The plethory of operations in complex topological K-theory
aa r X i v : . [ m a t h . K T ] J a n THE PLETHORY OF OPERATIONS IN COMPLEX TOPOLOGICAL K -THEORY WILLIAM MYCROFT AND SARAH WHITEHOUSE
Abstract.
We provide a concrete introduction to the topologised, graded analogue ofan algebraic structure known as a plethory, originally due to Tall and Wraith. Stacey andWhitehouse showed this structure is present on the cohomology operations for a suitablegeneralised cohomology theory. We compute an explicit expression for the plethory ofoperations for complex topological K -theory. This is formulated in terms of a plethoryenhanced with structure corresponding to the looping of operations. In this context weshow that the familiar λ -operations generate all the operations. Introduction
Cohomology operations provide a very powerful piece of structure associated with a gen-eralised cohomology theory and over the years they been used to prove many deep resultsin algebraic topology. However, despite the ubiquity of cohomology operations, there aresome challenges in identifying the appropriate algebraic framework in which to encode therich structure the operations admit.Historically, cooperations , the homological analogue of operations, have often been theobjects of interest and in many cases of interest they encode the same information. Theseare amenable to study via the medium of Hopf rings and many useful results have beenproved that way. Unfortunately, a Hopf ring contains no algebraic structure which naturallycorresponds to composition of operations. To address this, Boardman, Johnson and Wilsonintroduced the notion of an enriched Hopf ring which enhances the structure with an externalaction encoding the missing information [7]. Enriched Hopf rings of cooperations have beencomputed for many interesting theories, including complex K -theory [7, Theorem 17.14].However, this structure can be somewhat cumbersome for performing computations as theenrichment is not easily expressed in terms of generators and relations.An alternative approach proves fruitful. Roughly speaking, operations act non-linearlyon cohomology algebras, and this determines the structure in the same way that (not nec-essarily commutative) k -algebras are precisely the objects which act on k -modules. Therelevant abstract algebraic structure was first introduced in 1970 by Tall and Wraith [15],and subsequently studied by Bergman and Hausknecht [4], and by Borger and Wieland [8]who coined the term plethory . A priori, cohomology operations do not naturally fit intothis framework due to considerations of grading and topologies on the algebraic structures.However, as shown by Stacey and Whitehouse [12], in sufficiently nice cases the operationsadmit the structure of a graded topologised plethory and this acts on completed cohomologyalgebras. A related approach by Bauer considers formal plethories , thus avoiding comple-tion issues [3]. All this should be viewed as an algebraic shadow of corresponding structure Date : January 7, 2020.2010
Mathematics Subject Classification.
Key words and phrases. topological K -theory, K -theory operations, lambda operations, lambda ring,plethory. in the world of spectra and there is current work towards developing a theory of spectralplethories.Of course, in the case of complex topological K -theory, there is a long tradition of workwith cohomology operations, often formulated in terms of λ -operations or Adams operations.The ring of symmetric functions provides a basic example of a plethory, whose algebras are λ -rings. Yau has related the enriched Hopf ring approach to that of filtered λ -rings [16],restricting attention to the degree zero part of complex K -theory. Working with p -adiccoefficients, Bousfield’s theory of p -adic θ -rings captures the structure [9], and work of Rezk,again in a p -complete setting, extends this to exhibit the relevance of plethories to poweroperations at higher chromatic heights [11].The main aim of this paper is to give a concise full description of the integral operationsof complex topological K -theory in plethystic terms. We first give a direct proof of theapplication of plethories to cohomology operations which illuminates exactly where topolog-ical issues arise. We then extend our algebraic gadgets to encode the looping of operationsarising in the topological context. Applying our technical framework to the study of theoperations of complex topological K -theory yields our main result, in particular showinghow the λ -operations generate all K -theory operations.The main result is Theorem 5.7. This describes the operations as a Z / Z -plethorywith looping, in terms of the plethory of symmetric functions and the plethory of set mapsfrom Z to Z .This paper is organised as follows. Section 2 covers plethories in a graded and topologisedcontext. The (completed) plethory structure of set maps from a ring to itself, such as Set ( Z , Z ), is discussed here. Section 3 covers the plethory of operations in ungraded K -theory and looping is discussed in Section 4. The main result appears in Section 5.Throughout, rings and algebras will be assumed to be (graded) commutative and unitalunless stated otherwise. 2. Topological plethories
We generalise the theory of plethories [15, 8] to a suitably graded and topologised context.This variant is needed to capture the structure on cohomology operations. We assumefamiliarity with [8] and our focus is on the differences in the graded, topologised case.Fix a commutative monoid Z , typically ( Z , +) or ( Z / , +), used for grading.Let k and k ′ be Z -graded rings. Let Alg k be the category of k -algebras and let CAlg k bethe category of filtered k -algebras which are complete Hausdorff under the filtration topology,meaning that the completion map A → b A = lim ← A/F a A is an isomorphism. Morphisms arecontinuous k -algebra maps of degree zero. We write b ⊗ for the completed tensor productover k . Further details can be found in [6, Section 6]. Definition 2.1.
The category
CBiring k,k ′ of complete Hausdorff k - k ′ -birings is the cate-gory of co- k ′ -algebra objects in CAlg k . To be explicit, an object in this category consistsof a Z -graded collection of complete Hausdorff k -algebras B • = ( B n ) n ∈ Z together with HE PLETHORY OF OPERATIONS IN COMPLEX TOPOLOGICAL K -THEORY 3 continuous k -algebra maps for each n ∈ Z ∆ + : B n → B n b ⊗ B n (co-addition) ε + : B n → k (co-zero) σ : B n → B n (co-additive inverse) ∆ × : B n → Y i + j = n B i b ⊗ B j (co-multiplication) ε × : B → k (co-unit)and for each κ ∈ k ′ , γ ( κ ) : B → B (co- k ′ -linear structure)satisfying the usual relations for a co- k ′ -algebra object [15, 6].We use the above notation for the co-algebraic structure maps of a biring throughout thispaper.It is customary [6, 8] to consider the equivalent formulation of the co- k ′ -linear structuregiven by β ( κ ) = ε × ◦ γ ( κ ), where we set ε × ( b ) = 0 for b ∈ B n with n = 0. Endowing Alg k ( B • , k ) with the Z -graded ring structure determined by the other maps above, thisyields a map of Z -graded rings β : k ′ → Alg k ( B • , k ). (However, this alternative descriptionis unavailable in the case of non-(co-unital) co- k ′ -algebra objects, where we have no ε × .)A complete Hausdorff k - k ′ -biring B is naturally Z - Z -bigraded, with gradings induced bythe gradings on k and k ′ . By an element x ∈ B , we mean x ∈ B n for some n ∈ Z . For x ∈ B n , we define the • -degree by deg • ( x ) = n and the ∗ -degree to be deg ∗ ( x ) = | x | ∈ Z ,the degree of x in the graded k -algebra B n . We can recover the ungraded context as thespecial case Z = 0, the trivial monoid.We make extensive use of sumless Sweedler notation [14], writing∆ + ( x ) = x (1) ⊗ x (2) , ∆ × ( x ) = x [1] ⊗ x [2] . In the untopologised case, the algebro-geometric viewpoint of k - k ′ -birings as representablefunctors Alg k → Alg k ′ turns out to give very useful intuition. This naturally generalisesto the topologised setting via the language of formal schemes. We only need affine schemes,so we use the following definitions [13], where we use varieties of algebras in the sense ofuniversal algebra [5]. Definition 2.2. A k -scheme is a covariant representable functor X : Alg k → Set . Given avariety of algebras V , if X has a specified lift to a functor Alg k → V , we say the lift, oftenalso denoted X , is a V -algebra k -scheme . If A denotes the representing k -algebra, we write X = Spec k ( A ) = Alg k ( A, − ).A formal k -scheme is a functor X : Alg k → Set which is a small filtered colimit of k -schemes. If X has a specified lift to a functor Alg k → V , we say the lift, often also denoted X , is a V -algebra formal k -scheme . Given a filtered k -algebra A , we define the formal k -scheme Spf k ( A ) = lim −→ a Alg k ( A/F a A, − ) . This construction is functorial, giving a contravariant functor Spf k from filtered k -algebrasto formal k -schemes. WILLIAM MYCROFT AND SARAH WHITEHOUSE
It is worth noting that Spf k ( A ) = Spf k ( b A ), i.e. Spf k ( − ) is blind to completions. Withoutgoing into detail, we remark that both k -schemes and formal k -schemes preserve complete-ness and Hausdorff properties; see [12]. For example, if X : Alg k → Alg k ′ is a (formal) k ′ -algebra k -scheme then X restricts to a functor CAlg k → CAlg k ′ . Definition 2.3.
A formal k -scheme X is solid if it is isomorphic to Spf k ( A ) for some filtered k -algebra A . Proposition 2.4.
The functor
Spf k gives an anti-equivalence between complete Hausdorff k - k ′ -birings B and solid formal k ′ -algebra k -schemes.Proof. The functor Spf k is an anti-equivalence from CAlg k to the category of solid formal k -schemes and the result follows by restricting to co- k ′ -algebra objects. (cid:3) Example 2.5.
As in [8, Example 1.2(1)], k is the initial k - k ′ -biring, with all structure mapsgiven by the identity map of k . The corresponding functor is the constant functor at thezero ring. Example 2.6.
When Z = Z , the identity functor Alg k → Alg k can be expressed as Spf k ( I )where I n = ( k [ ι n ] n evenΛ k [ ι n ] n odd , | ι n | = n and each I n has the discrete topology. The structure maps are given by∆ + ( ι n ) = 1 ⊗ ι n + ι n ⊗ ε + ( ι n ) = 0 σ ( ι n ) = − ι n ∆ × ( ι n ) = X r + s = n ι r ⊗ ι s ε × ( ι n ) = ( n = 00 otherwiseand for each κ ∈ k , β ( κ )( ι n ) = ( κ | κ | = n Example 2.7.
The collection of set maps
Set ( k ′ , k ) endowed with the topology arising fromthe pro-finite filtration { ker ( Set ( k ′ , k ) → Set ( k ′ a , k )) | k ′ a ⊆ k ′ , k ′ a finite subring } naturally admits the structure of an ungraded complete Hausdorff k - k ′ -biring. The k -algebrastructure is induced by the k -algebra structure on k and the co- k ′ -algebra structure is inducedby the k ′ -algebra structure on k ′ .For example, the co-addition is given by the map Set ( k ′ , k ) Set (+ ,k ) −−−−−−→ Set ( k ′ × k ′ , k ) ∼ = Set ( k ′ , k ) b ⊗ Set ( k ′ , k ) . HE PLETHORY OF OPERATIONS IN COMPLEX TOPOLOGICAL K -THEORY 5 The formal k ′ -algebra k -scheme Spf k ( Set ( k ′ , k )) is naturally isomorphic to the functor of complete orthogonal idempotents given on k -algebras by COI k ′ ( A ) = ( ( x i ) ∈ Y i ∈ k ′ A | X i x i = 1 , x i = x i , x i x j = 0 for i = j ) . The addition and multiplication are specified by π l (( x i ) + ( y j )) = X i + j = l x i y j π l (( x i )( y j )) = X ij = l x i y j , where l ∈ k ′ and π l denotes the canonical projection Q i ∈ k ′ A → A to the component indexedby l . The zero in COI k ′ ( A ) is ( δ i ) i ∈ k ′ and the 1 is ( δ i ) i ∈ k ′ , where δ id is the Kronecker deltafunction. The topology is given by the filtration ideals consisting of sequences containingfinitely many non-zero elements. The identification Spf k ( Set ( k ′ , k )) ∼ = COI k ′ ( − ) is givenby the natural isomorphism which sends χ d to ( δ id ) i ∈ k ′ , where χ d is the indicator functoron { d } ⊆ k ′ .When A contains no zero divisors, we have COI k ′ ( A ) ∼ = k ′ . In fact, Spf k ( Set ( k ′ , k )) is thenearest solid formal k ′ -scheme to the constant k ′ -algebra scheme A k ′ ; see [3, Section 4].In various applications, we frequently encounter non-(co-unital) k - k ′ -birings , correspond-ing to representable functors from Alg k to Alg ! k ′ , the category of non-unital k ′ -algebras ortheir topological generalisations. At the level of algebras, it is standard to remedy the lackof a unit via unitalisation : given a non-unital k -algebra R , one forms the k -module k ⊕ R together with the obvious multiplication. More generally, if S is unital and R is additionallyan S -module then the coproduct of k -modules S ⊕ R is naturally a unital k -algebra withmultiplication given by( s + r )( s + r ) = ( s s + r r + s · r + s · r )and unit 1 S + 0 R .This construction has an analogue in the context of algebra schemes. If Spec k ( B ) isa representable non-unital k ′ -algebra scheme, Spec k ( B ′ ) is a unital k ′ -algebra scheme, andSpec k ( B )( A ) is naturally a Spec k ( B ′ )( A )-module, then the functor Spec k ( B ′ ⊗ B ) is a unital k ′ -algebra scheme given, up to natural isomorphism, on objects by A Spec k ( B ′ )( A ) ⊕ Spec k ( B )( A ) . At the level of the representing objects this translates to a B ′ -comodule structure on B .The comultiplication ∆ × on B ′ ⊗ B is given by the image of the identity map of B ′ ⊗ B ⊗ B ′ ⊗ B under the compositeSpec k ( B ′ ⊗ B ⊗ B ′ ⊗ B )( A ) ∼ = Spec k ( B ′ ⊗ B )( A ) × Spec k ( B ′ ⊗ B )( A ) µ −→ Spec k ( B ′ ⊗ B )( A ) , where µ is the multiplication and A = B ′ ⊗ B ⊗ B ′ ⊗ B . Using this, one can compute anexplicit formula for ∆ × and similarly for the counit ǫ × . Denoting the coaction B → B ′ ⊗ B by y y { } ⊗ y { } , we find the following formulas for the comultiplication and counit of the k - k ′ -biring structure on B ′ ⊗ B .∆ × ( x ⊗ y ) = x [1] y (2) { } ⊗ y (1)[1] y (3) { } ⊗ x [2] y (3) { } ⊗ y (1)[2] y (2) { } ε × ( x ⊗ y ) = ε × ( x ) ε + ( y ) . This construction generalises without difficulty to our topologised framework, replacingschemes with formal schemes and completing tensor products.
WILLIAM MYCROFT AND SARAH WHITEHOUSE
Example 2.8.
Let B be a non-(co-unital) k - k -biring and let A be a k -algebra. The non-unital k -algebra Spf k ( B )( A ) naturally admits a Spf k ( Set ( k, k ))( A )-module structure which,after identifying Spf k ( Set ( k, k ))( A ) with COI k ( A ), is given by(( a λ ) · φ )( b ) = X λ φ ( γ ( λ )( b )) a λ . Here φ ∈ Spf k ( B )( A ), λ ∈ k , ( a λ ) ∈ COI k ( A ), b ∈ B and γ specifies the co- k -linear structureof B .This translates to a Set ( k, k )-comodule structure on B given by b X λ χ λ ⊗ γ ( λ )( b )and thus, Set ( k, k ) b ⊗ B is naturally a k - k -biring with structure maps ∆ × , ε × determined by∆ × ( χ d ⊗ b ) = X rs = d χ r ⊗ b (1)[1] γ ( s )( b (2) ) ⊗ χ s ⊗ b (1)[2] γ ( r )( b (3) ) ε × ( f ⊗ b ) = ε × ( f ) ε + ( b ) . We generalise the composition product ⊙ [8, 15], which represents the composition offunctors, to the graded topologised setting in two stages, first adding the grading and thenthe topology. The grading will allow us to model operations between graded objects andthe topological setting allows us to consider only the continuous operations.Just as with the tensor product of algebras, the composition product of a complete Haus-dorff biring with a complete Hausdorff algebra is not necessarily complete Hausdorff withrespect to the canonical filtration. As with the tensor product, this is remedied by takingthe completion. Definition 2.9.
For a complete Hausdorff k - k ′ -biring B and complete Hausdorff k ′ -algebra A , we define the complete Hausdorff composition product b ⊙ as follows. First, take thequotient of B ⊙ A by the ideal generated by the relations b ⊙ a = 0 whenever deg • ( b ) = | a | .The grading on B ⊙ A is specified by | b ⊙ a | = deg ∗ ( b ) = | b | . Now define B b ⊙ A to be thecomplete Hausdorff k -algebra B b ⊙ A = lim ←− α,β BF β B ⊙ AF α A together with the canonical filtration where F α A and F β B denote the filtration ideals on A and B respectively.The defining properties of the composition product generalise without difficulty to thegraded topological setting and we have a bifunctor b ⊙ : CBiring k,k ′ × CAlg k ′ → CAlg k .If B and B ′ are complete Hausdorff birings, the bigrading on the composition product isdefined • -componentwise in the sense that ( B b ⊙ B ′ ) n = B b ⊙ B ′ n . Proposition 2.10.
Let B be a complete Hausdorff k - k ′ -biring. The functor B b ⊙− : CAlg k ′ → CAlg k is left adjoint to Spf k ( B ) : CAlg k → CAlg k ′ . (cid:3) Proposition 2.11.
For a complete Hausdorff k - k ′ -biring B and complete Hausdorff k ′ -algebra A , the formal scheme Spf k ( B b ⊙ A ) is given by the composition CAlg k Spf k ( B ) −−−−−→ CAlg k ′ Spf k ′ ( A ) −−−−−→ Set . HE PLETHORY OF OPERATIONS IN COMPLEX TOPOLOGICAL K -THEORY 7 Hence, b ⊙ lifts to a functor CBiring k,k ′ × CBiring k ′ ,k ′′ → CBiring k,k ′′ and ( CBiring k,k , b ⊙ , I ) forms a monoidal category. (cid:3) Proceeding as in the discrete case, we can now define structure which precisely modelscomposition of operations.
Definition 2.12.
We define the category of complete Hausdorff k -plethories CPlethory k to be the category of monoids in CBiring k,k . Explicitly, a complete Hausdorff k -plethory isa complete Hausdorff k - k -biring P together with two additional complete Hausdorff biringmorphisms ◦ : P b ⊙ P → P (composition) u : I → P (identity)satisfying the usual relations for a monoid. Example 2.13.
The initial complete Hausdorff k -plethory is the complete Hausdorff k - k -biring I of Example 2.6 together with the canonical structure maps. Example 2.14.
The complete Hausdorff k - k -biring Set ( k, k ) together with composition ofmaps and the identity map forms an ungraded complete Hausdorff k -plethory. We use ι todenote the identity on k and 1 to denote the constant map k → k sending κ to 1 for all κ ∈ k . Example 2.15.
As detailed in [8], in the discrete setting we have a free functor fromthe category of k - k -birings to the category of k -plethories, analogous to the tensor algebraconstruction over a k -module. In the topological setting, we define T b ⊙ ( B ), the free completeHausdorff k -plethory over a complete Hausdorff k - k -biring B by T b ⊙ ( B ) = [ O n ≥ B b ⊙ n together with the obvious identity and composition.We wish to encode not only the composition of operations, but the actions of operationson suitable algebras. This leads to a result which proves useful for calculations. Definition 2.16.
For a complete Hausdorff k -plethory P , we define the category of com-plete Hausdorff P -algebras to be the category of algebras over the monad P b ⊙− : CAlg k → CAlg k . We write r ( x ) for the image of r ⊙ x under the action map P b ⊙ A → A . Example 2.17.
For a space X , the degree zero complex K -theory, K ( X ) = [ X, Z × BU ],admits the structure of a Set ( Z , Z )-algebra. The action of f ∈ Set ( Z , Z ) sends the class of x : X → Z × BU to the class of the composite X x −→ Z × BU f × −−−→ Z × BU.
Proposition 2.18.
For a complete Hausdorff k -plethory P , the structure maps are completeHausdorff P -algebra maps and so the co-algebraic structure is determined by the action oncomplete Hausdorff P -algebras. For example, if r ∈ P then r ( xy ) = r [1] ( x ) r [2] ( y ) for all x, y in any complete Hausdorff P -algebra A if and only if ∆ × r = r [1] ⊗ r [2] .Proof. See [15, Section 4] for the discrete case, which generalises without difficulty. (cid:3)
We can now give a direct proof of a key result of Stacey and Whitehouse [12, Corollary5.4]. The original proof is an application of a very abstract, but more general result.
WILLIAM MYCROFT AND SARAH WHITEHOUSE
Theorem 2.19.
Let E ∗ ( − ) be a multiplicative cohomology theory. If E ∗ ( E n ) is a free E ∗ -module for each n ∈ Z then E ∗ ( E • ) is a complete Hausdorff E ∗ -plethory. Moreover, for anyspace X the completed cohomology b E ∗ ( X ) is naturally a E ∗ ( E • ) -algebra.Proof. Since each E ∗ ( E n ) is a free E ∗ -module, we have suitable K¨unneth isomorphisms andthus the E ∗ -algebra object structure on ( E n ) n ∈ Z induces a co- E ∗ -algebra structure on thecollection of complete Hausdorff E ∗ -algebras E ∗ ( E n ). Thus, E ∗ ( E • ) is a complete Hausdorff E ∗ - E ∗ -biring. We define a composition ◦ : E ∗ ( E • ) b ⊙ E ∗ ( E • ) → E ∗ ( E • ) by r ◦ s = s ∗ ( r ) andthe unit u : I → E ∗ ( E • ) by u ( ι n ) = ι n ∈ E ∗ ( E n ), the universal class. These maps make E ∗ ( E • ) a complete Hausdorff E ∗ -plethory by construction. (cid:3) As this theory is set up to work with completed cohomology algebras we lose some infor-mation. In general, the completion of a cohomology algebra contains strictly less informationthan the uncompleted algebra. In forming the completion, we take the quotient by the phan-tom classes : those which are zero on any finite subcomplex. In [3], Bauer shows that we canavoid this issue by working with the entire pro-system of cohomology algebras. However inmany cases of interest, there are results that preclude the existence of phantom classes andthus E ∗ ( X ) = b E ∗ ( X ).We introduce some theory of non-(co-unital) birings which will prove useful. For brevity,we focus on the discrete, ungraded case but remark that these constructions generalisewithout difficulty to the topologised, graded setting. Definition 2.20.
We define the non-(co-unital) composition product B (cid:26) A of a non-(co-unital) k - k ′ -biring and a non-unital k ′ -algebra A to be the free unital k -algebra on thesymbols b (cid:26) a , for b ∈ B , a ∈ A , quotiented by the relations enforcing that b b (cid:26) a is a k -algebra map together with the relations b (cid:26) ( a + a ) = ( b (1) (cid:26) a )( b (2) (cid:26) a ) b (cid:26) ( a a ) = ( b [1] (cid:26) a )( b [2] (cid:26) a ) b (cid:26) ( κa ) = γ ( κ )( b ) (cid:26) ab (cid:26) ε + ( b )for all a, a , a ∈ A , b ∈ B and κ ∈ k ′ . Proposition 2.21. If B is a non-(co-unital) k - k ′ -biring, the functor B (cid:26) − : Alg ! k ′ → Alg k is left adjoint to Spec k ( B ) : Alg k → Alg ! k ′ .Proof. This is the same argument as in the co-unital setting. (cid:3)
For k - k ′ -birings B, B ′ and k ′ -algebras A, A ′ we have natural isomorphisms B ⊙ ( A ⊗ A ′ ) ∼ = ( B ⊙ A ) ⊗ ( B ⊙ A ′ ) , ( B ⊗ B ′ ) ⊙ A ∼ = ( B ⊙ A ) ⊗ ( B ′ ⊙ A ) ,k ⊙ B ∼ = k ∼ = B ⊙ k ′ . These have analogues in the non-(co-unital) setting.Let
R, S be non-unital k -algebras. Recall the coproduct R ⊠ S is given by the k -module R ⊕ S ⊕ ( R ⊗ S ) together with multiplication specifed by the product of r + s + r ′ ⊗ s ′ and r + s + r ′ ⊗ s ′ being given by r r + s s + r ⊗ s + r ⊗ s + r r ′ ⊗ s ′ + r ′ r ⊗ s ′ + r ′ ⊗ s s ′ + r ′ ⊗ s ′ s + r ′ r ′ ⊗ s ′ s ′ . Proposition 2.22.
Let B be a non-(co-unital) k - k ′ -biring and A, A ′ non-unital k ′ -algebras.We have isomorphisms B (cid:26) ( A ⊠ A ′ ) ∼ = ( B (cid:26) A ) ⊗ ( B (cid:26) A ′ ) and k (cid:26) A ∼ = k . HE PLETHORY OF OPERATIONS IN COMPLEX TOPOLOGICAL K -THEORY 9 Proof.
For any k -algebra X we have isomorphisms Alg k ( B (cid:26) ( A ⊠ A ′ ) , X ) ∼ = Alg ! k ′ ( A ⊠ A ′ , Spec k ( B )( X )) ∼ = Alg ! k ′ ( A, Spec k ( B )( X )) × Alg ! k ′ ( A ′ , Spec k ( B )( X )) ∼ = Alg k ( B (cid:26) A, X ) × Alg k ( B (cid:26) A ′ , X ) ∼ = Alg k (( B (cid:26) A ) ⊗ ( B (cid:26) A ′ ) , X ) . As in Example 2.5, k is the initial k - k ′ -biring corresponding to the constant functor at thezero ring and the isomorphism k (cid:26) A ∼ = k is trivial. (cid:3) Proposition 2.23.
Suppose B is a non-(co-unital) k - k ′ -biring and a co- B ′ -module where B ′ is a k - k ′ -biring. For an augmented k ′ -algebra A , we have an isomorphism ( B ′ ⊗ B ) ⊙ A ∼ = ( B ′ ⊙ A ) ⊗ ( B (cid:26) IA ) , where IA denotes the augmentation ideal of A .Proof. For any k -algebra X we have isomorphisms Alg k (( B ′ ⊗ B ) ⊙ A, X ) ∼ = Alg k ′ ( A, Spec k ( B ′ ⊗ B )( X )) ∼ = Alg k ′ ( A, Spec k ( B ′ )( X ) ⊕ Spec k ( B )( X )) ∼ = Alg k ′ ( A, Spec k ( B ′ )( X )) × Alg ! k ′ ( IA,
Spec k ( B )( X )) ∼ = Alg k ( B ′ ⊙ A, X ) × Alg k ( B (cid:26) IA, X ) ∼ = Alg k (( B ′ ⊙ A ) ⊗ ( B (cid:26) IA ) , X ) . (cid:3) We write b (cid:26) for the non-(co-unital) composition product in the completed setting.3. Ungraded K -theory operations The study of the operations of ungraded K -theory is a classical subject in algebraictopology [2] and it is well known that the degree zero K -cohomology, K ( X ) = K ( X ), ofa space X naturally forms a λ -ring. In this section we exhibit a concise description of theoperations in a plethystic setting.The classifying space BU of the infinite unitary group is central to the study of K -theory and admits the structure of a non-unital ring space, with abelian group structurecorresponding to the direct sum of vector bundles, and (non-unital) multiplication inducedby the tensor product. Thus, since K ( BU ) is free as a Z -module, K ( BU ) naturally admitsthe structure of a non-(co-unital) complete Hausdorff Z - Z -biring by the ungraded and non-unital analogue of Theorem 2.19. Theorem 3.1.
We have an isomorphism of non-(co-unital) complete Hausdorff Z - Z -birings (1) K ( BU ) ∼ = Z [[ λ ι, λ ι, . . . ]] where ι is represented by the inclusion BU ≃ { } × BU ⊆ Z × BU . The filtration idealsare given by the kernels of the projection maps Z [[ λ ι, λ ι, . . . ]] → Z [[ λ ι, . . . , λ n ι ]] , and thenon-(co-unital) biring structure is determined by ∆ + ( λ k ι ) = X i + j = k λ i ι ⊗ λ j ι ∆ × ( λ k ι ) = P k ( λ ι ⊗ , . . . , λ k ι ⊗
1; 1 ⊗ λ ι, . . . , ⊗ λ k ι ) , where the P k are the universal polynomials arising in the theory of λ -rings, see [17, Definition1.10] . Proof.
The description of K ( BU ) as a power series ring in the lambda operations is well-known and the remaining structure follows directly from the theory of λ -rings. (cid:3) Since Z × BU is the representing space for ungraded K -theory, studying the operationscorresponds to understanding the complete Hausdorff Z -plethory K ( Z × BU ). Proposition 3.2.
We have an isomorphism of (ungraded) complete Hausdorff Z - Z -birings, K ( Z × BU ) ∼ = Set ( Z , Z ) b ⊗ K ( BU ) where the Z - Z -biring structure is specified in Example 2.8.Proof. By the K¨unneth theorem, we have an isomorphism of rings. We write θ : Set ( Z , Z ) b ⊗ K ( BU ) → K ( Z × BU )for this isomorphism. Since the abelian group structure on Z × BU is given by the productstructure, this is an isomorphism of Hopf algebras. It remains to show that θ respectsthe co-multiplicative structure. The element χ d ⊗ x ∈ Set ( Z , Z ) b ⊗ K ( BU ) corresponds to π ∗ χ d π ∗ x under the K¨unneth isomorphism θ , where π , π denote the canonical projections.By Proposition 2.18, we can compute the comultiplication by considering the action of π ∗ χ d π ∗ x on general α, β ∈ K ( X ). Assume that X is connected and thus has a unique upto homotopy choice of base point. Denote the map induced by the inclusion of the basepoint by ε : K ( X ) → Z . The case of general X will follow by considering each connectedcomponent individually. For f ∈ Set ( Z , Z ) and x ∈ K ( BU ), we have π ∗ f ( α ) = f ( ε ( α )) and π ∗ x ( α ) = x ( α − ε ( α )). In K ( X ),( π ∗ χ d π ∗ x )( αβ )= χ d ( ε ( α ) ε ( β )) x [ αβ − ε ( α ) ε ( β )]= X rs = d χ r ( ε ( α )) χ s ( ε ( β )) x [( α − ε ( α )( β − ε ( β )) + ε ( α )( β − ε ( β )) + ε ( β )( α − ε ( α ))]= X rs = d χ r ( ε ( α )) χ s ( ε ( β )) π ∗ (cid:2) x (1)[1] γ ( ε ( β ))( x (3) ) (cid:3) ( α ) π ∗ (cid:2) x (1)[2] γ ( ε ( α ))( x (2) ) (cid:3) ( β )= X rs = d χ r ( ε ( α )) χ s ( ε ( β )) π ∗ (cid:2) x (1)[1] γ ( s )( x (3) ) (cid:3) ( α ) π ∗ (cid:2) x (1)[2] γ ( r )( x (2) ) (cid:3) ( β )= X rs = d (cid:0) π ∗ χ r π ∗ (cid:2) x (1)[1] γ ( s )( x (3) ) (cid:3)(cid:1) ( α ) (cid:0) π ∗ χ s π ∗ (cid:2) x (1)[2] γ ( r )( x (2) ) (cid:3)(cid:1) ( β )where the fourth equality follows since χ i ( j ) = δ ij , the Kronecker delta. Hence∆ × ( π ∗ χ d π ∗ x ) = X rs = d π ∗ χ r π ∗ (cid:2) x (1)[1] ( γ ( s )( x (3) )) (cid:3) ⊗ π ∗ χ s π ∗ (cid:2) x (1)[2] ( γ ( r )( x (2) )) (cid:3) . Therefore the K¨unneth isomorphism respects the comultiplication ∆ × . To see that theco-unit is preserved, notice that ( π ∗ f π ∗ x )(1) = f (1) x (0) = ε × ( f ) ε + ( x ). (cid:3) Recall that for a based space X , the reduced K -theory, which we denote K ( X, o ), is thekernel of the augmentation given by the map induced by the inclusion of the basepoint.
Proposition 3.3.
We have a map of rings K ( BU ) b (cid:26) K ( BU, o ) → K ( BU ) determined by λ i ι ◦ λ j ι = P i,j ( λ ι, . . . , λ ij ι ) , where the P i,j are the universal polynomials arising in the theory of λ -rings, see [17, Defini-tion 1.10] .Proof. This is immediate from the properties of λ -rings. (cid:3) HE PLETHORY OF OPERATIONS IN COMPLEX TOPOLOGICAL K -THEORY 11 For based spaces
X, Y , the cohomological K¨unneth isomorphism induces an isomorphismof non-unital rings on reduced cohomology K ( X × Y, o ) ∼ = K ( X, o ) b ⊠ K ( Y, o ) . Recall that the co-zero map, which defines the augmentation ideal, on
Set ( Z , Z ) is givenby the evaluation map ε + : Set ( Z , Z ) → Z , with ε + ( f ) = f (0). We have an isomorphism I (cid:0) Set ( Z , Z ) b ⊗ K ( BU ) (cid:1) ∼ = I Set ( Z , Z ) b ⊠ K ( BU, o ) . We now define the appropriate composition on
Set ( Z , Z ) b ⊗ K ( BU ) by the following se-quence of maps, where φ R : Set ( Z , Z ) b ⊙ Z → Z and φ L : Z b (cid:26) I Set ( Z , Z ) → Z denote thecanonical isomorphisms. (cid:0) Set ( Z , Z ) b ⊗ K ( BU ) (cid:1) b ⊙ (cid:0) Set ( Z , Z ) b ⊗ K ( BU ) (cid:1)(cid:0) Set ( Z , Z ) b ⊙ Set ( Z , Z ) (cid:1) b ⊗ (cid:0) Set ( Z , Z ) b ⊙ K ( BU ) (cid:1) b ⊗ (cid:0) K ( BU ) b (cid:26) I Set ( Z , Z ) (cid:1) b ⊗ (cid:0) K ( BU ) b (cid:26) K ( BU, o ) (cid:1)(cid:0) Set ( Z , Z ) b ⊙ Set ( Z , Z ) (cid:1) b ⊗ (cid:0) Set ( Z , Z ) b ⊙ Z (cid:1) b ⊗ (cid:0) Z b (cid:26) I Set ( Z , Z ) (cid:1) b ⊗ (cid:0) K ( BU ) b (cid:26) K ( BU, o ) (cid:1) Set ( Z , Z ) b ⊗ Z b ⊗ Z b ⊗ K ( BU ) Set ( Z , Z ) b ⊗ K ( BU ) ∼ =1 b ⊗ b ⊙ ε + b ⊗ ε + b (cid:26) b ⊗ ◦ b ⊗ φ R b ⊗ φ L b ⊗◦∼ = On the level of elements, for d ∈ Z , g ∈ Set ( Z , Z ), x, y ∈ K ( BU ), this reads as( χ d ⊗ x ) ◦ ( g ⊗ y ) = X rs = d χ r ( ε + ( y )) χ s ◦ g ⊗ γ ( s )( x ) ◦ (cid:0) y − ε + y (cid:1) , with identity given by 1 ⊗ λ ι + ι ⊗
1. Note that composition respects sums on the left soit is enough to specify it on the above elements.
Theorem 3.4.
We have an isomorphism of ungraded complete Hausdorff Z -plethories K ( Z × BU ) ∼ = Set ( Z , Z ) b ⊗ K ( BU ) . Proof.
By Proposition 3.2, we have an isomorphism of birings and it remains to checkcompatibility with composition. Let d ∈ Z , g ∈ Set ( Z , Z ), x, y ∈ K ( BU ) and α ∈ K ( X ).We have θ ( χ d ⊗ x ) ◦ θ ( g ⊗ y ) = ( π ∗ χ d π ∗ x ) ◦ ( π ∗ gπ ∗ y )( α )= ( π ∗ χ d π ∗ x )( g ( ε ( α )) y ( α − ε ( α )))= ( π ∗ χ d )( g ( ε ( α )) y ( α − ε ( α )))( π ∗ x )( g ( ε ( α )) y ( α − ε ( α )))= (cid:2) χ d ( g ( ε ( α )) ε + ( y )) (cid:3) (cid:2) γ ( g ( ε ( α )))( x ) ◦ (cid:0) y − ε + ( y ) (cid:1) ( α − ε ( α )) (cid:3) = X rs = d (cid:2) χ r ( ε + ( y )) χ s ( g ( ε ( α ))) (cid:3) (cid:2) γ ( g ( ε ( α )))( x ) ◦ (cid:0) y − ε + ( y ) (cid:1) ( α − ε ( α )) (cid:3) = X rs = d (cid:2) χ r ( ε + ( y )) χ s ( g ( ε ( α ))) (cid:3) (cid:2) γ ( s )( x ) ◦ (cid:0) y − ε + ( y ) (cid:1) ( α − ε ( α )) (cid:3) = X rs = d π ∗ (cid:2) χ r ( ε + ( y )) χ s ◦ g (cid:3) π ∗ (cid:2) γ ( s )( x ) ◦ (cid:0) y − ε + ( y ) (cid:1)(cid:3) ( α )= θ (( χ d ⊗ x ) ◦ ( g ⊗ y )) . Finally, we note that ( π ∗ π ∗ λ ι + π ∗ ιπ ∗ α ) = α − ε ( α ) + ε ( α ) = α . (cid:3) Plethories with looping
The standard definition [6] of a (graded) generalised cohomology theory is a Z -graded col-lection of well-behaved functors E n ( − ) : Ho → Ab together with suspension isomorphisms .For a based space X , the corresponding reduced cohomology groups are denoted E n ( X, o )and are defined as the kernel of the map induced by inclusion of the base point, as we alreadysaw in the case of K -theory. The theory is extended to pairs, by defining the cohomology ofa pair to be the reduced cohomology of the quotient space. The suspension isomorphismscan be expressed as isomorphisms of abelian groups Σ : E n ( X ) → E n +1 ( S × X, o × X )for all spaces X and all n ∈ Z , or equivalently Σ : E n ( X, o ) ∼ = E n +1 (Σ X, o ) on reducedcohomology groups where Σ X = S ∧ X denotes the reduced suspension.The suspension isomorphisms impose additional structure on the algebras over a plethoryof unstable cohomology operations. Since plethories are precisely the structure which actson algebras, we will need extra structure to encode this information.Recall that for a based operation r : E n ( − ) E m ( − ), there is the looped operation Ω r : E n − ( − ) E m − ( − ) defined by the following commutative diagram. E n − ( X ) E n ( S × X, o × X ) E m − ( X ) E m ( S × X, o × X ) ΣΩ r r Σ Definition 4.1.
Let P be a complete Hausdorff k -plethory. We define the augmentationideal IP , primitives Add( P ) and indecomposables Ind( P ) by IP = ker ǫ + , Add( P ) = { x ∈ P | ∆ + ( x ) = 1 ⊗ x + x ⊗ } , Ind( P ) = IP ( IP ) . The additional structure of a plethory induces additional structure on these familiar con-structions from Hopf algebra theory as detailed in the ungraded setting in [8] and the gradedsetting in [10]. These constructions carry over to the topological context without difficulty.
Definition 4.2.
We define a k -plethory with looping to be a complete Hausdorff k -plethory P equipped with a continuous bidegree ( − , − k -module map Ω : IP → IP satisfying thefollowing properties.(1) Ω is zero on ( IP ) and takes values in primitives. That is, Ω factors as IP π −→ Ind( P ) → Add( P ) ⊆ IP , where π denotes the quotient map.(2) For r ∈ IP , ∆ × (Ω r ) = ( − deg ∗ ( r [1] ) σ deg • ( r [1] ) r [1] ⊗ Ω r [2] .(3) For r, s ∈ IP , Ω( r ◦ s ) = Ω r ◦ Ω s .(4) For all n ∈ Z , Ω( ι n ) = ι n − .A map of plethories f : P → P ′ is a map of k -plethories with looping if Ω f ( r ) = f Ω( r )for all r ∈ P . We denote the category of k -plethories with looping by ΩPlethory k . Theorem 4.3.
Let E ∗ ( − ) be a graded cohomology theory. If E ∗ ( E n ) is a free E ∗ -modulefor each n ∈ Z then E ∗ ( E • ) is an E ∗ -plethory with looping.Proof. Looping of operations is defined for based maps and so gives a map from IE ∗ ( E • )to IE ∗ ( E • ), of bidegree ( − , − HE PLETHORY OF OPERATIONS IN COMPLEX TOPOLOGICAL K -THEORY 13 from the definition. For (2), let x, y ∈ b E ∗ ( X ) for some space X and let π : S × X → X denote the canonical projection. To determine the comultiplication of a looped operation,we consider the action on products. By definition we haveΣ(Ω r )( xy ) = r (Σ( xy ))= r (cid:16) ( − | x | ( π ∗ x )Σ y (cid:17) = r [1] (cid:16) π ∗ (( − | x | x ) (cid:17) r [2] (Σ y )= π ∗ (cid:16) ( σ | x | r [1] )( x ) (cid:17) Σ(Ω r [2] )( y )= Σ (cid:16) ( − deg ∗ ( r [1] ) ( σ | x | r [1] )( x )(Ω r [2] )( y ) (cid:17) and thus ∆ × (Ω r ) = ( − deg ∗ ( r [1] ) σ deg • ( r [1] ) r [1] ⊗ Ω r [2] . (cid:3) Definition 4.4. An ideal of a k -plethory with looping is an ideal J of a k -plethory suchthat Ω x ∈ J for all x ∈ J .It is immediate that if J ⊆ P is an ideal of a k -plethory with looping then the canonicalmap P → P/ J is a map of k -plethories with looping.In many settings, we obtain interesting collections of operations by considering loopingsand composites of a small set of operations. Definition 4.5.
Let P be a complete Hausdorff k -plethory. We define the complete Haus-dorff k - k -biring P Ω to be the free k -algebra generated by the symbols Ω x for x ∈ P togetherwith Ω l x for x ∈ IP and l >
0, quotiented by the ideal generated by the relationsΩ ( x + y ) = Ω ( x ) + Ω ( y )Ω ( xy ) = (Ω x )(Ω y )Ω ( κ ) = κ, for κ ∈ k Ω l ( x + y ) = Ω l ( x ) + Ω l ( y )Ω l ( xy ) = ε + ( x )Ω l ( y ) + ε + ( y )Ω l ( x ) . The bigrading is determined by deg ∗ (Ω k x ) = deg ∗ ( x ) − k and deg • (Ω k x ) = deg • ( x ) − k .The identification x Ω x ∈ P Ω yields a canonical k -algebra map P ֒ → P Ω . The biringstructure on P Ω is given by defining the elements Ω k x to be primitive for k >
0, the canonicalmap P → P Ω to be a monomorphism of k - k -birings together with the following formulae for k >
0. ∆ × (Ω k x ) = ( − k deg ∗ ( x [1] ) σ k deg • ( x [1] ) x [1] ⊗ Ω k x [2] ε × (Ω k x ) = ( − k deg ∗ ( x [1] ) ε × (cid:16) σ k deg • ( x [1] ) x [1] (cid:17) βλ (Ω k x ) = ( βλ )( x [1] ) ε × (Ω k x [2] )We define Ω P , the free k -plethory with looping on P to be the complete Hausdorff k -plethory T b ⊙ ( P Ω ) quotiented by the relationsΩ k x ◦ Ω k y = Ω k ( x ◦ y )Ω k ι n = ι n − k . The looping in Ω P is given by Ω(Ω k x ) = Ω k +1 x and a map f : P → P ′ of complete Hausdorff k -plethories induces a map of k -plethories with looping Ω P → Ω P ′ by f (Ω k x ) = Ω k f ( x ).This construction defines a functor Ω : CPlethory k → ΩPlethory k . Proposition 4.6.
The forgetful functor U : ΩPlethory k → CPlethory k is right adjointto Ω .Proof. A map of complete Hausdorff k -plethories f : P → U P ′ defines a map of k -plethorieswith looping ˆ f : Ω P → P ′ by ˆ f (Ω k x ) = Ω k ( f ( x )). Conversely, a map of k -plethories withlooping Ω P → P ′ restricts to a map of complete Hausdorff k - k -plethories P → U P ′ via thecanonical inclusion P → Ω P . (cid:3) K -theory operations as a free plethory with looping We briefly study the K -theory operations of odd source degree. Since complex K -theoryis represented in odd degrees by the infinite unitary group U , this is tantamount to un-derstanding the Hopf algebra K ∗ ( U ). We then relate these results to the λ -operations andshow that in a suitable context, the λ -operations generate all K -theory operations.Write Λ k : U ( n ) → U (cid:0) nk (cid:1) ⊆ U for the exterior power representation of the unitary groupand let µ kn ∈ K − ( U ( n )) denote the class represented by Λ k . Theorem 5.1 ([2, Theorem 2.7.17]) . We have an isomorphism of K ∗ -algebras K ∗ ( U ( n )) ∼ = Λ K ∗ [ µ n , . . . , µ nn ] . Moreover, if i : U ( n − → U ( n ) denotes the standard inclusion map then i ∗ ( µ kn ) = µ kn − + µ k − n − . We remark that the choice of degree for the elements µ kn ∈ K ∗ ( U ( n )) is arbitrary and wecould choose any odd degree. Our selection is motivated by a relation to the even degreeoperations: the looping of the λ -operations will be expressible in terms of the µ kn and wechose the λ -operations to lie in cohomological degree zero.To understand the relationship between the µ kn and our choice of generators of K ( BU )it proves fruitful to understand the representing maps of the λ -operations. By a classicalresult of Anderson [1], there are no phantom operations in K -theory and thus K ( BU ) ∼ =lim ←− n K ( BU ( n )). Let β kn ∈ K ( BU ( n ) , o ) be represented by B Λ k : BU ( n ) → BU ≃ { } × BU ⊆ Z × BU.
Proposition 5.2.
Define λ kn = P ki =0 (cid:0) − ni (cid:1) β k − in ∈ K ( BU ( n ) , o ) . The following hold.(1) For j = Bi : BU ( n ) → BU ( n + 1) , we have j ∗ λ kn +1 = λ kn .(2) The element λ k ι ∈ K ( BU, o ) ∼ = lim ←− n K ( BU ( n ) , o ) corresponds to the inverse limitof the λ kn ∈ K ( BU ( n ) , o ) .Proof. The first result follows immediately since j ∗ β kn +1 = β kn + β k − n . For the second,let X be a compact Hausdorff space, so the representing map for x ∈ K ( X, o ) factors via Z × BU ( n ) for some n . Let x = [ ξ ] − n ∈ K ( X, o ). Now the composition X x −→ Z × BU ( n ) Z × λ kn −−−−→ Z × BU represents the virtual bundle P ki =0 (cid:0) − ni (cid:1) h Λ k − i [ ξ ] − (cid:0) nk − i (cid:1)i = P ki =0 (cid:0) − ni (cid:1) Λ k − i [ ξ ] = ( λ k ι )( x ). (cid:3) This linear combination of generators allows us to compute K ∗ ( U ) = lim ←− K ∗ ( U ( n )) in aform closely related to our description of K ( BU ). Proposition 5.3.
Let l kn = k − P i =0 (cid:0) − ni (cid:1) µ k − in ∈ K − ( U ( n )) for k ≤ n . HE PLETHORY OF OPERATIONS IN COMPLEX TOPOLOGICAL K -THEORY 15 (1) If i : U ( n − → U ( n ) is the inclusion map as above then i ∗ ( l kn ) = l kn − . (2) We have an isomorphism of K ∗ -algebras K ∗ ( U ( n )) ∼ = Λ K ∗ [ l n , . . . , l nn ] . (3) We have an isomorphism of K ∗ -algebras K ∗ ( U ) ∼ = Λ K ∗ [ l , l , . . . ] where if ι : U ( n ) → U denotes the inclusion then ι ∗ l k = l kn .Proof. This follows directly from Theorem 5.1. (cid:3)
The following result is now an immediate consequence and the motivation for the defini-tion of the odd degree operations l k . Corollary 5.4.
The composition
Set ( Z , Z ) b ⊗ K ( BU ) θ −→ K ( Z × BU ) = K ( Z × BU ) Ω −→ K − ( U ) , is determined by f ⊗ λ k ι f (0) l k , for f ∈ Set ( Z , Z ) .Proof. Since Ω( Z × BU ) = Ω( { } × BU ), it suffices to consider the restriction of π ∗ f π ∗ ( λ k ι )to { } × BU ≃ BU which is f (0) λ k ι ∈ K ( BU ) and so Ω( f ⊗ λ k ι ) = f (0)Ω( λ k ι ). Now, byProposition 5.2, λ k ι is represented by the inverse limit of the maps k X i =0 (cid:18) − ni (cid:19) B Λ k − i : BU ( n ) → BU.
Since Ω B ≃
1, we see that Ω( λ k ι ) is represented by the inverse limit of the maps k X i =0 (cid:18) − ni (cid:19) Λ k − i : U ( n ) → U and hence Ω( λ k ι ) = l k . (cid:3) The remaining piece of structure to understand is the looping of the odd degree operations.
Definition 5.5.
Let P l ∈ Z [ x , . . . , x l ; y , . . . , y l ] denote the universal polynomial encodingthe action of the λ -operation λ l on products in a λ -ring [17, Definition 1.10]. We define the left-linearisation , P Ll , of P l to be the polynomial given by the sum of the monomials of P l containing a single x i . Concretely, if we define | x i | = 1 , | y j | = 0, for all i, j , then P Ll is thedegree 1 homogeneous part of P l . Proposition 5.6.
For l k ∈ K − ( U ) , we have Ω l k = 1 ⊗ P Lk (1 , − , . . . , ( − k − ; λ ι, . . . , λ k ι ) ∈ K − ( Z × BU ) . Proof.
By Corollary 5.4, Ω l k = Ω ( π ∗ f π ∗ λ k ι ) for any f with f (0) = 1. Now let α ∈ K ( X )and denote the degree 2 suspension element by u = [ ξ ] − ∈ K ( S , o ) where ξ is thecanonical line bundle over S ≃ C P . Then we have(Σ Ω l k )( α ) = Σ Ω ( π ∗ f π ∗ λ k ι )( α )= ( π ∗ f π ∗ λ k ι )( u × α )= f ( ε ( u ) ε ( α )) λ k ( u × α )= P k ( λ ( u ) × , . . . , λ k ( u ) ×
1; 1 × λ ( α ) , . . . , × λ k ( α ))= u × P Lk (1 , − , . . . , ( − k − ; λ ( α ) , . . . , λ k ( α ))= Σ P Lk (1 , − , . . . , ( − k − ; λ ι, . . . , λ k ι )( α )where the penultimate equality follows since λ i ( u ) = ( − i − u , and ( u ) = 0. (cid:3) We are now in a position to prove our main result.
Theorem 5.7.
We have an isomorphism of Z / -graded Z -plethories with looping, K ∗ ( K • ) ∼ = Ω( Set ( Z , Z ) b ⊗ K ( BU )) I , where I is the plethystic ideal with looping generated by the relations Ω( f ⊗ λ p ι ) = f (0)Ω(1 ⊗ λ p ι ) , Ω ( f ⊗ λ p ι ) = f (0) ⊗ P Lp (1 , − , . . . , ( − p − ; λ ι, . . . , λ p ι ) , for all p ≥ .Proof. From Theorem 3.4 we have seen that we have an isomorphism of complete Hausdorff Z -plethories θ : Set ( Z , Z ) b ⊗ K ( BU ) ∼ = −→ K ( K ) ⊆ K ∗ ( K • ) . By Proposition 4.6 this extends to a map of Z -plethories with loopingΩ( Set ( Z , Z ) b ⊗ K ( BU )) → K ∗ ( K • ) , which is surjective by Proposition 5.3 and Corollary 5.4. By Corollary 5.4 and Proposition 5.6the kernel of this map is precisely I . (cid:3) References [1] D. W. Anderson. There are no phantom cohomology operations in K -theory. Pacific J. Math. ,107(2):279–306, 1983.[2] M. Atiyah.
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E-mail address : [email protected] (S. Whitehouse) School of Mathematics and Statistics, University of Sheffield, S3 7RH, Eng-land
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