z p splits into the product × p G ( p ) of operations on CK ∗ Z p , where each G ( p ) is one of the Ψ c p b p or e Ψ c p b p . Let P will be the setof prime numbers and J ⊂ P be the subset of those primes, for which ( b p , c p ) = (0 , G ( p ) is e Ψ. Then the data ( J, b, c ), where the p -components of b, c ∈ b Z are b p and c p ,determines our operation G . Let us call it J Ψ cb . Here ( J, b, c ) runs over all possible triplessatisfying: 1) b p = 0 ⇒ c p ∈ Z × p and 2) ( b p , c p ) = (0 , ⇒ p / ∈ J .The operations ∅ Ψ b are (non-twisted) Adams operations with ϕ G = id, which naturallyform a ring isomorphic to b Z . These operations commute with every other operation. Theoperations ∅ Ψ c are invertible and form a group isomorphic to b Z × . Below we will suppress J = ∅ from notations and will denote the respective operations simply as Ψ cb .The formulas in Example 5.3 show that the monoid of multiplicative operations isnon-commutative.5.2. Multiplicative operations in K gr over Z . For A ∗ = B ∗ = K ∗ gr we have: A = B = Z [ t, t − ], F A = F B = x + y − txy . Similar calculations as in the previous section showthat the coefficient a n in (5.4) will belong to Z [ t, t − ] for every n , if and only if b ( t ) c ( t ) islinear in t . Thus, c ( t ) = ct l , for c = ± l ∈ Z , and b ( t ) = bt − l , for some b ∈ Z .Then the coefficient a n is ( − n − t n − l (cid:0) bcn (cid:1) c . Denote this operation as l Ψ cb . It scales the grading on K ∗ gr by the coefficient l . So, onlythe operations Ψ cb are homogeneous.The case c ( t ) = t and b ( t ) = b , that is, Ψ b corresponds to the Adams operation Ψ b -see [11, Sect. 6.3]. In this case ϕ G = id . The operation − Ψ is an automorphism of order2 acting identically on K gr and mapping t to t − .We will omit l and c from the notation l Ψ cb when these will be equal to 1.6. Stable operations To be able to discuss stability of operations, we need the notion of a suspension. Follow-ing Voevodsky and Panin-Smirnov we can introduce the category of pairs SmOp whoseobjects are pairs ( X, U ), where X ∈ Sm k and U is an open subvariety of X - see [11, Def.3.1], with the smash product:( X, U ) ∧ ( Y, V ) := ( X × Y, X × V ∪ U × Y ) , and the natural functor Sm k → SmOp given by X ( X, ∅ ). Then suspension can bedefined as: Σ T ( X, U ) := ( X, U ) ∧ ( P , P \ . Any theory A ∗ extends from Sm k to SmOp by the rule: A ∗ (( X, U )) := Ker( A ∗ ( X ) → A ∗ ( U )) . Any additive operation A ∗ → B ∗ on Sm k extends uniquely to an operation on SmOp .An element ε A = c A ( O (1)) ∈ A ∗ (( P , P \ σ AT : A ∗ (( X, U )) ∼ = −→ A ∗ +1 (Σ T ( X, U )) , PERATIONS IN CONNECTIVE K -THEORY 29 given by x x ∧ ε A . Definition 6.1. For any additive operation G : A ∗ → B ∗ we define its desuspension asthe unique operation Σ − G : A ∗ → B ∗ such that G ◦ σ AT = σ BT ◦ Σ − G. Definition 6.2. A stable additive operation G : A ∗ → B ∗ is the collection { G ( n ) | n > } of operations A ∗ → B ∗ such that G ( n ) = Σ − G ( n +1) . Proposition 6.3. Suppose, G : A ∗ → B ∗ is a multiplicative operation with γ G ( x ) ≡ bx modulo x for some b ∈ B ∗ ( k ) . Then Σ − G = b · G .Proof. We have: G ( σ AT ( u )) = G ( u ∧ ε A ) = G ( u ) ∧ G ( ε A ) = G ( u ) ∧ ( b · ε B ) = σ BT ( b · G ( u )). (cid:3) We call a multiplicative operation G stable if the constant sequence ( G, G, G, . . . ) isstable. By Proposition 6.3, G is stable if and only if the linear coefficient of γ G is equalto 1 - cf. [11, Proposition 3.8].For a commutative ring R define the operatorΦ = Φ R : R [[ x ]] → R [[ x ]] , Φ( G ) = ( x − 1) d G d x , Stable operations in CK over b Z . Recall that in the case A ∗ = B ∗ = CK ∗ ˆ Z , thegroup OP n,n b Z of additive operations for n n > b Z [[ x ]],respectively, x b Z [[ x ]]. Proposition 6.4. The desuspension operator Σ − : OP n,n b Z → OP n − ,n − b Z is given by therule Σ − ( G ) = (cid:26) Φ( G ) , if n ; ∂ (Φ( G )) = Φ( G ) − Φ( G )(0) , if n > .Proof. The Adams operation Ψ k is identified with the power series A k ( x ) = (1 − x ) k if n − x ) k − n > 0. By Proposition 6.3, Σ − Ψ k = k Ψ k , so the formulaholds for G = Ψ k .The map Σ − is continuous in τ o and the map Φ is continuous in τ s . Hence both mapsare continuous as the maps τ s → τ o . Since τ o is Hausdorff (as τ w is), it follows thatthe set of power series where Σ − and Φ coincide is closed in τ s . But the set of linearcombinations of Adams operations is everywhere dense in τ s . (cid:3) It follows from Proposition 6.4 that the desuspension map Σ − is injective and yields atower of injective maps in the other direction:(6.5) b Z [[ x ]] = OP , b Z Σ − ←−− OP , b Z Σ − ←−− . . . Σ − ←−− OP n,n b Z Σ − ←−− . . . . Moreover, the group OP st b Z of homogeneous degree 0 stable operations CK ∗ b Z → CK ∗ b Z thatis the limit of the sequence 6.5 is naturally isomorphic to the group S := ∩ n Im(Φ n ) = ∩ n Im((Σ − ) n ) ⊂ b Z [[ x ]] . Indeed, if { G ( n ) | n > } is a stable operation, then G (0) = Φ n ( G ( n ) ) for every n , hence G (0) ∈ S . Conversely, given G ∈ S , write G = Φ n ( H ( n ) ) for every n . Since Ker(Φ n )consists of constant power series only, the sequence G ( n ) = Φ( H ( n +1) ) is a stable operation. Lemma 6.6. Let G ∈ x b Z [[ x ]] and n > . Then (1) ∂ n ( G ) has coefficients in Z if and only if ∂ n − (Φ( G )) has coefficients in Z . (2) v ( ∂ n ( G )) > m for some m if and only if v ( ∂ n − (Φ( G ))) > m − .Proof. ( ⇒ ) Follows from Proposition 2.5 for both (1) and (2).( ⇐ ) Simply write H k for ( x − k d k G d x k . We claim that ∂ n − ( H k ) has coefficients in Z incase (1) and v ( ∂ n − ( H k )) > m − k in case (2) for every k > 1. We prove the statementsby induction on k .( k ⇒ k + 1) We have H k +1 = Φ( H k ) − kH k , hence ∂ n − ( H k +1 ) = ∂ n − (cid:0) Φ( H k ) (cid:1) − k∂ n − ( H k ) . Then k∂ n − ( H k ) has coefficients in Z in case (1) and v ( k∂ n − ( H k )) > m − k in case (2)by the induction hypothesis. As the derivative ∂ n ( H k ) has coefficients in Z in case (1)and v ( ∂ n ( H k )) > m − k in case (2), it follows from Proposition 2.5, applied to the powerseries H k , that ∂ n − (Φ( H k )) also has coefficients in Z in case (1) and v (cid:0) ∂ n − (Φ( H k )) (cid:1) > m − k − ∂ n − ( H k +1 ) has coefficients in Z in case (1) and v ( ∂ n − ( H k +1 )) > m − k − H k are divisible by k ! in b Z . It follows that the power series k ! ∂ n − ( H k ) have coefficients in Z in case (1). By Proposition 2.5, ∂ n ( G ) has coefficientsin Z in case (1) and v ( ∂ n ( G )) > m in case (2). (cid:3) In particular, we can describe the integral operations OP n,m as follows. Proposition 6.7. Let G ∈ x b Z [[ x ]] and m > n > . Then G ∈ OP n,m if and only if Φ n ( G ) ∈ Z [[ x ]] and v (Φ n ( G )) > m − n .Proof. Theorem 4.12 and iterated application of Lemma 6.6 show that G ∈ OP n,m if andonly if ∂ (Φ n − ( G )) ∈ Z [[ x ]] and v ( ∂ (Φ n − ( G ))) > m − n + 1. Thus, it suffices to provethe following for a power series H ∈ b Z [[ x ]] and integer k > ∂ ( H ) ∈ Z [[ x ]] ⇐⇒ Φ( H ) ∈ Z [[ x ]],2. v ( ∂ ( H )) > k + 1 ⇐⇒ v (Φ( H )) > k .If ∂ ( H ) ∈ Z [[ x ]], then clearly Φ( H ) ∈ Z [[ x ]]. Conversely, if Φ( H ) ∈ Z [[ x ]], then ∂ ( H ) ∈ Q [[ x ]] ∩ b Z [[ x ]] = Z [[ x ]]. The second statement follows from the obvious equality v ( ∂ ( H )) = v (Φ( H )) + 1. (cid:3) Let m a positive integer. It follows from Lemma 6.6(2) that there is a tower of inclusionsas in (6.5):(6.8) x m b Z [[ x ]] = OP ,m b Z ← ֓ OP ,m +1 b Z ← ֓ . . . ← ֓ OP n,n + m b Z ← ֓ . . . . and for every n the intersection of OP n,n + m b Z and OP n +1 ,n +1 b Z in OP n,n b Z coincides with OP n +1 ,n + m +1 b Z . Therefore, we obtain: Proposition 6.9. The group of homogeneous degree m stable operations CK ∗ b Z → CK ∗ + m b Z is naturally isomorphic to the intersection x max(0 ,m ) b Z [[ x ]] ∩ S . PERATIONS IN CONNECTIVE K -THEORY 31 The map Φ : b Z [[ x ]] → b Z [[ x ]] is continuous in τ w and the space b Z [[ x ]] is compactHausdorff. Hence Im(Φ n ) is closed in b Z [[ x ]] for any n . It follows that the set S is alsoclosed in b Z [[ x ]] in the topology τ w and hence in τ o and τ s .It follow from Proposition 4.26 and Lemma 6.6 that the topology on OP st b Z induced by τ o is generated by the neighborhoods of zero W m consisting of all collections { G ( n ) | n > } such that G ( n ) acts trivially on varieties of dimension < n + m . We still denote thistopology by τ o .Let A r ( x ) = (1 − x ) r ∈ b Z [[ x ]] for r ∈ b Z . Note that Φ( A r ) = r · A r . In particular, if r isinvertible in b Z , then A r ∈ S .We can describe the set S via divisibility conditions on the coefficients of the powerseries. Theorem 6.10. The set S = ∩ r Im(Φ r ) ⊂ b Z [[ x ]] consists of all power series G = P i > a i x i satisfying the following property: for every prime p and every positive inte-gers n and m such that m is divisible by p n , for every nonnegative j < m divisible by p ,the sum P m − i = j (cid:0) ij (cid:1) a i is divisible by p n .Proof. Let n be a positive integer, G ∈ S and write G = Φ n ( H ) for some H ∈ b Z [[ x ]].Consider the ideal I = ( p n , x m ) ⊂ b Z [[ x ]], where m is divisible by p n . Note that Φ( I ) ⊂ I since p n divides m .Let G ′ be the x m -truncation of G and H ′ the x m -truncation of H . As G − G ′ ∈ I and H − H ′ ∈ I , we have G ′ − Φ n ( H ′ ) ∈ I . Since G ′ and Φ n ( H ′ ) are polynomials of degreeless than m , we conclude that G ′ and Φ n ( H ′ ) are congruent modulo p n .We write G ′ and H ′ as polynomials in y = x − 1. Since Φ n ( y i ) = i n y i , the y i -coefficientsof Φ n ( H ′ ) are divisible by p n for all i divisible by p . It follows that the same propertyholds for G ′ . As G ′ = m − P i =0 a i x i = m − P i =0 a i ( y + 1) i = m − P i =0 a i i P j =0 (cid:18) ij (cid:19) y j = m − P j =0 y j m − P i = j (cid:18) ij (cid:19) a i , the divisibility condition holds.Conversely, as b Z = Q Z p , it suffices to prove the statement over Z p . Let G ∈ Z p [[ x ]]satisfy the divisibility condition in the theorem. Choose n and m such that m is divisibleby p n and set I = ( p n , x m ) ⊂ Z p [[ x ]] as above. Recall that Φ( I ) ⊂ I . Let F be the x m -truncation of G . By assumption, we can write F ≡ P b i y i modulo p n , where the sumis taken over i < m that are prime to p . In particular, G ≡ P b i y i modulo I .Choose r > F ′ = P b i i r y i . Then Φ r ( F ′ ) = P b i y i ≡ G modulo I , i.e., G isin the image of Φ r modulo I . As Im(Φ r ) is closed in Z p [[ x ]] in the topology τ w , we have G ∈ Im(Φ r ) for all r , i.e., G ∈ S . (cid:3) Stable operations in CK over Z . Now we turn to the study of stable operationsover Z . Proposition 6.11. The pre-image of OP n,n under Σ − : OP n +1 ,n +1 b Z → OP n,n b Z is equal to OP n +1 ,n +1 for every n > .Proof. As Σ − = ∂ ◦ Φ, for n > 1, and Σ − = Φ, for n = 0, this follows immediately fromProposition 6.7. (cid:3) Thus, we have a tower Z [[ x ]] = OP , ← ֓ OP , ← ֓ . . . ← ֓ OP n,n ← ֓ . . . , given by the desuspension and the group OP st of stable homogeneous degree 0 integraloperations is identified with S := S ∩ Z [[ x ]], where S is described by Theorem 6.10.Applying Proposition 6.7 again we get: Proposition 6.12. The group of homogeneous degree m stable operations CK ∗ → CK ∗ + m is naturally isomorphic to the intersection x max(0 ,m ) Z [[ x ]] ∩ S . We would like to determine the structure of S . Lemma 6.13. For every n > there is a positive integer d such that dx n ∈ S + x n +1 b Z [[ x ]] .Proof. Choose distinct elements r , . . . , r n ∈ b Z × such that r i − r j ∈ Z for all i and j . The x i -coefficients with i = 0 , , . . . , n of the power series A r j ( x ) = (1 − x ) r j ∈ S form an( n + 1) × ( n + 1) Van der Monde type matrix (cid:2) ( − i (cid:0) r j i (cid:1)(cid:3) . Its determinant d is a nonzerointeger since all r i − r j are integers. It follows that there is a b Z -linear combination of the A r j ’s that is equal to dx n modulo x n +1 . (cid:3) Note that any ideal in b Z that contains a non-zero integer is generated by a positiveinteger (the smallest positive integer in the ideal). It follows from Lemma 6.13 that forevery n > d n such that the ideal of all a ∈ b Z withthe property ax n ∈ S + x n +1 b Z [[ x ]] is generated by d n . We will determine the integers d n below.For every n > G n ∈ S such that G n ≡ d n x n modulo x n +1 . Lemma 6.14. Let G = P i > a i x i ∈ S be such that a , . . . , a n − ∈ Z . Then there exist b i ∈ b Z for all i > n such that G − P i > n b i G i ∈ S .Proof. Find an integer a ′ n such that a n − a ′ n is divisible by d n , thus, a n = a ′ n + d n b n forsome b n ∈ b Z . Then the x i -coefficients of G − b n G n are integer for i = 0 , . . . , n . Continuingthis procedure, we determine all b i for i > n , so that all coefficients of G − P i > n b i G i areintegers. (cid:3) Theorem 6.15. For all n > there are power series F n ∈ S such that F n ≡ d n x n modulo x n +1 . Moreover, (1) The group S consists of all infinite linear combinations P n > a n F n with a n ∈ Z . (2) The group of homogeneous degree m stable operations CK ∗ → CK ∗ + m is naturallyisomorphic to the group of all infinite linear combinations P n > max(0 ,m ) a n F n with a n ∈ Z .Proof. Fix an n > 0. The coefficient d n of G n is integer. Applying Lemma 6.14, we find b i ∈ b Z for i > n + 1 such that F n := G n − P i > n +1 b i G i ∈ S . Statements (1) and (2) areclear. (cid:3) PERATIONS IN CONNECTIVE K -THEORY 33 The integers d n . Our next goal is to determine the integers d n . Let n > r write L r for the n -tuple of binomial coefficients: (cid:16) (cid:18) r (cid:19) , (cid:18) r (cid:19) , . . . , (cid:18) rn − (cid:19) (cid:17) = (1 , r, . . . ) ∈ Z n . For a n -sequence ¯ a = ( a , . . . , a n ) of positive integers let d (¯ a ) be the determinant of the n × n matrix with columns L a , L a , . . . , L a n . We have(6.16) d (¯ a ) = (cid:0) Q s>t ( a s − a t ) (cid:1) / n − Q k =1 k ! ∈ Z . Let p be a prime integer. An n -sequence ¯ a is called p -prime if all its terms are primeto p . Let ¯ a ( n )min be the “smallest” strictly increasing p -prime n -sequence(1 , , . . . , p − , p + 1 , . . . ) . Lemma 6.17. Let ¯ a be a p -prime n -sequence that differs from ¯ a ( n )min at one term only.Then d (¯ a ( n )min ) divides d (¯ a ) in the ring of p -adic integers Z p .Proof. Suppose ¯ a is obtained from ¯ a ( n )min by replacing a term a by b . It follows from (6.16)that d (¯ a ) /d (¯ a ( n )min ) = Q ( b − a ′ ) / Q ( a − a ′ ) , where the products are taken over all terms a ′ of ¯ a min but a . Since a is prime to p , theproduct Q ( a − a ′ ) generates the same ideal in Z p as a !( c − a )!, where c is the last term of¯ a min . Similarly, as b is prime to p , the product Q ( b − a ′ ) generates the same ideal in Z p as b ( b − · · · ( b − a + 1) · ( b − a − · · · ( b − c + 1)( b − c ) = ( − c − a a !( c − a )! (cid:18) ba (cid:19)(cid:18) c − bc − a (cid:19) . (cid:3) Corollary 6.18. The integer d (¯ a ( n )min ) divides d (¯ a ( n +1)min ) in Z p .Proof. In the cofactor expansion (Laplace’s formula) of the determinant d (¯ a ( n +1)min ) alongthe last row all minors are divisible by d (¯ a ( n )min ) in view of Lemma 6.17. (cid:3) Write M n for the Z p -submodule of ( Z p ) n generated by the tuples L a , L a , . . . , L a n ,where ( a , a , . . . , a n ) = ¯ a ( n )min . Lemma 6.19. Let b be an integer prime to p . Then the n -tuple L b is contained in M n .In others words, the Z p -submodule of ( Z p ) n generated by L b for all integers b > primeto p coincides with M n .Proof. By Cramer’s rule, the solutions of the equation L b = x L a + . . . + x n L a n are givenby the formula x i = d (¯ a ( i ) ) /d (¯ a ( n )min ), where the sequence ¯ a ( i ) is obtained from ¯ a ( n )min byreplacing the i -th term with b . By Lemma 6.17, we have x i ∈ Z p . (cid:3) The following statement is a generalization of Lemma 6.17. Corollary 6.20. Let ¯ a be any p -prime n -sequence. Then d (¯ a ( n )min ) divides d (¯ a ) in Z p . (cid:3) Set d n = d ( p ) n := d (¯ a ( n +1)min ) /d (¯ a ( n )min ) . By Corollary 6.18, d n ∈ Z p .Write n in the form n = ( p − k + i , where i = 0 , . . . , p − k = ⌊ np − ⌋ . Then itfollows from (6.16) that(6.21) d n Z p = p k · k ! n ! Z p , or, equivalently v p ( d n ) = k + v p ( k !) − v p ( n !) , where v p is the p -adic discrete valuation.Note that v p (( n + k )!) = v p (( pk + i )!) = v p (( pk )!) = k + v p ( k !), hence d n Z p = ( n + k )! n ! Z p .Observe that the function n v p ( d n ) is not monotonic. Proposition 6.22. An ( n + 1) -tuple (0 , , . . . , , d ) is contained in M n +1 if and only if d is divisible by d n in Z p .Proof. As in the proof of Lemma 6.19, (0 , , . . . , , d ) ∈ M n +1 if and only if d · d (¯ a ( i ) )is divisible by d (¯ a ( n +1)min ) in Z p for all i , where the sequence ¯ a ( i ) is obtained from ¯ a ( n )min bydeleting the i -th term in ¯ a ( n +1)min . We have d (¯ a ( i ) ) = d (¯ a ( n )min ) if i = n + 1 and by Corollary6.20, all d (¯ a ( i ) ) are divisible by d (¯ a ( n )min ), whence the result. (cid:3) For an integer r , let as before A r ( x ) = (1 − x ) r . Note that the n -tuple L r is the tuple ofcoefficients (after appropriate change of signs) of the x n -truncation of the polynomial A r .Denote by N ( p ) the Z p -submodule of Z p [ x ] generated by A r for all integers r > p . We get an immediate corollary from Lemma 6.19 and Proposition 6.22: Proposition 6.23. Let d ∈ Z p and n > . Then dx n ∈ N ( p ) + x n +1 Z p [ x ] if and only if d isdivisible by d n . Moreover, there is a Z p -linear combination G n of the Adams polynomials A a , A a , . . . , A a n +1 , where ( a , a , . . . , a n +1 ) = ¯ a ( n +1)min , such that G n ≡ d n x n ( mod x n +1 ) . (cid:3) Proposition 6.24. The set S Z p = ∩ r Im(Φ r Z p ) contains a power series ≡ dx n ( mod x n +1 ) if and only if d is divisible by d n in Z p .Proof. Suppose that G ∈ S Z p and G ≡ dx n modulo x n +1 . Choose integers k > p k is divisible by d (¯ a ( n +1)min ) and m > n divisible by p k and consider the ideal I =( p k , x m ) ⊂ Z p . We have G = Φ k ( G ′ ) for some G ′ ∈ Z p [[ x ]] and write G ′ = m − P i =0 b i A i modulo x m Z p [[ x ]]for some b i ∈ Z p . Applying Φ k and taking into account the equality Φ k ( A i ) = i k A i , weget G = Φ k ( G ′ ) ∈ N ( p ) + I . Taking the x n +1 -truncations, we see that dx n ∈ N ( p ) + x n +1 Z p [ x ] + p k Z p [ x ] . As p k is divisible by d (¯ a ( n +1)min ), we conclude that p k Z p [ x ] ⊂ N ( p ) + x n +1 Z p [ x ], hence dx n ∈ N ( p ) + x n +1 Z p [ x ]. By Proposition 6.23, d is divisible by d n . (cid:3) PERATIONS IN CONNECTIVE K -THEORY 35 Now we turn to the ring b Z . The integers d n defined before Lemma 6.14 are the productsof primary parts of d n = d ( p ) n determined as above for every prime p . In view of (6.21) wehave v p ( d n ) = ⌊ np − ⌋ + v p ( ⌊ np − ⌋ !) − v p ( n !)for every prime p . For example, d = 1, d = 2, d = 2 · d = 2 , d = 2 · · d = 2 · d = 2 · · d = 2 · .Propositions 6.23 and 6.24 yield: Theorem 6.25. Let n > be an integer. Then (1) There is a b Z -linear combination G n of the Adams polynomials A a , A a , . . . , A a n +1 for some a , a , . . . , a n +1 ∈ b Z × such that G n ≡ d n x n modulo x n +1 . (2) The set S = ∩ r Im(Φ r b Z ) contains a power series ≡ dx n ( mod x n +1 ) if and only if d is divisible by d n in b Z . Remark 6.26. It follows from Proposition 6.23 that a , a , . . . , a n +1 ∈ b Z × can be chosenso that for every prime p , we have (( a ) p , ( a ) p , . . . , ( a n +1 ) p ) = ¯ a ( n +1)min with respect to p .In particular, a = 1. Proposition 6.27. The set S = ∩ r Im(Φ r ) is the closure in the topology τ s , and hence,in the topologies τ o and τ w of the set of all (finite) b Z -linear combinations of the powerseries A r for r ∈ b Z × .Proof. Denote as T s , T w , T o the closures of the mentioned set of linear combinations in ourthree topologies. As S is closed in τ w , we have T s ⊂ T o ⊂ T w ⊂ S .Let G ∈ x k b Z [[ x ]] ∩ S . Then by Theorem 6.25, G ≡ dx k ( mod x k +1 ), where d = d k · c ,for some c ∈ b Z . We know that there exists a b Z -linear combination G k of the powerseries A a , A a , . . . , A a k (with invertible a i ’s) such that G k ≡ d k x k ( mod x k +1 ). Hence, G − c · G k ∈ x k +1 b Z [[ x ]] ∩ S . Applying this inductively, we obtain that, for any G ∈ S and any positive integer m , there exists a finite b Z -linear combination H of invertible A r ’s,such that G − H ∈ x m b Z [[ x ]] ∩ S . Therefore, T s = S and hence T s = T o = T w = S . (cid:3) Stable operations in K gr . In section 4.7 we defined the bi-algebra A of co-operationsin K gr . Recall that for a commutative ring R , the bi-algebra of operations OP n,nR ( K gr ) = OP , R (CK) = R [[ x ]] is dual to A . The same proof as in Proposition 6.4 shows that thedesuspension operatorΣ − : R [[ x ]] = OP n,nR ( K gr ) → OP n − ,n − R ( K gr ) = R [[ x ]]coincides with Φ. It follows that OP stR ( K gr ) = lim( R [[ x ]] Φ ←− R [[ x ]] Φ ←− R [[ x ]] Φ ←− . . . ) . Lemma 6.28. The desuspension operator Φ is dual to the multiplication by s in A .Proof. As Φ((1 − x ) m ) = m (1 − x ) m , in view of Lemma 4.35 we have h e n , Φ((1 − x ) m ) i = h e n , m (1 − x ) m i = m · e n ( m ) = h se n , (1 − x ) m i . (cid:3) The localization A [ s ] can be identified with colim( A s −→ A s −→ . . . ). Therefore, OP stR ( K gr ) ≃ Hom( A h s i , R ) , i.e., the bi-algebra OP stR ( K gr ) of stable operations is dual to A [ s ].The bi-algebra A [ s ] coincides with the algebra of degree 0 stable operations K ( K )in topology (see [6, Proposition 3] and [2]). Moreover, A [ s ] is a free abelian group ofcountable rank [1, Theorem 2.2] and can be described as the set of all Laurent polynomials f ∈ Q [ s, s − ] such that f ( ab ) ∈ Z [ ab ] for all integers a and b = 0.It follows that the bi-algebra A [ s ] admits an antipode s s − that makes A [ s ] a Hopfalgebra. It follows that OP stR ( K gr ) is a (topological) Hopf algebra. Remark 6.29. We have a diagram of homomorphisms of bi-algebras and its dual: Z [ s ] (cid:15) (cid:15) / / A (cid:15) (cid:15) R [0 , ∞ ) R [[ x ]] b o o Z [ s, s − ] / / A [ s ] R Z O O OP stR ( K gr ) O O o o The bottom maps are homomorphisms of Hopf algebras. The antipode of R Z takes asequence r i to r − i .The group of degree 0 stable operations OP st b Z ( K gr ) coincides with OP st b Z (CK) = S whosestructure was described in Theorem 6.25. Our nearest goal is to determine the structureof OP st Z ( K gr ). We remark that this group is different from OP st Z (CK) = S ∩ Z [[ x ]].Let R be one of the following rings: Z , Z p or b Z . Recall that we have an injectivehomomorphism b R : R [[ x ]] → R [0 , ∞ ) taking (1 − x ) m to the sequence (1 , m, m , . . . ).The operation Φ on R [[ x ]] corresponds to the shift operation Π on R [0 , ∞ ) defined byΠ( a ) i = a i +1 .An n -interval of a sequence a in R [0 , ∞ ) or R Z is the n -tuple ( a i , a i +1 , . . . , a i + n − ) forsome i . We say that this interval starts at i .For every n > 1, let M n be the R -submodule of R n generated by the n -tuples ¯ r :=(1 , r, r , . . . , r n − ) for all integers r > 0. Note that M n is of finite index in R n . Lemma 6.30. A sequence a ∈ R [0 , ∞ ) belongs to the image of b R if and only if for every n > , the n -interval of a starting at is contained in M n .Proof. The implication ⇒ is clear. For the converse note that by assumption a is containedin the closure of Im( b R ). On the other hand, if R = Z p or b Z , the space R [[ x ]] is compactin τ w and R [0 , ∞ ) is Hausdorff, hence Im( b R ) is closed, i.e., a ∈ Im( b R ). If R = Z , itfollows from the case R = b Z that a = b b Z ( G ) for some G ∈ b Z [[ x ]]. Since at the same time G ∈ Q [[ x ]], we have G ∈ Z [[ x ]]. (cid:3) Let T R ⊂ R Z be the R -submodule of all sequences a ∈ R Z such that every n -intervalof a is contained in M n for all n > 1. If a ∈ T R , by Lemma 6.30, for every n > G n ∈ R [[ x ]] such that b R ( G n ) = ( a − n , a − n +1 , . . . ). Since Φ( G n +1 ) = G n , the PERATIONS IN CONNECTIVE K -THEORY 37 sequence ( G n ) n > determines an element in OP stR ( K gr ). This construction establishes anisomorphism OP stR ( K gr ) ≃ T R . Note that T b Z = OP st b Z (CK) = S = ∩ r Im(Φ r ).For every n > 1, let N n be the R -submodule of R n generated by the n -tuples ¯ r for all r ∈ R × . Then N n is of finite index in R n if R = Z p or b Z .Note that every n -tuple ¯ r with r ∈ R × extends to the sequence a with a i = r i that iscontained in T R . Lemma 6.31. N n ⊂ M n for all n > .Proof. It suffices to consider the case R = Z p . Choose an integer m > p m · Z np ⊂ M n . Let r ∈ Z × p . Find an integer r ′ > r modulo p m . Thenthe tuple ¯ r = (1 , r, r , . . . , r n − ) is congruent to ¯ r ′ modulo p m . Hence ¯ r = ¯ r ′ + (¯ r − ¯ r ′ ) ∈ M n + p m Z np ⊂ M n . (cid:3) It follows from Lemma 6.31 that every element in N n is an n -interval of a sequence in T R . Proposition 6.32. If R = Z p or b Z , the R -module T R consists of all sequences a ∈ R Z such that every n -interval of a is contained in N n for all n > .Proof. We may assume that R = Z p . Let a ∈ T R . In view of Lemma 6.31 it suffices toshow that every n -interval v of a starting at i is contained in N n for all n > 1. Take aninteger m > n + m )-interval w of Π − m ( a ) starting at i , so that v is thepart of w on the right. Write w as a (finite) linear combination P t r ¯ r over positive integers r , where t r ∈ Z p and ¯ r = (1 , r, r , . . . , r n + m − ) ∈ M n + m − . Applying Π m to Π − m ( a ) wesee that v = P t r r m ˆ r , where ˆ r = (1 , r, r , . . . , r n − ) ∈ M n − . As r m is divisible by p m if r is divisible by p , it follows from the definition of N n that v ∈ N n + p m M n . Since N n is offinite index in M n , we can choose m such that p m M n ⊂ N n , hence a ∈ N n . (cid:3) Denote by θ : R Z → R Z the reflection operation taking a sequence a to the sequence θ ( a ) i = a − i . Corollary 6.33. The module T R is invariant under θ .Proof. In the case R = Z p or b Z it suffice to notice that if r ∈ R × , the symmetric n -tuple( r n − , r n − , . . . , r, 1) = r n − (1 , r − , ( r − ) , . . . , ( r − ) n − ) is contained in N n . If R = Z thestatement follows from the equality T Z = T b Z ∩ Z Z . (cid:3) Now let R = b Z and n > 0. The ideal of all t ∈ b Z such that (0 , . . . , , t ) ∈ N n is generatedby a (unique) positive integer ˜ d n = n ! · d n , where the integers d n were introduced in Section6.3. We know that v p ( ˜ d n ) = v p (( n + k p )!)for all primes p , where k p = ⌊ np − ⌋ . By Theorem 6.15, there are power series F n ∈ S = S ∩ Z [[ x ]] such that F n ≡ d n x n modulo x n +1 .Let f ( n ) ∈ T b Z be the image of F n under the map S = OP st b Z ( K gr ) → b Z Z . Thus,(0 , . . . , , ˜ d n ) is the n -interval of f ( n ) starting at 0. For example, we can choose: f (0) = ( . . . , , , , , . . . ) , f (1) = ( . . . , , , , , . . . ) . As in the proof of Theorem 6.15, modifying f ( n ) by adding multiples of the shifts of f ( m ) for m > n and their reflections we can obtain f ( n ) ∈ Z Z for all n . Theorem 6.34. Every sequence a ∈ T Z ≃ OP st Z ( K gr ) can be written in the form a = ∞ X i =0 h b i Π − i θ ( f (2 i ) ) + b i +1 Π i ( f (2 i +1) ) i for unique b , b , . . . ∈ Z .Proof. We determine the integers b , b , . . . inductively so that for every m > P mi =0 of the terms in the right hand side and the sequence a have the same 2 m +2-intervalsstarting at − m . (cid:3) Remark 6.35. Observe that { Π − i θ ( f (2 i ) ) , Π i ( f (2 i +1) ) | i ∈ Z > } is also a topological basisof OP st b Z ( K gr ). Note that, at the same time, { f ( j ) | j ∈ Z > } form a topological basis for OP st Z (CK) and OP st b Z (CK).6.5. Stable multiplicative operations. We first consider stable multiplicative opera-tions on CK ∗ b Z . From Proposition 5.3 we obtain: Proposition 6.36. Stable multiplicative operations CK ∗ b Z → CK ∗ b Z are exactly operations Ψ c , for c ∈ b Z × . These are invertible and form a group isomorphic to b Z × . Similarly, stablemultiplicative operations on CK ∗ form a group isomorphic to Z × . Restricted to CK b Z , the operation Ψ c is given by G = 1 (as it is multiplicative and so,maps 1 to 1), while G ( tx ) = G ( t ) G ( x ) = ct · γ G ( x ) = 1 − (1 − tx ) c and so, our operationcorresponds to the power series A c = (1 − x ) c . In other words, on CK b Z , operation Ψ c coincides with the Adams operation Ψ c . Then on CK n b Z it is equal to c − n · Ψ c .Proposition 6.27 gives: Corollary 6.37. The set of homogeneous stable additive operations on CK ∗ b Z is the clo-sure in the topology τ o of the set of (finite) b Z -linear combinations of stable multiplicativeoperations. Remark 6.38. Note that the respective statement for Z -coefficients is not true, as thereare only two stable multiplicative operations on CK ∗ , namely, Ψ and Ψ − , while thegroup of stable additive operations there has infinite (uncountable) rank.Now we consider stable multiplicative operations on K ∗ gr over Z . Proposition 6.39. Stable multiplicative operations K ∗ gr → K ∗ gr are exactly operations Ψ c ,for c = ± . These are invertible and form a group isomorphic to Z × ∼ = Z / Z .Proof. The linear coefficient of γ G for the operation l Ψ cb is t − l b - see 5.2. This will beequal to 1 exactly when l = 1 and b = 1. (cid:3) As above, the operation Ψ c corresponds to the power series A c = (1 − x ) c . 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Oriented cohomology theories of algebraic varieties. vol. 30. 2003, pp. 265–314. Specialissue in honor of Hyman Bass on his seventieth birthday. Part III.[9] Smirnov, A. Orientations and transfers in cohomology of algebraic varieties. St. Petersburg Math.J. 18 , 2 (2007), 305–346.[10] Strong, M.-J., and Whitehouse, S. Integer-valued polynomials and K -theory operations. Proc.Amer. Math. Soc. 138 , 6 (2010), 2221–2233.[11] Vishik, A. Stable and unstable operations in algebraic cobordism. Ann. Sci. ´Ec. Norm. Sup´er. (4)52 , 3 (2019), 561–630. Alexander Merkurjev, Department of Mathematics, University of California, LosAngeles, CA, USA E-mail address : [email protected] Alexander Vishik, School of Mathematical Sciences, University of Nottingham, Uni-versity Park, Nottingham, NG7 2RD, United Kingdom E-mail address ::