The congruence criterion for power operations in Morava E-theory
aa r X i v : . [ m a t h . A T ] D ec THE CONGRUENCE CRITERION FOR POWER OPERATIONS INMORAVA E -THEORY CHARLES REZK
Abstract.
We prove a congruence criterion for the algebraic theory of power operationsin Morava E -theory, analogous to Wilkerson’s congruence criterion for torsion free λ -rings.In addition, we provide a geometric description of this congruence criterion, in terms ofsheaves on the moduli problem of deformations of formal groups and Frobenius isogenies. Introduction
The purpose of this paper is to prove a congruence criterion for the algebraic theory ofpower operations acting on the homotopy of a K ( n )-local commutative E -algebra spectrum,where E denotes a spectrum of “Morava E -theory” at height n . This criterion is bestunderstood as being a higher chromatic analogue of Wilkerson’s congruence criterion for λ -rings.1.1. Algebraic theories of power operations.
By an “algebraic theory of power opera-tions” for a commutative ring spectrum A , we mean an algebraic category which models allthe algebraic structure which naturally adheres to π ∗ A , the homotopy groups of A . As an ex-amplar of this notion, consider the description of the natural operations on the homotopy of acommutative H F p -algebra spectrum, as given by McClure (see [BMMS86] especially § IX.2).In this work, it is shown (in modern language) that if A is a commutative H F p -algebra,then π ∗ A is an algebra for a certain monad C on graded F p -vector spaces. Furthermore,an algebraic description for C -algebras is provided: a C -algebra amounts to a graded com-mutative F p -algebra, together with the structure of a module over the May-Dyer-Lashofalgebra, which (i) is compatible with multiplication, in the sense of having a suitable Cartanformula, and which (ii) satisfies an “instability” relation, which includes the fact that the“top” Dyer-Lashof operation is equal to the p th power map. That this is the right answeris justified by the existence of a natural isomorphism C ( π ∗ M ) ≈ π ∗ ( P M ), where M is an H F p -module and P is the free H F p -algebra functor.In this paper, we use the work of Ando, Hopkins and Strickland to develop an analogoustheory for the homotopy of a K ( n )-local commutative algebra over a Morava E -theoryspectrum. Let E denote the cohomology theory associated to the universal deformations ofa height n formal group G over a perfect field k of characteristic p . There is a monad T onthe category Mod E ∗ of graded E ∗ = π ∗ E -modules such that(1) T E ∗ ≈ L m ≥ E ∧∗ B Σ m , where E ∧∗ ( − ) denotes K ( n )-localized homology;(2) T ( M ∗ ⊕ N ∗ ) ≈ T M ∗ ⊗ E ∗ T N ∗ for any E ∗ -modules M ∗ and N ∗ ; Date : November 12, 2018.The author was supported under NSF grant DMS-0505056. (3) There is a natural map T ( π ∗ M ) → π ∗ P M for E -module spectra M , where P M denotes the free commutative E -algebra on M . Furthermore, this map induces anisomorphism T ( π ∗ M ) ≈ [ π ∗ P M ] ∧ m , where N ∧ m denotes completion of an E ∗ -module N with respect to the maximal ideal m ⊂ π E .The category of algebras for this monad is denoted Alg ∗ T ; the monad T and its category ofalgebras are described in § § Description of
Alg ∗ T . The next goal is to give a workable description of Alg ∗ T . Everyobject of Alg ∗ T is (in particular) a graded commutative ring; it is also a right-module for acertain associative ring Γ, which we may call a “Dyer-Lashof algebra” (by analogy with theMay-Dyer-Lashof algebra for ordinary mod p homology).The ring Γ is defined in §
6; effectively, Γ is the ring of endomorphisms of the forgetfulfunctor Alg ∗ T → Ab which sends an T -algebra to its degree 0 part, viewed as an abeliangroup. Explicitly, Γ is a direct sum of the E -linear duals of the rings E B Σ p k / (transfers).There is a ring homomorphism η : E → Γ (the image of which is not typically central).The ring Γ is very nearly a Hopf algebra (more precisely, it is a “twisted bialgebra”, see § ⊗ : Mod ∗ Γ × Mod ∗ Γ → Mod ∗ Γ , in such a way that the underlying E ∗ -module of M ⊗ N is precisely M ⊗ E ∗ N .Let Alg ∗ Γ denote the category of commutative monoid objects in graded Γ-modules. Thereis a forgetful functor U : Alg ∗ T → Alg ∗ Γ . The first main result in this paper describes theessential image of the restriction of U to torsion free objects. Before stating the result, weconsider the motivating example.1.3. Example.
Let E be p -adic K -theory. In this case, E ≈ Z p , and Γ ≈ Z p [ ψ ]. Theelement ψ ∈ Γ corresponds to the p th Adams operation in the K -theory of a space. AΓ-algebra is precisely a ψ -ring , i.e., a graded commutative Z p -algebra B together with aring homomorphism ψ : B → B . In this case, the structure of Alg ∗ T can also be completelyunderstood, using the work of McClure in [BMMS86]. In particular, an object in Alg ∗ T isessentially what Bousfield [Bou96] calls a Z / -graded θ -ring . Thus, an object Alg ∗ T isstrictly commutative graded Z p -algebra B ∗ , together with a function θ : B ∗ → B ∗ whichsatisfies certain axioms. The operation ψ is recovered from θ using the identity ψ ( x ) = x p + pθ ( x )for all x ∈ B . The “Wilkerson criterion” states that a torsion free ψ -ring B admits thestructure of a θ -ring (necessarily uniquely), if and only if ψ ( x ) ≡ x p mod pB for all x ∈ B . (This result is a “ p -typicalization” of the original theorem of Wilkerson [Wil82], whichcharacterizes the torsion free λ -rings in terms of congruences on the Adams operations atall primes.)Our first result is a generalization of Wilkerson’s criterion. Its statement involves a rep-resentative σ ∈ Γ of a certain conjugacy class in Γ /p Γ, which is described in § B satisfies the congruence condition if for all x ∈ B , xσ ≡ x p mod pB. Theorem A.
An object B ∈ Alg ∗ Γ which is p -torsion free admits the structure of a T -algebra(necessarily uniquely), if and only if it satisfies the congruence condition. The proof of Theorem A is completed in § HE CONGRUENCE CRITERION FOR POWER OPERATIONS IN MORAVA E -THEORY 3 Interpretation in terms of formal groups.
The second result of this paper givesa reinterpretation of the above theorem in terms of formal groups, and in doing so explainsthe significance of the element σ ∈ Γ.Fix a perfect field k of characteristic p >
0, and a formal group G of finite height over k .Let E be the Morava E -theory associated to the universal deformation of G in the sense ofLubin-Tate.Given a complete local ring R , there is a category Def R , whose objects are deformationsof G to R , and whose morphisms are isogenies of formal groups which are “deformations ofa power of Frobenius”; that is, morphism are isogenies which are identified with some powerof the Frobenius isogeny when we base change to to the residue field of R . The collection ofcategories Def R for suitable rings R and base change functors f ∗ : Def R → Def R ′ describe akind of moduli problem. We let Sh(Def , Alg) ∗ denote the category of quasi-coherent sheavesof graded commutative O -algebras over Def. An object A of this category is (modulo someissues of grading) a pseudonatural transformation of pseudofunctors Def → Alg ∗ ; moreconcretely, an object A consists of data { A R , A f } , where for each ring R there is a functor A R : Def R → Alg R , and for each local homomorphism f : R → R ′ a natural isomorphism A f : A R ′ f ∗ → f ∗ A R , satisfying a collection of coherence relations. The precise definitions ofDef R and Sh(Def , Alg) ∗ , including correct treatment of the grading, are given in § A satisfies the Frobenius congruence if (roughly), forevery F p -algebra R and every object G ∈ Def( R ), we have A R ( G Frob −−−→ φ ∗ G ) = ( A R ( G ) Frob −−−→ φ ∗ A R ( G )), where “Frob” denotes the relative Frobenius isogeny on formal groups (on theleft-hand side) and on R -algebras (on the right-hand side). (See § Theorem B.
There is an equivalence of categories
Sh(Def , Alg) ∗ ≈ Alg ∗ Γ . Under this equiva-lence, sheaves which satisfy the Frobenius congruence exactly correspond to graded Γ -algebraswhich satisfy the congruence condition. As described below, the equivalence of categories of Theorem B is well known to expertsin this area, and amounts to an interpretation of some theorems of Strickland. The proof ofTheorem B is completed in § The work of Ando, Hopkins, and Strickland.
The structure of power operationson Morava E -theory is largely understood, thanks to work of Matt Ando, Mike Hopkins,and Neil Strickland. The hard results that underlie the version of the theory I will describeare theorems of Neil Strickland, and are proved in [Str97] and [Str98] (the latter corrected in[Str99]); this work in turn uses crucially some results of Kashiwabara [Kas98]. Unfortunately,there is no complete statement yet in print of the picture of operations on E -algebra spectra.Strickland provides a very brief sketch in [Str97, § § §
11 in this paper. Unfortunately, the level structure approach of[AHS04] is not convenient for describing the congruence condition.Their unpublished work has some overlap with what we discuss in this paper. In partic-ular, they constructed the algebra Γ and perceived the equivalence of Alg Γ ≈ Sh(Def , Alg)of Theorem B (a large part of this is accomplished in Strickland’s papers). They also un-derstood that the difference between the categories Alg T and Alg Γ was precisely an issue CHARLES REZK of understanding certain congruences, and that these congruences were generated by oneswhich are detected in the E -homology of the classifying space B Σ p . The precise economicalstatement of Theorem A is new, as is the treatment of gradings.1.6. Treatment of gradings.
We should note that we deal with gradings in a somewhatnovel way. Since Morava E -theory is an even periodic theory, we can regard the homotopygroups of a K ( n )-local commutative E -algebra spectrum as being a Z / π E -module,rather than a Z -graded π ∗ E -module. This would not be a viable procedure, except for thefact that we can modify the tensor product structure on Z / ω = π E (see § Z / ω to a tensor category.This point of view turns out to be very convenient for dealing with power operations, andwe believe it is worthy of attention.1.7. Completion.
There is a piece of structure on the homotopy π ∗ B of a K ( n )-local com-mutative E -algebra spectrum B which is not encoded in our algebraic model Alg ∗ T ; namely,the fact that π ∗ B is usually complete with respect to the maximal ideal of π E (or moreprecisely, that π ∗ B is always L -complete, see § Calculations.
As a companion piece to this paper, I’ve made available calculations ofthe structure of Γ and Alg ∗ T in a particular case for height n = 2 at the prime p = 2 [Rez08].1.9. Acknowledgments.
I’d like to thank Matt Ando and Mike Hopkins for helpful dis-cussions relating to issues in this paper. I would especially like to thank Haynes Miller andMIT, for allowing me to teach a course on power operations as a visiting instructor there inthe spring of 2006; the results of this paper were conjectured and proved while at MIT, andpresented in the course I gave there.2.
Twisted Z / -graded categories In this section, we describe a procedure for constructing Z / ω of the original category. This procedure will be used in the rest of this paperto handle issues related to the “odd” degree gradings of Morava E -theory. It gives a veryconcise way to explain the graded nature of the objects in question, and we’ll use it bothfor odd degrees in Morava E -theory (see § § Symmetric objects.
Let ( C , ⊗ , k ) be an additive tensor category; that is, an additivecategory C equipped with a symmetric monoidal structure ⊗ with unit object k , such that ⊗ distributes over finite sums.Let τ M,N : M ⊗ N → N ⊗ M denote the interchange isomorphism of the symmetricmonoidal structure on C . Say that an object ω ∈ C is symmetric if τ ω,ω = id ω ⊗ ω ; this isequivalent to requiring that the symmetric group act trivially on ω ⊗ m for all m ≥ HE CONGRUENCE CRITERION FOR POWER OPERATIONS IN MORAVA E -THEORY 5 Twisted tensor product.
Let C ∗ be the category of Z / -graded objects of C ; anobject of C ∗ is a pair M ∗ = { M , M } of objects of C , and a morphism f : M ∗ → N ∗ is apair f i : M i → N i , i = 0 ,
1, of morphisms of C . We define a functor ⊗ : C ∗ × C ∗ → C ∗ asfollows. If M ∗ and N ∗ are objects in C ∗ , we define an object M ∗ ⊗ N ∗ ∈ C ∗ by( M ∗ ⊗ N ∗ ) = ( M ⊗ N ) ⊕ ( M ⊗ N ⊗ ω ) , ( M ∗ ⊗ N ∗ ) = ( M ⊗ N ) ⊕ ( M ⊗ N ) , where the tensor products on the right-hand side are taken in C . We refer to this as the ω -twisted tensor product on C ∗ .Let k = { k , } as an object of C ∗ ; it serves as the unit object of the monoidal structurevia the evident isomorphisms k ⊗ M ∗ ≈ M ∗ ≈ M ∗ ⊗ k . Define an interchange map τ ∗ : M ∗ ⊗ N ∗ ∼ −→ N ∗ ⊗ M ∗ by τ : ( M ⊗ N ) ⊕ ( M ⊗ N ⊗ ω ) → ( N ⊗ M ) ⊕ ( N ⊗ M ⊗ ω )( m ⊗ n , m ⊗ n ⊗ x ) ( τ ( m ⊗ n ) , − τ ( m ⊗ n ) ⊗ x ) τ : ( M ⊗ N ) ⊕ ( M ⊗ N ) → ( N ⊗ M ) ⊕ ( N ⊗ M )( m ⊗ n , m ⊗ n ) ( τ ( m ⊗ n ) , τ ( m ⊗ n )) . (I’ve written these formulas using “element” notation, but they are easily converted into“arrow theoretic” formulas, meaningful in any additive tensor category C .)Finally, there is an associativity isomorphism α : ( M ∗ ⊗ N ∗ ) ⊗ P ∗ → M ∗ ⊗ ( N ∗ ⊗ P ∗ ),which I won’t write out in detail. It is defined using the associativity and interchangeisomorphisms for C , with the interchange map used when needed to move the extra factorof ω into the “correct” position. For instance, (( M ∗ ⊗ N ∗ ) ⊗ P ∗ ) contains a summand ofthe form ( M ⊗ N ⊗ ω ) ⊗ P , while the corresponding summand of ( M ∗ ⊗ ( N ∗ ⊗ P ∗ )) is M ⊗ ( N ⊗ P ⊗ ω ); the map α maps one to the other by switching ω and P using theinterchange map τ .2.3. Proposition. If ω ∈ C is a symmetric object, then the above structure makes C ∗ intoan additive tensor category. The functor C → C ∗ defined by M
7→ { M, } identifies C with afull monoidal subcategory of C ∗ .Proof. The only delicate point is to check the commutativity of the pentagon which com-pares the associativity isomorphisms of four-fold tensor products; that this commutes makesessential use of the fact that ω is symmetric. (cid:3) We will typically identify C with its essential image in C ∗ without comment.2.4. The odd square root of ω . Let ω / denote the object of C ∗ defined by ω / = { , k } .Then ω / ⊗ ω / ≈ { ω, } ≈ ω . Furthermore, the interchange map τ ∗ on ω / ⊗ ω / isequal to − id. Every object M ∗ of C ∗ is isomorphic to one of the form M ⊕ ( M ⊗ ω / ),where M , M ∈ C ⊂ C ∗ . Thus we can think of C ∗ as the additive tensor category obtainedfrom C by “adjoining an odd square-root” of ω . CHARLES REZK
Functors from a twisted Z / -graded category. Given an additive tensor category C , and a symmetric object ω of C , we define the groupoid Sqrt( ω ) of odd square-roots of ω as follows. The objects of Sqrt( ω ) are pairs ( η, f ), where η is an object of C such that τ η,η = − id η ⊗ η , and f : η ⊗ η → ω is an isomorphism, and the morphisms ( η, f ) → ( η ′ , f ′ ) ofSqrt( ω ) are isomorphisms g : η → η ′ such that f ′ ( g ⊗ g ) = f .Now suppose that C is an additive tensor category with symmetric object ω , that D is anadditive tensor category, and that F : C → D is a symmetric monoidal functor; thus F ( ω ) isa symmetric object of D . Let C ∗ denote the Z / C and ω , and identify C with its essential image in C ∗ . Let G denote the groupoid whose objectsare additive symmetric monoidal functors F ∗ : C ∗ → D such that F ∗ | C = F , and whosemorphisms F ∗ → F ∗ are monoidal natural isomorphisms which restrict to the identity mapover C .2.6. Proposition.
There is an equivalence of categories
G →
Sqrt( F ( ω )) , defined by F ∗ ( F ∗ ( ω / ) , g ) , where g : F ∗ ( ω / ) ⊗ F ∗ ( ω / ) ∼ −→ F ∗ ( ω / ⊗ ω / ) F ∗ ( f ) −−−→ F ∗ ( ω ) is the compositeof the coherence map of D and the map F ∗ ( f ) , where f : ω / ⊗ ω / → ω is the tautologicalisomorphism in C ∗ . Examples.
Example. If ω = k , then C ∗ is just the “usual” Z / C .2.9. Example.
Suppose the symmetric object ω is also ⊗ -invertible in C , i.e., there existsan object ω − ∈ C and an isomorphism ω ⊗ ω − ≈ k . Then we can define a Z -gradedcommutative ring object R ∗ of C by R k = ω ⊗ k and R k +1 = 0 for all k ∈ Z . It isstraightforward to check that C ∗ is then equivalent to the additive tensor category of Z -graded modules over R ∗ . This equivalence associates a Z -graded R ∗ module M ∗ with theobject M ∗ = { M , M − } ≈ M ⊕ ( M − ⊗ ω / ) in C ∗ .2.10. E ∗ -modules. Let E ∗ = π ∗ E , the coefficient ring of an even periodic ring spectrum,and let ω = π E viewed as an E = π E -module. We can, and will, identify the categoryMod E ∗ of Z -graded E ∗ -modules with the Z / ∗ E , where ω = π E = E S = π Σ − E is used as the symmetric object. Observe that under the equivalence of Z -graded and Z / E ∗ -modules described above, π ∗ Σ q E ≈ E ∗ S q is naturally identifiedwith ω − q/ . The K¨unneth isomorphism E ∗ S i ⊗ E ∗ E ∗ S j → E ∗ ( S i ∧ S j ) ≈ E ∗ ( S i + j ) producesa canonical isomorphism κ : ω − i/ ⊗ ω − j/ → ω − ( i + j ) / .If M = { M , M } is an object of Mod ∗ E , we can recover the Z -graded E ∗ -module M ∗ associated to it by M q = Hom Mod ∗ E ( ω − q/ , M ).In the examples we’ve given above, the symmetric object ω was ⊗ -invertible. Later in thispaper, in sections § §
11, we will consider ω -twisted tensor categories in cases where thesymmetric object ω is not ⊗ -invertible. It is in these non-invertible cases that the formalismof twisted Z / Morava E -theory and extended powers Morava E -theory. We fix for the rest of the paper a perfect field k of characteristic p , and a formal group G over k of height n , with 1 ≤ n < ∞ . Let E denote the Morava E -theory associated to the universal deformation of the formal group G . HE CONGRUENCE CRITERION FOR POWER OPERATIONS IN MORAVA E -THEORY 7 It is a theorem of Goerss, Hopkins, and Miller, that E is a commutative S -algebra in anessentially unique way [GH04].3.2. E -modules and E ∗ -modules. Let Mod E denote the category of E -module spectraas in [EKMM95], and let h Mod E denote its homotopy category. We write M ∧ E N andhom E ( M, N ) for the smash product and function spectrum of E -module, and also for theirderived versions on h Mod E .We write E = π E and ω = π E as an E -module.Taking homotopy groups defines a functor π ∗ : Mod E → Mod E ∗ . In what follows we aregoing to regard Mod E ∗ as the Z / § π ∗ is explicitly described by π ∗ M = { π M, π − M } . Recall that according to our conventions, there are natural isomorphisms π ∗ Σ q M ≈ ω − q/ ⊗ π ∗ M for E -modules M , for all q ∈ Z . (The reader may prefer to regard Mod E ∗ as the usualcategory of modules over a Z -graded ring; doing so should not cause any confusion untilsection § The completion functor.
Let K ( n ) denote the n th Morava K -theory spectrum.Let L : Mod E → Mod E denote the Bousfield localization functor with respect to thehomology theory on E -modules defined by smashing with the module E ∧ K ( n ). It comesequipped with a natural coaugmentation map j : M → LM , which is a K ( n )-homologyequivalence. The localization functor L descends to a functor h Mod E → h Mod E on thehomotopy category of E -module spectra, which we also denote L . We have the followingequivalent descriptions of L .3.4. Proposition.
We have the following equivalences of coaugmented functors h Mod E → h Mod S , where h Mod S denotes the homotopy category of spectra.. (1) L ≈ L K ( n ) , where L K ( n ) denotes Bousfield localization of spectra with respect toMorava K -theory. (2) L ≈ L F ( n ) , where L F ( n ) denotes Bousfield localization of spectra with respect to atype n -finite spectrum F ( n ) . (3) LM ≈ holim( E ∧ M ( i , . . . , i n − )) ∧ E M , where { M ( i , . . . , i n − ) } denotes a cer-tain inverse system of finite spectra, constructed so that π ∗ E ∧ M ( i , . . . , i n − ) ≈ E ∗ / ( p i , u i , . . . , u i n − n − ) .Proof. Statement (1) is [EKMM95, Prop. VIII.1.7].Statements (2) and (3) follow from [HS99, Prop. 7.10]; the proof of (2) uses the fact that E is an L n -local spectrum, whence the underlying spectrum of every E -module is K ( n )-local. (cid:3) Because of the last equivalence in this list, we can think of L as a “completion” functor.3.5. Derived functors of m -adic completion. Thus, let L s : Mod E ∗ → Mod E ∗ denotethe s th left derived functor of M M ∧ m . There is a map L M → M ∧ m , which is notgenerally an isomorphism since completion is not right exact.3.6. Proposition.
The functors L s vanish identically if s > n . If M ∗ is either a flat E ∗ -module, or a finitely generated E ∗ -module, then L ( M ∗ ) ≈ ( M ∗ ) ∧ m and L s ( M ∗ ) = 0 for s > . CHARLES REZK
Proof.
The first statement is [HS99, Thm. A.2(d)]. The statement about flat modules followsfrom [HS99, Thm. A.2(b)], and the statement about finitely generated modules follows from[HS99, Thm. A.6(e)]. (cid:3)
The functor L : Mod E ∗ → Mod E ∗ is equipped with natural transformations M i −→ L M j −→ M ∧ m . In general, i : L M → M ∧ m is a surjection, but not an isomorphism, while L ( i ) is anisomorphism. The groups L s M can be identified with certain local cohomology groups ofthe modules M , as described in [HS99, Appendix A].3.7. Proposition. [Hov08, Prop. 2.3] . There is a conditionally and strongly convergentspectral sequence of E ∗ -modules E s,t = L s π t M = ⇒ π s + t LM, which vanishes for s > n . As a consequence, we have3.8.
Corollary.
The map π ∗ M → π ∗ LM factors through a natural transformation L π ∗ M → π ∗ LM of functors h Mod E → Mod E ∗ , which is a natural isomorphism whenever π ∗ M is flat. For a spectrum X , the completed E -homology of X is defined by E ∧∗ ( X ) def = π ∗ L ( E ∧ X ).3.9. Complete E -modules. A complete E -module is an E -module M which is K -local,i.e., one such that j : M → LM is an equivalence. Note that(a) If M and N are E -modules, and N is complete, then hom E ( M, N ) is a complete E -module.(b) If M and N are complete E -modules, M ∧ E N need not be complete. However, L ( M ∧ E N ) is a complete E -module, called the completed smash product .3.10. Finite and finite free modules.
We write hom E ( M, N ) for the function spectrumin Mod E .3.11. Proposition.
Let M be an E -module spectrum. The following are equivalent. (1) π ∗ M is a finitely generated (resp. finitely generated free) E ∗ -module. (2) π ∗ hom E ( M, E ) is a finitely generated (resp. finitely generated free) E ∗ -module.If either of (1) or (2) hold, then M ≈ LM and M ≈ hom E (hom E ( M, E ) , E ) .In particular, if X is a spectrum, E ∗ X is finitely generated (resp. free) if and only if E ∧∗ X is so.Proof. See [HS99, § (cid:3) We say a module M is finite if π ∗ M is a finitely generated E ∗ -module. A module is finiteif and only if it is contained in the thick subcategory of h Mod E generated by E . Accordingto the above proposition, finite modules are complete.Say that a E -module M is finitely generated and free , or finite free for short, if π ∗ M is a finitely generated free π ∗ E -module. All such modules are equivalent to ones of the form W ki =1 Σ d i E . Note that if M and N are finite (resp. finite free), then so are M ∧ E N andhom E ( M, N ). Also, retracts of finite free modules are also finite free.Let Mod ff E denote the full subcategory of Mod E consisting finite free modules. Let h Mod ff E denote the full subcategory of h Mod E spanned by the finite free modules. HE CONGRUENCE CRITERION FOR POWER OPERATIONS IN MORAVA E -THEORY 9 Proposition.
The functor π ∗ : h Mod ff E → Mod ff E ∗ which associates to an E -module itshomotopy groups is an equivalence of categories.Proof. A straightforward consequence of the observation that h Mod E (Σ d E, M ) ≈ π d M . (cid:3) Flat modules.
Say that an E -module M is flat if π ∗ M is flat as a graded E ∗ -module.We will need the following analogue of Lazard’s theorem on flat modules over a ring, whichis a variant of an observation of Lurie [Lur07, § C be a category enriched over spaces. Say that C is filtered if the following hold.(1) For every finite set of objects X , . . . , X k in C , there exists an object Y and morphism X j → Y in C for all j = 1 , . . . , k .(2) For all k ≥
0, all
X, Y objects of C , and all maps S k → C ( X, Y ), there exists a map g : Y → Z and a dotted arrow making the diagram S k / / (cid:15) (cid:15) C ( X, Y ) C ( X,g ) (cid:15) (cid:15) D k +1 / / C ( X, Z )commute.For a category C enriched over spaces, let π C denote the ordinary (enriched over sets)category with the same objects as C , whose morphisms are the sets of path components ofthe mapping spaces of C . If C is filtered in the above sense, then the ordinary category π C is filtered in the usual sense.3.14. Proposition.
An object M ∈ Mod E is flat if and only if it is weakly equivalent to thehomotopy colimit of some continuous functor F : C → Mod ff E , where C is filtered.Proof. This is proved in much the same way as [Lur07, Theorem 4.6.19], though with changesof detail, since the notions of “flat” and “finite free” we use differ than the ones Lurie uses.We briefly sketch the ideas here.For the if direction, it suffices to note that taking homotopy groups commutes with takinghomotopy colimit over a filtered diagram.For the only if direction, consider the comma category Mod E /M , which admits the struc-ture of a topological closed model category. Choose a set S of fibrant-and-cofibrant repre-sentatives of weak equivalence classes of objects ( F, f : F → M ) for which F is finite free,and let C be the full topological subcategory of Mod E /M spanned by S . Now one showsthat if M is flat, then C is a filtered topological category in the sense described above.Given this, it is clear that M is equivalent to the homotopy colimit of the canonical functor C → Mod E . (cid:3) Completed extended powers. If M is an E -module, the m th symmetric power is the quotient ( M ∧ E m ) h Σ m of the m th smash power by the evident symmetric group action.The free commutative E -algebra on M is the coproduct W m ≥ P m ( M ).In this paper, we will deal mainly with the m th extended powers ; we write P m ( M ) def =( M ∧ E m ) h Σ m for this; the extended power also passes to a functor on the homotopy category h Mod E , also denoted P m . We recall that, in the EKMM model for S -modules, if we choose a q -cofibrant model for the commutative S -algebra E , then the symmetric powers of cellcofibrant R -modules are homotopy equivalent to extended powers [EKMM95, III.5].We write P ( M ) = W m ≥ P m ( M ). If G ⊆ Σ m is a subgroup, we write P G ( M ) def =( M ∧ E m ) hG . The functor P defines a monad on the homotopy category of E -modules. We willassume that the reader is familiar with properties of these functors, for instance as describedin [BMMS86, Ch. 1]. In particular, we note that P defines a monad on the the homotopycategory h Mod E of E -modules, and any commuative E -algebra results in an algebra for thismonad.3.16. Proposition.
The functors P m preserve K -homology isomorphisms. In particular, P m ( j ) : P m ( M ) → P m ( LM ) is a K -homology isomorphism, and thus there is a natural iso-morphism L P m ( j ) : L P m → L P m L of functors on h Mod E . The functor L P : h Mod E → h Mod E admits a unique monad structure with the property that j is a map of monads.Proof. The functors P are homology isomorphisms for any homology theory; the remainingstatements are straightforward. (cid:3) The goal of this section is to prove3.17.
Proposition. If M is an E -module which is finite free, then L P m ( M ) is also finitefree. This is well-known in the case that π ∗ M is concentrated in even degree (see [HKR00, Thm.D]).3.18. Proposition. If G contains a p -Sylow subgroup of Σ m , then P G ( M ) → P m ( M ) admitsa section for any E -module M .Proof. If G is a subgroup of a group H , with index prime to p , the map of spectraΣ ∞ + ( H/G ) ( p ) → Σ ∞ + ( H/H ) ( p ) admits a retraction in the homotopy category of spectraequipped with a H action. (cid:3) Recall that p is the characteristic of the residue field of E ∗ . Let ρ C p denote the realregular representation of the cyclic group C p , and let BC cρ Cp p denote the Thom spectrum ofthe virtual representation cρ C p , where c ∈ Z .3.19. Lemma. If c ∈ Z , then E ∧∗ BC cρ Cp p is a finitely generated free π ∗ E -module.Proof. First suppose c = 0. Then the cofiber sequence BC p ≈ S ( λ ⊗ p ) → BS → ( BS ) λ ⊗ p associated to the universal line bundle λ over BS , together with and the Thom isomorphismfor E -theory, give 0 ← E ∗ BC p ← E [[ x ]] [ p ]( x ) ←−−− E [[ x ]] ← , and in particular since [ p ]( x ) ≡ x p n mod m , E ∗ BC p is free over E ∗ on 1 , . . . , x p n − . Thushom(Σ ∞ + BC p , E ) is a finitely generated free E -module, and therefore so is L ( E ∧ Σ ∞ + BC p )by (3.11). HE CONGRUENCE CRITERION FOR POWER OPERATIONS IN MORAVA E -THEORY 11 The Thom isomorphism for E -theory immediately gives the result for even c , by identifying2 dρ C p with the complex bundle dρ C p ⊗ C . It remains to check the case of odd c , and theThom isomorphism allows us to reduce to the case c = 1.If p is odd, there is a splitting ρ C p ≈ R ⊕ ¯ ρ C p of real C p -representations, where thereal representation ¯ ρ C p admits a complex structure. The result follows using the Thomisomorphism, since E is complex orientable.If p = 2, then ρ C ≈ R ⊕ ¯ ρ C , and ¯ ρ C is the sign representation, so that as spaces, BC ρ C ≈ Σ BC ¯ ρ C ≈ Σ BC . Stably, the latter is a retract of Σ(Σ ∞ + BC ), whose completed E -homology is finite free asnoted above. (cid:3) Proof of (3.17) . Since ( S c ) ∧ phC p ≈ BC cρ Cp p , (3.19) implies that L P C p (Σ c E ) is finitely gener-ated free. Thus (3.18) shows that L P p ( M ) is a retract of L P C p ( M ), so that L P p (Σ c E ) isfinitely generated free.The “binomial formula” for P p says that P p ( M ∨ N ) ≈ _ i + j = p P i ( M ) ∧ E P j ( N ) , and since P i ( M ) is a retract of M ∧ E i by (3.18), we conclude that L P p ( M ) takes finite freesto finite frees.We have that P G P H ( M ) ≈ P H ≀ G ( M ) . If Σ ≀ rp denotes the r -fold wreath power, we have shown that L P Σ ≀ rp = L P p · · · P p preservesfinite frees.Finally, for m ≥ m = P a i p i , with a i ∈ { , . . . , p − } , the group Σ m containsa subgroup G = Q i (Σ ≀ ip ) × a i , which acts on m in the evident way, and which contains a p -Sylow subgroup of G . By (3.18), P m ( M ) is a retract of the smash product of finitely many P Σ ≀ ip ( M ), and therefore we are done. (cid:3) Remark.
The above proof shows a little bit more. Namely, if M is a finite free modulewith π ∗ M concentrated in even degree, then π ∗ L P m ( M ) is also concentrated in even degrees.There is no corresponding result when π ∗ M is concentrated in odd degree, although the proofof (3.17) implies the following result.3.21. Corollary.
The graded module π ∗ L P p (Σ c E ) is concentrated in odd degree if c is odd. We also note the following interesting consequence (a generalization of an observation ofMcClure).3.22.
Proposition. If A is a K ( n ) -local commutative E -algebra, then the multiplication on π ∗ A is strictly graded commutative, in the sense that if x ∈ π q A with q odd, then x = 0 .Proof. It is clear that we only need to prove something in the 2-local case. Let f : Σ q E → A be the E -module map which represents x . Then x ∈ π q A is the image of an element in π q L P (Σ q E ) under the map L P (Σ q E ) → L P (Σ q A ) → A . But π ∗ L P (Σ q E ) is concentratedin odd degree by (3.21). (cid:3) Power operations.
Let R be a commutative E -algebra spectrum. For any space X and any m ≥
0, we obtain an operation P m : R X → R ( X × B Σ m ) , defined so that an E -module map x : E ∧ Σ ∞ + X → R is sent to the composite E ∧ Σ ∞ + ( X × B Σ m ) → E ∧ Σ ∞ + X mh Σ m ≈ P m ( E ∧ Σ ∞ + X ) P m ( x ) −−−−→ P m R → R. The operation P m is called the m th power operation . It has the property that P m ( xy ) = P m ( x ) P m ( y ). Since E B Σ m is a finite free E -module, there is an isomorphism R ( X × B Σ m ) ≈ R X ⊗ E E B Σ m .If R X denotes the spectrum of functions from Σ ∞ + X to R , which is a commutative E -algebra, then the power operation P m : ( R X ) ( ∗ ) → ( R X ) ( B Σ m ) coincides with poweroperation on R X .Let J = X
The map P m is a ring homomorphism. Let i : X → X × B Σ m denote the map induced by inclusion of a basepoint in B Σ m .3.25. Proposition.
The composite map R ( X ) P m −−→ R ( X × B Σ m ) i ∗ −→ R ( X ) sends x x m . Relative power operations.
There is a “relative” version of the power operation,which we will need in §
12. Let (
X, A ) be a CW-pair of spaces, let D m ( X, A ) ⊆ X × m denotethe space which is the union of the subspaces of the form X i × A × X m − i − , and considerthe diagram ˜ D m ( X, A ) g / / (cid:15) (cid:15) X / / diag (cid:15) (cid:15) C ( g ) (cid:15) (cid:15) D m ( X, A ) f / / X × m / / C ( f )where ˜ D m ( X, A ) is the homotopy pullback of f along the diagonal inclusion, and C ( f ) and C ( g ) are homotopy cofibers. The group Σ m acts on every space in this diagram. We definea pointed space B m ( X, A ) def = C ( g ) ∧ mh Σ m ;it comes with a map B m ( X, A ) → C ( f ) ∧ mh Σ m ≈ ( X/A ) ∧ mh Σ m . Note that B m ( X, ∅ ) ≈ ( X × B Σ m ) + . Given such a pair ( X, A ), we define P m : ˜ R ( X/A ) → ˜ R B m ( X, A ) HE CONGRUENCE CRITERION FOR POWER OPERATIONS IN MORAVA E -THEORY 13 so that an E -module map x : E ∧ Σ ∞ X/A → R is sent to the composite E ∧ Σ ∞ B m ( X, A ) → E ∧ Σ ∞ ( X/A ) ∧ mh Σ m ≈ P m ( E ∧ Σ ∞ ( X/A )) P m ( x ) −−−−→ P m R → R. The diagram ˜ R ( X/A ) P m / / (cid:15) (cid:15) ˜ R B m ( X, A ) (cid:15) (cid:15) R X P m / / R X × B Σ m commutes.We are mainly interested in pairs of the form ( D ( V ) , S ( V )), where V → X is a real vectorbundle. In this case, we see that the relative power operation amounts to a map P m : ˜ R X V → ˜ R ( X × B Σ m ) V ⊠ ρ m , where ρ m → B Σ m is the real vector bundle associated to the real permutation representation, V ⊠ ρ m is the external tensor product bundle, and the spaces are Thom spaces.3.27. Power operations in non-zero degree.
The definition of § q ∈ Z , there is a function P m : R q X → R q ( X + ∧ B Σ − q ¯ ρ m m ) , where ¯ ρ m denotes the real permutation representation; if q >
0, then − q ¯ ρ m is a virtualbundle, and thus the target of P m is really R q (Σ ∞ + X ∧ B Σ − q ¯ ρ m m ). The function P m is definedso that an E -module map x : E ∧ Σ ∞ + X ∧ S − q → R is sent to the composite E ∧ Σ ∞ + X ∧ B Σ − q ¯ ρ m m ∧ S − q ≈ E ∧ Σ ∞ + X ∧ B Σ − qρ m m → E ∧ (Σ ∞ + X ∧ S − q ) h Σ m ≈ P m ( E ∧ Σ ∞ + X ∧ S − q ) → P m R → R. Let X be a pointed space. If m >
0, then the operator P m defined above restricts to afunction P ′ m : ˜ R q X → ˜ R q ( X ∧ B Σ − q ¯ ρ m m ) . Proposition.
Let X be a pointed space, and q ∈ Z . The diagram ˜ R q X P ′ m / / ∼ susp. (cid:15) (cid:15) ˜ R q ( X ∧ B Σ − q ¯ ρ m m ) ˜ R q (id ∧ e ) / / ˜ R q ( X ∧ B Σ − ( q +1)¯ ρ m m ) susp. ∼ (cid:15) (cid:15) ˜ R q +1 ( S ∧ X ) P ′ m / / ˜ R q +1 ( S ∧ X ∧ B Σ − ( q +1)¯ ρ m m ) commutes, where the vertical maps are the suspension isomorphisms, and e : B Σ − ( q +1)¯ ρ m m → B Σ − q ¯ ρ m m is the map of Thom spectra induced by the inclusion ⊂ ρ m of vector bundles.Proof. A straightforward calculation, using the definitions. (cid:3)
The following corollary will be crucial for our treatment of gradings in §
6. It relates theaction of power operations on R q S q with the action of power operations on R ( ∗ ). Corollary.
For all q ≥ , the diagram ˜ R S P ′ m / / ∼ (cid:15) (cid:15) ˜ R ( S ∧ B Σ m ) ˜ R (id ∧ e ) / / ˜ R B Σ − q ¯ ρ m m ∼ (cid:15) (cid:15) ˜ R q S q P ′ m / / ˜ R q ( S q ∧ B Σ − q ¯ ρ m m ) commutes, where e : B Σ − q ¯ ρ m m → B Σ m . Approximation functors
In this section, we are going to produce a monad T on the category Mod E ∗ , called the algebraic approximation functor , and thus a category Alg ∗ T of algebras for the monad T . There will be dotted arrow Alg ∗ T (cid:15) (cid:15) Alg
E π ∗ / / π ∗ : : Mod E ∗ making the diagram commute up to natural isomorphism. Furthermore, the lift is be“nearly” optimal, in the sense that for a flat E -module spectrum M , the object π ∗ L P M is be “nearly” isomorphic to the free T -algebra on π ∗ M . In the above sentence, “nearly”indicates that the isomorphism holds only after a suitable completion.The category Alg ∗ T has a number of nice properties. Most notably, the forgetful functorto commutative E ∗ -algebras U : Alg ∗ T → Alg E ∗ admits both a left and a right adjoint, so that U preserves both limits and colimits.4.1. Left Kan extension.
Recall that given functors F : I → D and U : I → C , a left Kanextension of F along U is the initial example of a pair ( E, δ ), where E : C → D is a functorand δ : F → EU is natural transformation of functors I → D . We denote the left Kanextension (if it exists) by colim U F , and we write β : F → (colim U F ) U for the canonicalnatural transformation. The universal property of the left Kan extension is equivalent to thefollowing: if G : C → D is any functor, then there is a one-to-one correspondence between(natural transformations F → GU ) ⇐⇒ (natural transformations colim U F → G ) . Note that if U is fully faithful, then β : F → (colim U F ) U is a natural isomorphism.Given essentially small I , choose a set of objects S of I which spans its isomorphismclasses. For q ∈ Z , define a functor B q = B F,Uq : C → D by B q ( X ) def = a I →···→ I q ∈ I UI q → X ∈ C F ( I ) , where the direct sum is taken over all diagrams I → · · · → I q in I where the objects are in S ,together with all morphisms U I q → X in C . Then the left Kan extension is the coequalizer HE CONGRUENCE CRITERION FOR POWER OPERATIONS IN MORAVA E -THEORY 15 of the evident pair of arrows(4.2) B ( X ) ⇒ B ( X ) → (colim U F )( X ) . Now suppose that we are additionally given a functor V : C → C ′ . Then we can formthe left Kan extension of F along V U : I → C ′ , and the canonical transformation F → (colim V U F ) V U corresponds (according to the universal property of colim U ) to a naturaltransformation γ : colim U F → (colim V U F ) V .4.3. Lemma.
Given a diagram of categories and functors I U / / F (cid:15) (cid:15) C V / / colim U F qqqqq x x qqqqq C ′ colim V U F t t hhhhhhhhhhhhhhhhhhhhhhhhhh D with I essentially small, if for all objects I in I and X in C the map C ( U I, X ) → C ′ ( V U I, V X ) is a bijection, then the transformation γ : colim U F → (colim V U F ) V is anisomorphism.Proof. Using the coequalizer (4.2), it suffices to check that the evident maps B F,Uq ( X ) → B F,V Uq ( V X ) are isomorphims, which is immediate from the hypothesis. (cid:3)
Construction of T m and T . In this section, we construct functors T m : Mod E ∗ → Mod E ∗ , called algebraic approximation functors, We will define T ( M ) def = L m ≥ T m ( M ).The idea is to define T m first on finitely generated free E ∗ -modules using the equivalenceof categories π ∗ : h Mod ff E → Mod ff E ∗ . Thus, for a finite free E ∗ -module M ∗ , we should set T m ( M ∗ ) = π ∗ L P m ( M ), where M is an E -module such that π ∗ M ≈ M ∗ . Then we extend T m to all E ∗ -modules via left Kan extension.We will refer to the following diagram of functors h Mod ff E i / / π ∗ ∼ (cid:15) (cid:15) π ∗ L P m i + + h Mod Eπ ∗ (cid:15) (cid:15) e T m / / Mod E ∗ Mod ff E ∗ j / / Mod E ∗ T m : : in which the left-hand square commutes (on the nose), the functors i and j are inclusions offull subcategories, and the vertical arrow on the left is an equivalence of categories.We define T m : Mod E ∗ → Mod E ∗ be the left Kan extension of the functor π ∗ L P m i : h Mod ff E → Mod E ∗ along the functor π ∗ i = jπ ∗ : h Mod ff E → Mod E ∗ ; this existsbecause h Mod ff E is essentially small.The functor π ∗ i is fully faithful, and so the tautological transformation β : π ∗ L P m i → T m π ∗ i is an isomorphism.4.5. Lemma.
The natural map κ : T m → colim j T m j adjoint to the identity id : T m j → T m j is an isomorphism. Proof.
Since β is an isomorphism and π ∗ i = jπ ∗ , we have natural isomorphisms T m = colim π ∗ i π ∗ L P i ≈ colim jπ ∗ T m jπ ∗ . Since π ∗ : h Mod ff E → Mod ff E ∗ is an equivalence of categories, we see that colim jπ ∗ T m jπ ∗ ≈ colim j T m j . (cid:3) Construction of the approximation map.
Let e T m : h Mod E → Mod E ∗ be the leftKan extension of the functor π ∗ L P m i : h Mod ff E → Mod E ∗ along the inclusion i : h Mod ff E → h Mod E . There are natural isomorphisms T m π ∗ i β ←− π ∗ L P m i id −→ π ∗ L P m i, where β is the tautological natural transformation for T m , which are associated (since e T m is a left Kan extension along i ), to natural transformations T m π ∗ γ ←− e T m ˜ α −→ π ∗ L P m of functors h Mod E → Mod E ∗ .4.7. Lemma.
The map γ is a natural isomorphism.Proof. We are precisely in the setup of (4.3), where we use the fact that π ∗ : h Mod E ( F, M ) → Mod E ∗ ( π ∗ F, π ∗ M ) is an isomorphism for all F in h Mod ff E (cid:3) The natural transformation α m : T m ( π ∗ M ) → π ∗ ( L P m ( M )) , is defined by α m = ˜ αγ − .4.8. Proposition.
When π ∗ M is a finite free E ∗ -module, the map α m : T m ( π ∗ M ) → π ∗ L P m ( M ) is an isomorphism. The natural transformation α : T ( π ∗ M ) → π ∗ L P ( M )is defined by M m T m ( π ∗ M ) α m −−→ M m π ∗ L P m M → π ∗ L _ m L P m M ! ≈ π ∗ L P M. Note that the analogue to (4.8) does not hold for α .4.9. T is a monad. Proposition.
The functor T : Mod E ∗ → Mod E ∗ admits the structure of a monad,compatibly with the monad structure of L P , in the sense that the diagrams π ∗ M / / % % JJJJJJJJJJ T ( π ∗ M ) α (cid:15) (cid:15) TT ( π ∗ M ) / / α ◦ T α (cid:15) (cid:15) T ( π ∗ M ) (cid:15) (cid:15) α (cid:15) (cid:15) π ∗ L P ( M ) π ∗ L P L P ( M ) / / π ∗ L P ( M ) commute, the unlabeled maps being the ones describing the monad structure. HE CONGRUENCE CRITERION FOR POWER OPERATIONS IN MORAVA E -THEORY 17 Proof.
The structure maps I → T and TT → T of the monad are defined on finite freemodules using the maps π ∗ X → π ∗ L P X and π ∗ L P L P X → π ∗ L P X for X ∈ Mod ff E , together with the equivalence L PP → L P L P as in (3.16). (cid:3) Colimits.
Proposition.
The functors T m commute with filtered colimits and reflexive coequaliz-ers.Proof. Observe that for q = 0 ,
1, the functor M B q ( M ) = M F →···→ F q ∈ h Mod ff E π ∗ F q → M ∈ Mod E ∗ π ∗ L P m F from Mod E ∗ → Mod E ∗ preserves filtered colimits, since the objects π ∗ F q of Mod E ∗ are small ,in the sense that hom Mod E ∗ ( π ∗ F q , − ) preserves filtered colimits. The filtered colimit part ofthe result follows using (4.2).The functors M B q ( M ) also reflexive coequalizers, since the objects π ∗ F q of Mod E ∗ are projective , in the sense that hom Mod E ∗ ( π ∗ F q , − ) carries epimorphisms to surjections. Thusthe reflexive coequalizer part of the result follows using (4.2). (cid:3) Tensor products.
Let k ≥
0, and M , . . . , M k ∈ Mod E ∗ . We define a natural map γ k : T ( M ) ⊗ · · · ⊗ T ( M k ) → T ( M ⊕ · · · ⊕ M k )as follows. As in the proof of (4.12), let B q ( M ) = M F →···→ F q ∈ Mod ff E ∗ F q → M ∈ Mod E ∗ M m π ∗ L P m F , so that T ( M ) ≈ H B ( M ). We have maps B q ( M ) ⊗ · · · ⊗ B q k ( M k ) s −→ B q ( M ) ⊗ · · · ⊗ B q ( M k ) t −→ B q ( M ⊕ · · · ⊕ M k )for q = P q i , where s is the Eilenberg-Mac Lane shuffle map, and t is the map constructedin the evident way from the “exponential isomorphism” maps π ∗ L P m F ⊗ · · · ⊗ π ∗ L P m k F k → π ∗ L P m + ··· + m k ( F ∨ · · · ∨ F k ) . These are maps of chain complexes, and taking the 0th homology group gives the desiredmap γ k .For the following, we will need to make use of comma categories. Given a category C and an object X of C , the comma category C /X is the category whose objects are pairs( Y, f : Y → X ) where Y is an objects of C , and morphisms ( Y, f ) → ( Y ′ , f ′ ) are maps g : Y → Y ′ such that f ′ g = f .4.14. Proposition.
The map γ k is an isomorphism. Proof.
It is standard that H ( s ) is an isomorphism, so it suffices to show that H ( t ) is anisomorphism. Consider the comma categories C = ( Y π ∗ : ( h Mod ff E ) k → (Mod kE ∗ )) / ( M , . . . , M k )and D = ( π ∗ : h Mod ff E → Mod E ∗ ) / ( M ⊕ · · · ⊕ M k ) , and let ρ : C → D be the functor sending a tuple ( F i , f i : π ∗ F i → M i ) i =1 ,...,k to( ∨ F i , ( f i ) : π ∗ ( ∨ F i ) → ⊕ M i ). Let R : C →
Mod E ∗ be the functor sending ( F i , f i ) to L π ∗ L P m F ⊗ · · · π ∗ L P m k F k , let S : D →
Mod E ∗ be the functor sending ( F, f ) to L π ∗ L P m F . Let h : R → Sρ be the evident natural isomorphism. It is clear that H ( t )is isomorphic to the map colim C R ≈ colim C Sρ η −→ colim D S, and the result follows from the observation that ρ admits a left adjoint and therefore η is anisomorphism. (cid:3) As a consequence, T ( M ) has a natural structure of a commutative ring, with productdefined by δ : T ( M ) ⊗ T ( M ) γ −→ T ( M ⊕ M ) T ((id M , id M )) −−−−−−−−→ T ( M ).The naturality of the construction of γ k shows the following.4.15. Corollary.
The natural isomorphisms γ k give T the structure of a symmetric monoidalfunctor (Mod E ∗ , , ⊕ ) → (Mod E ∗ , E ∗ , ⊗ ) . Furthermore, the monad structure maps η : I → T and µ : TT → T are maps of monoidal functors.Proof. Reduce to the case of free modules. (cid:3)
Completed approximation functor.
We construct completed approximation functors, which are better approximations to the homotopy of the K ( n )-localization of afree E -algebra, but which are less convenient to deal with algebraically. Thus we define b T ( M ) def = L T ( M ) where L is the functor of (3.7), and we let b α : b T ( π ∗ M ) → π ∗ L P ( M ) bethe unique factorization of α through b T ( M ).4.17. Proposition. If M is a flat E -module, then b T ( π ∗ M ) → [ T ( π ∗ M )] ∧ m and b α : b T ( π ∗ M ) → π ∗ L P ( M ) are isomorphisms.Proof. Since M is a flat E -module, T π ∗ M is a flat E ∗ module (since T commutes with filteredcolimits (4.12)), and the first isomorphism follows using (3.6).Since M is a flat module, then M ≈ hocolim J M j for some filtered topological category J , where the M α are finite free (3.14). Let N denote the category whose objects are naturalnumbers, and which has no non-identity maps. Consider the E -module N = hocolim ( i,j ) ∈ N × J L P i M j . Since each L P i M j is finite free, the E -module N is flat, and thus the map β : π ∗ N → π ∗ LN factors through an isomorphism L π ∗ N → π ∗ LN by (3.8). It is then straightforward tocheck that β is isomorphic to the approximation mapcolim T i π ∗ M j ≈ T π ∗ M α −→ π ∗ L hocolim L P i M j ≈ π ∗ L P M. HE CONGRUENCE CRITERION FOR POWER OPERATIONS IN MORAVA E -THEORY 19 Note that in the above, we have assumed that L is a continuous functor. We may in factdo this, for instance using the description of L given in (3.4)(3). (cid:3) T -algebras. As we have observed above, the functor T : Mod E ∗ → Mod E ∗ is a monad.Let Alg ∗ T denote the category of T -algebras. Every object of Alg ∗ T is a graded commutative E ∗ -algebra, and so there is a forgetful functor U : Alg ∗ T → Alg E ∗ . (Note that the image of U is contained inside the strictly graded commutative E ∗ -algebras, by (3.22).4.19. Corollary.
The forgetful functor U : Alg ∗ T → Alg E ∗ commutes with colimits.Proof. It suffices to show that U commutes with filtered colimits, reflexive coequalizers, andfinite coproducts. That it commutes with the first two types of colimit is immediate from(4.12). (cid:3) Power operations, revisited.
Let B be a T -algebra. We obtain functions P m : B → hom Mod E ∗ ( T m E ∗ , B )which are defined by sending an E ∗ -module homomorphism b : E ∗ → B to the composite T m E ∗ → T m B → B. We call the map P m a power operation . If R is a commutative E -algebra spectrum, wesee that this operation is identified with the operation P m : R ( ∗ ) → R B Σ m defined earlier,via the natural isomorphism R B Σ m ≈ hom Mod E ∗ ( E ∧ B Σ m , R ).Let f : E ∗ [ x ] → T E ∗ be the map from the free commutative E ∗ -algebra on one generatorwhich sends x to the tautological generator of T E ∗ as a T -algebra. Let f m : E ∗ → T m E ∗ be the restriction of f to the degree m part of E ∗ [ x ]. In terms of topology, f m is the map i ∗ : E ∗ ≈ E ∧∗ ( ∗ ) → E ∧∗ B Σ m induced by basepoint inclusion.4.21. Lemma.
The composite map B P m −−→ hom Mod E ∗ ( T m E ∗ , B ) → hom Mod E ∗ ( E ∗ , B ) ≈ B is the map which sends x x m Proof.
Use (3.25). (cid:3)
Plethories and plethyistic functors.
Let C be an abelian tensor category withtensor product ⊗ and unit object k , and let A denote the category of commutative monoidobjects in C with respect to the tensor product. Say that a functor U : D → A is plethyistic if (1) U reflects isomorphisms (i.e., U ( f ) iso implies f iso), and(2) U admits both a left adjoint F and a right adjoint G .It is a consequence of this definition (using Beck’s theorem [Mac71]) that if M = U F and C = U G are the monad and comonad associated to these adjoint pairs, then D is equivalentto the categories of M -algebras and C -coalgebras.The basic example of a plethyistic functor occurs when C = Mod R for some commutativering R . Then U : D →
Alg R amounts to what Borger and Wieland call a plethory [BW05].(More precisely, Borger and Wieland define a plethory to be a commutative ring P equippedwith some additional structure; they extract a plethyistic functor from this data, in such a way that P = M ( R ). Furthermore, they show [BW05, Thm. 4.9] that a plethyistic functor U : D →
Alg R determines a plethory in their sense.)4.23. Proposition.
The functor U : Alg ∗ T → Alg E ∗ is plethyistic.Proof. It is clear from the definitions that U reflects isomorphisms, and preserves limits. Itclear from (4.12) and (4.14) that U preserves colimits.Next, we construct a left adjoint F to U . Let C denote the full subcategory of Alg E ∗ consisting of E ∗ -algebras A ∗ for which there exists an object F ′ ( A ∗ ) ∈ Alg ∗ T and a naturalisomorphism Alg ∗ T ( F ′ ( A ∗ ) , B ∗ ) ≈ Alg E ∗ ( A ∗ , U ( B ∗ )); the left adjoint F exists if C = Alg E ∗ .Since U preserves limits, C is closed under colimits in Alg E ∗ , and thus it suffices to showthat Sym ∗ ( E ) and Sym ∗ ( ω / ) are in C . But it is straightforward to check that we can take F ′ (Sym ∗ ( E )) ≈ T ( E ) and F ′ (Sym ∗ ( ω / )) ≈ T ( ω / ).It follows by Beck’s theorem that U is monadic, since U preserves colimits (4.19).Next, we construct a right adjoint G : Alg E ∗ → Alg ∗ T to U . For a T -algebra A , considerthe functor X : (Alg ∗ T ) op → Set defined by X ( B ) = Hom Alg E ∗ ( U B, A ) . Since U preserves colimits, X carries colimits to limits. The category Alg ∗ T is locally pre-sentable, and so X must be representable by an object GA in Alg ∗ T . (cid:3) Weight decomposition.
Suppose that U : D → C is a plethyistic functor, and thatthe forgetful functor
A → C is monadic. Then the composite forgetful functor
D → A → C is also monadic. We write T : C → C for this monad. For future reference, we note thefollowing structure carried by T , namely natural maps of functors C → C η : M → T ( M ) , µ : T T ( M ) → T ( M ) ,ι : k → T ( M ) , δ : T ( M ) ⊗ T ( M ) → T ( M ) , which define the monad and commutative ring structures on the functor T .A weight decomposition of T is a collection of functors T k : C → C and a natural iso-morphism T ≈ L k ≥ T k such that the dotted arrows exist in each of the following diagrams. k ι " " DDDDDDDDD / / T ( M ) (cid:15) (cid:15) (cid:15) (cid:15) T k ( M ) ⊗ T ℓ ( M ) (cid:15) (cid:15) (cid:15) (cid:15) / / T k + ℓ ( M ) (cid:15) (cid:15) (cid:15) (cid:15) T ( M ) T ( M ) ⊗ T ( M ) δ / / T ( M ) M η " " FFFFFFFFF / / T ( M ) (cid:15) (cid:15) (cid:15) (cid:15) T k T ℓ ( M ) (cid:15) (cid:15) (cid:15) (cid:15) / / T kℓ ( M ) (cid:15) (cid:15) (cid:15) (cid:15) T ( M ) T T ( M ) µ / / T ( M )In each case, the dotted arrow is unique if it exists.In the case of the plethyistic functor Alg ∗ T → Alg E ∗ , T is precisely the algebraic ap-proximation functor T . It is clear that the standard splitting T ≈ L k ≥ T k is a weightdecomposition for T . HE CONGRUENCE CRITERION FOR POWER OPERATIONS IN MORAVA E -THEORY 21 Suspension map.
There are natural “suspension maps” E q : Σ q L P m ( X ) → L P m (Σ q X )for q ≥
0. This gives rise to a map of E ∗ -modules E q : ω − q/ ⊗ T m ( M ) → T m ( ω − q/ ⊗ M )which we will also call a suspension map .We record some properties of the suspension map below. They all reduce to correspondingproperties of the extended power functors.4.26. Proposition.
The diagram ω − i/ ⊗ ω − j/ ⊗ T ( M ) id ⊗ E j / / κ ⊗ id (cid:15) (cid:15) ω − i/ ⊗ T ( ω − j/ ⊗ M ) E i / / T ( ω − i/ ⊗ ω − j/ ⊗ M ) T ( κ ⊗ id) (cid:15) (cid:15) ω − ( i + j ) / ⊗ T ( M ) E i + j / / T ( ω − ( i + j ) / ⊗ M ) commutes in Mod E ∗ for all i, j ≥ and all M ∈ Mod E ∗ . Proposition.
The diagrams ω − q/ ⊗ M id ⊗ η (cid:15) (cid:15) η ( ( QQQQQQQQQQQQQ ω − q/ ⊗ T ( M ) E q / / T ( ω − q/ ⊗ M ) and ω − q/ ⊗ TT ( M ) E q / / id ⊗ µ (cid:15) (cid:15) T ( ω − q/ ⊗ T ( M )) T ( E q ) / / TT ( ω − q/ ⊗ M ) µ (cid:15) (cid:15) ω − q/ ⊗ T ( M ) E q / / T ( ω − q/ ⊗ M ) commute in Mod E ∗ for all q ≥ and all M ∈ Mod E ∗ . Define ν : T ( M ⊗ N ) → T ( M ) ⊗ T ( N ) ≈ T ( M ⊕ N ) to be the unique T -algebra map whichextends η ⊗ η : M ⊗ N → T ( M ) ⊗ T ( N ).4.28. Proposition.
The diagram ω − i/ ⊗ ω − j/ ⊗ T ( M ⊗ N ) “ E i + j ” (cid:15) (cid:15) id ⊗ id ⊗ ν / / ω − i/ ⊗ ω − j/ ⊗ T ( M ) ⊗ T ( N ) id ⊗ τ ⊗ id (cid:15) (cid:15) T ( ω − i/ ⊗ ω − j/ ⊗ M ⊗ N ) T (id ⊗ τ ⊗ id) (cid:15) (cid:15) ω − i/ ⊗ T ( M ) ⊗ ω − j/ ⊗ T ( N ) E i ⊗ E j (cid:15) (cid:15) T ( ω − i/ ⊗ M ⊗ ω − j/ ⊗ N ) ν / / T ( ω − i/ ⊗ M ) ⊗ T ( ω − j/ ⊗ N ) commutes in Mod E ∗ for all i, j ≥ and all M, N ∈ Mod E ∗ , where “ E i + j ” means the mapdefined by composition along the top of the diagram in (4.26) . Twisted bialgebras
Let R be a commutative ring. An algebra under R is an associative ring Γ togetherwith a ring homomorphism η : R → Γ. Note that the image of η is not assumed to be central(in which case Γ is an R -algebra ).An twisted cocommutative R -bialgebra (or bialgebra ) is an algebra Γ under R whichis equipped with certain additional structures which make the category of right Γ-modulesinto a symmetric monoidal category, in such a way that this symmetric monoidal productcoincides with the tensor product over R of the underlying R -modules. In the case when R is central in Γ, then we have the conventional notion of a cocommutative R -bialgebra (likea cocommutative Hopf algebra, but without an antipode).The original definition of bialgebra is due to Sweedler [Swe74], as modified by [Tak77],under the name of “ × R -bialgebra”. (Sweedler described a somewhat more general situation,in which the monoidal structure is not required to be symmetric.) This notion as wellas various generalizations have been much studied in the literature on Hopf algebras andallied notions. It has entered topology in various guises. I first learned about the notionfrom [Voe03]; the motivic Steenrod algebra is naturally a twisted bialgebra over the motivichomology of the base scheme. Bialgebras can be thought of as a kind of “dual” version of thenotion of Hopf algebroids (or more precisely, of “affine category schemes”), an observationwe will make explicit below ( § Bimodules, multimorphisms, and functors.
Let R be a commutative ring, andsuppose that P and Q be R -bimodules. We write P R ⊗ R Q def = P ⊗ Q/ ( pr ⊗ q ∼ p ⊗ rq ) ,P R ⊗ Q R def = P ⊗ Q/ ( pr ⊗ q ∼ p ⊗ qr ) . Because R is commutative, each of these admit three possible R -modules structures. Forinstance, P R ⊗ Q R admits the following three R -module structures (two on the left, one onthe right): r · ( p ⊗ q ) = rp ⊗ q ; r · ( p ⊗ q ) = p ⊗ rq ; ( p ⊗ q ) · r = pr ⊗ q = p ⊗ qr. If P and Q , . . . , Q K are R -bimodules, then a k -multimorphism is a function f : P → ( Q ) R ⊗ · · ·⊗ ( Q k ) R which is a map of right R -modules, and which is a map of left R -modulesfor each of the k left R -module structures on the target. Note that a 0-multimorphism isjust a map P → R of right R -modules.Given an R -bimodule P , let H M : Mod R → Mod R denote the functor defined by H P ( M ) =hom R ( P, M ), where this means the set of homomorphisms with respect to the right R -modulestructure on P . Thus, f ∈ H P ( M ) is a additive map f : P → M such that f ( pr ) = f ( p ) r for all p ∈ P and r ∈ R . A bimodule map b : P → Q induces a natural transformation˜ b : H Q → H P , defined by˜ b : H Q ( M ) → H P ( M ) , f ( p f ( b ( p )) . HE CONGRUENCE CRITERION FOR POWER OPERATIONS IN MORAVA E -THEORY 23 More generally, a k -multimorphism b : P → ( Q ) R ⊗ · · · ⊗ ( Q k ) R induces a natural transfor-mations ˜ b : H Q ( M ) R ⊗ · · · ⊗ H Q k ( M k ) R → H P ( M ⊗ R · · · ⊗ R M k )( f ⊗ · · · ⊗ f k ) ( p X α f ( q α ) ⊗ · · · ⊗ f ( q αk )) , where b ( p ) = P α q α ⊗ · · · ⊗ q αk . In particular, a 0-multimorphism b : P → R induces˜ b : R → H P ( R ) , r ( p b ( rp )) . Remark.
The assignments b ˜ b described above satisfy a number of useful properties,relating to composition of functors and to the tensor product of R -modules; these can besafely left to the reader. The language of “multicategories” [Lei04, § B , additiveendofunctors Mod R → Mod R and multilinear natural transformations form a multicategory F , and there is a morphism H : B → F of multicategories. Furthermore, B and F admitan additional “monoidal” structure (corresponding to bimodule tensor product and functorcomposition respectively), and H is compatible with this monoidal structure (in the weaksense).5.3. Twisted commutative R -bialgebra. Let R be a commutative ring. An R -bialgebra (more precisely, a twisted cocommutative R -bialgebra is data (Γ , ǫ, ∆ , η, µ ), consistingof (a) an associative ring Γ;(b) a map η : R → Γ of rings;(c) a map ǫ : Γ → R of right R -modules such that ǫ ( η ( r )) = r for all r ∈ R and ǫ ( xy ) = ǫ ( η ( ǫ ( x )) y ) for all x, y ∈ Γ;(d) a map ∆ : Γ → Γ R ⊗ Γ R which is a 2-multimorphism of R -bimodules, which is iscoassociative and cocommutative with counit ǫ , and such that∆( xy ) = X x ′ i y ′ j ⊗ x ′′ i y ′′ j , where ∆( x ) = P x ′ i ⊗ x ′′ i and ∆( y ) = P y ′ j ⊗ y ′′ j .Note that the multiplication on Γ descends to a map µ : Γ R ⊗ R Γ → Γ.The functor H Γ is a comonad, with comonadic structure ˜ η : H Γ → I and ˜ µ : H Γ → H Γ H Γ induced by η and µ . The maps ǫ and ∆ induce natural transformations˜ ǫ : R → H Γ ( R ) , ˜∆ : H Γ ( M ) ⊗ R H Γ ( N ) → H Γ ( M ⊗ R N )which give H Γ the structure of a symmetric monoidal functor; this structure is compatiblethe monad structure, so that ˜ η and ˜ µ are transformations of monoidal functors.By a module over the R -bialgebra Γ, we mean a pair ( M, ψ M ) consisting of an R -module M , and a map ψ M : M → H Γ ( M ) of R -modules such that ˜ ηψ M = id M and H Γ ( ψ M ) ψ M =˜ µψ M . That is, a module is defined to be a coalgebra for the comonad ( H Γ , ˜ η, ˜ µ ). Let Mod Γ denote the category of modules over Γ.Note that what we are calling a module over Γ really does coincide with the usual notionof a right Γ-module: the map ψ M : M → H Γ ( M ) is adjoint to a map M R ⊗ R Γ → M defininga right Γ-module structure on M . Symmetric monoidal structure on
Mod Γ . There is a canonical Γ-module structureon R , defined by taking ψ R = ˜ ǫ : R → H Γ ( R ).Given M, N ∈ Mod Γ we define their tensor product to be ( M ⊗ R N, ψ M ⊗ R N ), where ψ M ⊗ R N def = ˜∆( ψ M ⊗ ψ N ) : M ⊗ R N → H Γ ( M ⊗ R N ) . That is is indeed a Γ-module follows from the fact that the comonad structure maps ˜ η and˜ µ are compatible with monoidal structures.5.5. Proposition.
The above defines a symmetric monoidal structure on the category
Mod Γ ,such that the forgetful functor U : Mod Γ → Mod R is a symmetric monoidal functor. -algebras. Let Γ be an R -bialgebra. By a Γ -algebra , we mean a commutative monoidobject B in the symmetric monoidal category of Γ-modules. That is, B is equipped withmaps i : R → B and m : B ⊗ B → B of Γ-modules which also provide it with the structureof a commutative ring. We write Alg Γ for the category of Γ-algebras.The free Γ -algebra on an R -module M is seen to have the form Sym ∗ R ( M ⊗ R Γ), wherethe symmetric powers are taken with respect to the right R -module structure on Γ.The forgetful functor Alg Γ → Alg R is plethyistic, in the sense of § Graded bialgebras and graded affine category schemes. A grading for an R -bialgebra Γ is a decomposition Γ ≈ L k ≥ Γ[ k ], such that µ (Γ[ k ] ⊗ Γ[ ℓ ]) ⊆ Γ[ k + ℓ ] , η ( R ) ⊆ Γ[0] , ∆(Γ[ k ]) ⊆ Γ[ k ] R ⊗ Γ[ k ] R . Thus in particular Γ is a graded ring, and each graded piece Γ[ k ] is a cocommutative coal-gebra.Suppose that each Γ[ k ] is projective and finitely generated as a right R -module. Let A [ k ] = H Γ[ k ] ( R ). The A [ k ]’s admit two different R -module structures, which we call the“source” and “target” module structures; the “source” module structure is defined by ( r · s f )( x ) = f ( xr ), while the “target” module structure is defined by ( r · t f )( x ) = f ( rx ), where f ∈ A k , r ∈ R , x ∈ Γ[ k ].We define µ : A [ k ] ⊗ R A [ k ] → A [ k ] , µ ( f ⊗ g )( x ) = X f ( x ′ i ) g ( x ′′ i ) ,s ∗ : R → A [ k ] , s ∗ ( r )( x ) = ǫ ( xr ) = ǫ ( x ) r,t ∗ : R → A [ k ] , t ∗ ( r )( x ) = ǫ ( rx ) ,i ∗ : A [0] → R, i ∗ ( f ) = f ( η (1)) ,c ∗ : A [ k + ℓ ] → A [ k ] s ⊗ R t A [ ℓ ] , X f ′′ i ( f ′ i ( x ) y ) = f ( xy ) , where ∆( x ) = P x ′ i ⊗ x ′′ i and we write c ∗ ( f ) = P f ′ j ⊗ f ′′ j . The map µ makes A [ k ] into acommutative ring, and the “source” and “target” R -module structures on A [ k ] are are thesame as those induced by the ring homomorphisms s ∗ and t ∗ : r · s f = f ( xr ) = µ ( s ∗ ( r ) ⊗ f ) , r · t f = f ( rx ) = µ ( t ∗ ( r ) ⊗ f ) . By a graded category , we mean a small category C equipped with a degree functiondeg : mor C → N , such that (i) deg( α ◦ β ) = deg( α )+deg( β ), and deg(id) = 0. The data A =( R, A [ k ] , s ∗ , t ∗ , i ∗ , c ∗ ) described above determine a graded affine category scheme ; that HE CONGRUENCE CRITERION FOR POWER OPERATIONS IN MORAVA E -THEORY 25 is, they corepresent a functor from commutative rings to graded categories. In particular, R represents the set of objects of C , and A [ r ] represents the set of morphisms of C of degree r . The maps s ∗ , t ∗ , i ∗ , c ∗ correspond to source, target, identity, and composition in C .If A = ( R, A [ r ] , s ∗ , t ∗ , i ∗ , c ∗ ) is the data representing an affine graded category schemeover R , let V : Mod R → Mod R be the functor defined by V ( M ) def = Y r ≥ A [ r ] s ⊗ R M, where V ( M ) obtains its R -module structure from the “target” module structure on the A [ r ]’s. The functor V admits the structure of a comonad, in an apparent way; the counitmap V M → M is the projection V M → A [0] s ⊗ R M i ∗ ⊗ id −−−→ R ⊗ R M ≈ M , and thecomultiplication V M → V V M is the map Y r A [ r ] s ⊗ R M → Y ℓ A [ k ] s ⊗ R t Y k A [ ℓ ] s ⊗ M ! ≈ Y k,ℓ A [ k ] s ⊗ R t A [ ℓ ] s ⊗ R M induced by the maps c ∗ . A coalgebra for the comonad V is called an A -comodule , and thecategory of A -comodules is denoted Comod A .Under our hypothesis on Γ, the natural transformation χ : V ( M ) → H Γ ( M ) of functorsMod R → Mod R defined by χ ( f ⊗ m ) ( x f ( x ) m )is an isomorphism for all M , and the comonad structure on V corresponds exactly to thecomonad structure on H Γ . Thus, we get the following.5.8. Proposition. If Γ is a graded bialgebra over R such that each graded piece Γ[ k ] isprojective and finitely generated as a right R -module, and if A is the corresponding gradedaffine category scheme, then there is an isomorphism of categories Mod Γ ≈ Comod A . The bialgebra of power operations
In this section, we are going to construct a twisted commutative E -bialgebra Γ, togetherwith a symmetric object ω in Mod Γ ; the bialgebra Γ will come with a grading in the senseof (5.7).From the point of view of topology, Γ is the “algebra of power operations in Morava E -theory”; more precisely, it is (up to issues of completion), the algebra of additive operationson π of a K ( n )-local E -algebra spectrum. The difficult part of the construction was givenin [Str98]. From the point of view of the theory of plethories, Γ is the “additive bialgebra”of the plethory associated to T , as described in [BW05, § Additive operations on T -algebras. Let P = Hom Alg ∗ T ( T ( E ∗ ) , T ( E ∗ )). The set P has the structure of a monoid, under composition of morphisms. Because there is anatural bijection Hom Alg ∗ T ( T ( E ∗ ) , A ) ≈ A , defined by evaluation at the canonical generator ι ∈ T ( E ), we see that P is naturally identified with the monoid of endomorphisms of theforgetful functor Alg T → Set which sends to A to the underlying set A of the degree 0 partof A . In particular, evaluation at ι ∈ T ( E ) defines a natural bijection π : P → [ T ( E ∗ )]
06 CHARLES REZK to the degree 0 part of T ( E ∗ ). We will use this bijection implicitly in what follows.Write ◦ : P × P → P for the monoid product on P , defined in terms of composition ofendomorphisms of the forgetful functor. If f, g ∈ P correspond to maps ˜ f , ˜ g : E → T E of E -modules, then f ◦ g corresponds to E g −→ T E T ˜ f −→ TT E µ −→ T E . Observe that if f ∈ T m E and g ∈ T n E , then f ◦ g ∈ T mn E ; this is a consequence of thefact that µ : TT → T carries T m T n into T mn ( § P has the structure of an abelian group, by addition of natural endomorphismsof the forgetful functor, or equivalently by addition in T ( E ∗ ). We observe that ◦ is “leftadditive”, in the sense that ( f + g ) ◦ h = ( f ◦ h ) + ( g ◦ h )for f, g, h ∈ P .6.2. The additive bialgebra associated to T . Let Γ ⊂ P denote the subset of elements f ∈ P which induce additive natural endomorphisms of the forgetful functor; that is, f ∈ Γif it induces an endormorphism of the forgetful functor Alg ∗ T → Ab. Thus, Γ is naturally anassociative ring, isomorphic to the ring of endomorphisms of the forgetful functor Alg ∗ T → Abto abelian groups.Define ∆ + : T ( E ∗ ) → T ( E ∗ ) ⊗ T ( E ∗ )to be the composite T ( E ) T ( ∇ ) −−−→ T ( E ⊕ E ) γ − −−→ ∼ T ( E ) ⊗ T ( E ) , where ∆ : E ∗ → E ∗ ⊕ E ∗ denotes the diagonal. Say an element f ∈ [ T ( E ∗ )] is primitive with respect to ◦ if ∆ + ( f ) = f ⊗ ⊗ f . It is a straightforward exercise to check thatunder the bijection P ∼ −→ [ T ( E ∗ )] , elements of Γ correspond precisely to primitive elements.Translating this into topology by means of the isomorphism T m ( E ∗ ) ≈ E ∧∗ B Σ m , we seethat Γ ≈ L k ≥ Γ[ k ], where Γ[ k ] ⊂ T p k ( E ) defined by0 → Γ[ k ] → E ∧ B Σ p k → M The submodule Γ[ k ] ⊂ T p k ( E ) is a direct summand. In particular, Γ[ k ] is a finitely generated free E -module.Proof. By standard considerations involving Sylow subgroups of Σ p k , it is straightforward toshow that for k ≥ 1, Γ[ k ] is the kernel of the single transfer map E ∧ B Σ p k → E ∧ B (Σ p k − ) p .Strickland [Str98] proves that cokernel of the cohomology transfer is a finite free E -module,so that there is an exact sequence E B (Σ p k − ) p τ −→ E B Σ p k π −→ R p k → HE CONGRUENCE CRITERION FOR POWER OPERATIONS IN MORAVA E -THEORY 27 where R p k is a finite free E -module. Taking E -linear duals gives the desired result. (cid:3) Observe that Γ[ k ] ◦ Γ[ ℓ ] ⊆ Γ[ k + ℓ ], whence Γ is a graded ring. The unit 1 ∈ Γ corresponds tothe canonical generator ι ∈ T ( E ∗ ). From now on we omit the “ ◦ ” notation when discussingthe product on Γ.The map E → P defined by r r · η : E → Γ, so that η ( r ) = π ( T ( λ r )), where λ r : E → E denotes multiplication by r ∈ E . The map η : E → Γis a ring homomorphism. Observe that the image of η need not be central in Γ (and in factit isn’t, except when k = F p and n = 1.)The ring Γ becomes a E -bimodule by η , and multiplication descends to a map µ : Γ E ⊗ E Γ → Γ.Let ǫ × : P → E and ∆ × : P → P ⊗ P be the maps isomorphic the morphisms of T -algebras T ( E ) → E , T ( E ) → T ( E ⊕ E )which correspond to the maps E −→ E , E ∼ −→ T ( E ) ⊗ T ( E ) ⊂ T ( E ) ⊗ T ( E )of E -modules. (In terms of topology, ∆ × and ǫ × are induced by the space-level diagonaland projection maps B Σ m → B Σ m × B Σ m and B Σ m → ∗ .)6.4. Proposition. The maps ǫ × and ∆ × restrict to functions ǫ : Γ → E , ∆ : Γ → Γ E ⊗ Γ E , and the data (Γ , ǫ, ∆ , η, µ ) constitutes a graded twisted commutative E -bialgebra.Proof. The map Γ ⊗ E → P ⊗ is an inclusion, by (6.3). Its image is the set of “interlinearelements”; an element f ∈ P ⊗ P is interlinear if the induced operation ¯ f : A × A → A on an T -algebra A is additive in each variable, and if ¯ f ( xc, y ) = ¯ f ( x, yc ) for all c ∈ E ; see [BW05,Prop. 10.2] for a proof. It is clear that ∆ × carries Γ into the set of interlinear elements, andthus we obtain the desired map ∆. The rest of the argument is as in [BW05]. (cid:3) Note that the diagonal map sends each graded piece to a single grading; that is, ∆(Γ[ k ]) ⊆ Γ[ k ] E ⊗ Γ[ k ] E . Thus, ǫ and ∆ give Γ[ k ] a cocommutative coalgebra structure, dual to thering structure on E B Σ p k / (transfer). Thus, Γ ≈ L Γ[ k ] is a graded bialgebra, in the senseof § Example. Let G be the multiplicative group over a perfect field k of characteristic p , andlet E be the corresponding Morava E -theory. In this case, E is the extension of p -complete K -theory to the Witt ring W k . Let a a σ be the lift to W k of the Frobenius automorphismof k . In this case, Γ is isomorphic to the ring W k h ψ p i , where ψ p is the p th Adams operation(viewed as a power operation on E -algebras), which satisfies the commutation relation ψ p · a = a σ · ψ p for a ∈ W k . Observe that W k is not central in Γ (unless k ≈ F p .) The grading on Γ is such that Γ[0] = W k ,and ψ p ∈ Γ[1]. Since ψ p is a multiplicative operation, we have ǫ ( ψ p ) = 1 , ∆( ψ p ) = ψ p ⊗ ψ p . Let Alg Γ denote the category of Γ-algebras. We define a functor U : Alg T → Alg Γ by U ( B ) = Hom Alg T ( P, B ) , equipped with the Γ-algebra structure induced by the action of Γ on P . The underlying E -module of U B is just the underlying E -module of B .6.6. The Γ -module ω . Let ω be the object in Mod Γ defined by the reduced cohomologyof CP ≈ S : ω = Ker[ U ( E ( CP )) → U ( E )] . It is free of rank 1 as an E -module, and thus is a symmetric object in Mod Γ . This is aslight abuse of notation; we already use ω to represent the kernel of E ∗ ( CP ) → E ∗ in thecategory of E ∗ -modules.7. Operation algebras for non-zero gradings In the previous section we constructed a bialgebra Γ, which acts naturally on the evendegree part of a T -algebra. In this section we describe what happens when we take oddgradings into account. In the end, we will show that underlying an T -algebra is an object ofMod ∗ Γ , the Z / ω -twisted tensor category of modules over Γ, in terms of the formalismof § 2. We will define Alg ∗ Γ to be the category of commutative ring objects in Mod ∗ Γ , and willobtain a “forgetful” functor U : Alg ∗ T → Alg ∗ Γ . The functor U is plethyistic (7.25), in the sense of § Z -graded theory. In particular, foreach q ∈ Z , we will construct Γ q as a set of additive operations sitting inside P q ≈ Hom Alg ∗ T ( T ( ω q/ ) , T ( ω q/ )) ≈ L π − q L P m (Σ − q E ), analogously to the construction of Γ in § 6. The ring Γ q naturally acts on π − q of a K ( n )-local E -algebra. The collection of { Γ q } forms a kind of “ Z -graded bialgebra”. In particular, there will be coproduct maps∆ i,j : Γ i + jE → Γ iE ⊗ Γ jE which determine tensor functors ⊗ : Mod Γ i × Mod Γ j → Mod Γ i + j . Furthermore, there are “forgetful” functors ˜ U i : Alg ∗ T → Mod Γ i . These will fit together intoa tensor functor ˜ U ∗ : Alg ∗ T → Mod Z ∗ ˜ U ∗ E ∗ where the target is a category of “ Z -graded ˜ U ∗ E ∗ -modules” built from the categories Mod Γ i (see § Z / § 2. First, the periodicity of the theory E ∗ produces natural equivalences Mod Γ i ≈ Mod Γ i +2 (7.11). Second, the “suspension map” defines an isomorphism Γ → Γ of rings (7.17).Using these observations, we reformulate the category Mod Z ∗ ˜ U ∗ E ∗ in terms of the bialgebraΓ, by producing (in § Z ∗ ˜ U ∗ E ∗ and the Z / ω -twisted category Mod ∗ Γ of Γ-modules. HE CONGRUENCE CRITERION FOR POWER OPERATIONS IN MORAVA E -THEORY 29 The rings Γ q . Let P q = Hom Mod E ∗ ( ω q/ , T ( ω q/ )) ≈ Hom Alg ∗ T ( T ( ω q/ ) , T ( ω q/ )). Asin § P q with the monoid of endomorphisms of thefunctor A Hom Mod E ∗ ( ω q/ , A ) : Alg ∗ T → Set . We let Γ q ⊂ P q denote the subset of elements which correspond to additive endomorphisms.Thus Γ q is isomorphic to the ring of endomorphisms of A Hom Mod E ∗ ( ω q/ , A ) : Alg ∗ T → Ab.Define ∆ + : P q → P q ⊗ P q by postcomposition with γ − ◦ T ( ∇ ) : T ( ω q/ ) → T ( ω q/ ) ⊗ T ( ω q/ ). As in § q corresponds to the set of primitives in P q with respectto ∆ + . Observe that Γ is just the ring Γ of § Proposition. There are non-cannonical isomorphisms of rings Γ q ≈ Γ q +2 k for all q, k ∈ Z .Proof. This is immediate from the definitions, and the fact that ω q/ ≈ ω ( q +2 k ) / as E ∗ -modules. (cid:3) In terms of topology, Γ q = L k ≥ Γ q [ k ], where Γ q [ k ] ⊆ Hom Mod E ∗ ( ω q/ , T p k ( ω q/ )) can beidentified with Hom Mod E ∗ ( E ∧ S − q , K q [ k ]), where K q [ k ] fits in the exact sequence0 → K q [ k ] → E ∧∗ B Σ − qρ pk p k → M For all q ∈ Z , the sub- E -module Γ q [ k ] ⊆ Hom Mod E ∗ ( ω q/ , T p k ( ω q/ )) isa direct summand, and thus in particular is a finite free E -module.Proof. (Compare [Str98, Prop. 5.6].) We have already addressed the case of q = 0 (6.3),and thus need only consider q = 1. Let B ∗ def = L m ≥ π ∗ L P m Σ E ; this is a commutativeHopf algebra, with product given by the ring structure of P Σ E , and coproduct given by P ( ∇ ) : P Σ E → P (Σ E ∨ Σ E ). The result follows from the observation that B ∗ is a primitivelygenerated exterior algebra, since B ∗ ≈ T ( ω / ). We prove this using the bar spectral sequence E = Tor E ∗ ∗ ( A ∗ , A ∗ ) = ⇒ B ∗ where A ∗ ≈ L m ≥ π ∗ L P m E . By a result of Strickland, following Kashiwabara [Str98, Prop.5.1], A ∗ is a polynomial algebra, finite type with respect to the grading determined by m .Thus Tor E ∗ ∗ ( A ∗ , A ∗ ) is an exterior algebra, with generators represented by the homologysuspensions of the generators of the polynomial ring A ∗ . These classes survive, since theydetect the image of the suspension maps π ∗ L P m E → π ∗ +1 L P m Σ E , and thus the spectralsequence collapses at E . (cid:3) We will need the following result in § Proposition. The inclusion of Γ q ⊆ Hom Mod E ∗ ( ω q/ , T ( ω q/ )) induces an isomorphism Sym E ∗ (Γ q ⊗ ω q/ ) ⊗ Q ≈ T ( ω q/ ) ⊗ Q of graded E ∗ -modules.Proof. By (7.3), Γ q ⊗ ω q/ is the primitives of the Hopf algebra T ( ω q/ ), and is a directsummand. Thus, (Γ q ⊗ ω q/ ) ⊗ Q is the primitives of T ( ω q/ ) ⊗ Q . The result follows fromthe structure theory of graded Hopf algebras. (cid:3) Remark. It may be helpful to consider the following picture. Fix k ≥ 1, and considerPrim q E ∧ q B Σ qρ pk p k ≈ π q L P p k Σ q E ։ Ind q , where Prim q and Ind q denote the part ofthe primitives and indecomposables of L m π ∗ L P m Σ q E associated the m = p k summandin dimension q ; the ring Γ q is the direct sum of the E -modules Prim − q as k varies. Thesuspension map π q L P p k Σ q E → π q +1 L P p k Σ q +1 E factors through Prim q → Ind q +1 ; from thesemaps we obtain the following diagram, which is 2-periodic.Prim − ∼ (cid:15) (cid:15) Prim f (cid:15) (cid:15) Prim ∼ (cid:15) (cid:15) Prim f (cid:15) (cid:15) ∼ : : uuuuuuuuuuu Ind − ∼ tttttttttt Ind ∼ : : uuuuuuuuu Ind ∼ : : uuuuuuuuu Ind ∼ ; ; vvvvvvvvvv The vertical maps Prim q → Ind q are isomorphisms for odd q using (7.3). The map Ind → Prim is an isomorphism by Theorems 8.5 and 8.6 of [Str98], where the result is statedin “dual” form; specifically, in that paper it is proved that the image of Prim E B Σ p k → Ind E B Σ p k is generated by the Euler class of ¯ ρ C p k . The map f is not an isomorphism, butis a monomorphism with torsion cokernel; the present paper is a essentially a meditation oncok f .The bijection Hom Alg ∗ T ( T ( ω q/ ) , T ( ω q/ )) → P q induces an associative monoid structureon P q , defined by composition, and thus descends to a multiplication on Γ q . There is a ringhomomorphism η q : E → Γ q , defined as for Γ.Finally, let ∆ × i,j : P i + j → P i ⊗ P j be the map induced by the “multiplicativity map” ν : T ( ω i/ ⊗ ω j/ ) → T ( ω i/ ⊕ ω j/ ). As before, we have ǫ × : P → E .7.6. Proposition. The maps ∆ × i,j restrict to maps ∆ i,j : Γ i + jE → Γ iE ⊗ Γ jE , which in turn induce functors ⊗ : Mod Γ i × Mod Γ j → Mod Γ i + j , which are unital, associative, and commutative in the sense that there are natural isomor-phisms E ⊗ M ≈ M ≈ M ⊗ E of functors Mod Γ i → Mod Γ i , M ⊗ ( M ⊗ M ) ≈ ( M ⊗ M ) ⊗ M , and M ⊗ M ≈ M ⊗ M , HE CONGRUENCE CRITERION FOR POWER OPERATIONS IN MORAVA E -THEORY 31 where E is regarded as a Γ -module. (Note that the interchange map introduces a sign whenapplied to elements of odd degree.) Define functors ˜ U q : Alg ∗ T → Mod Γ q by sending B ∈ Alg ∗ T to the right Γ q -moduleHom Alg ∗ T ( T ( ω q/ ) , B ) ≈ Hom Mod E ∗ ( ω q/ , B ).7.7. Remark. If A is a K ( n )-local commutative E -algebra, so that π ∗ A is a T -algebra, thentracing through the definitions reveals that the underlying E -module of ˜ U q ( π ∗ A ) is canon-ically identified with π − q A . Thus, we see that π − q A naturally carries the structure of aΓ q -module.7.8. Proposition. There are natural transformations ˜ U i ( M ) ⊗ ˜ U j ( N ) → ˜ U i + j ( M ⊗ N ) offunctors Alg ∗ T × Alg ∗ T → Mod Γ i + j .Proof. Straightforward. (cid:3) Z -graded Γ ∗ -modules and ˜ U ∗ E ∗ -modules. By a Z -graded Γ ∗ -module , we mean atuple M = ( M i ) i ∈ Z , where M i is a right Γ i -module. We write Mod Z ∗ Γ ∗ for the category of Z -graded Γ ∗ -modules. This category admits the structure of an additive symmetric monoidalcategory, by the tensor product functors of (7.6), so that M ⊗ N = ( P i ) i ∈ Z , where P i = L q M q ⊗ N i − q .Let ˜ U ∗ : Alg ∗ T → Mod Z ∗ Γ ∗ be the functor which sends a T -algebra B to the Z -graded Γ ∗ -module ( ˜ U i B ). By (7.8), the functor ˜ U ∗ is a lax symmetric monoidal functor; i.e., there isa coherent natural transformation ˜ U ∗ B ⊗ ˜ U ∗ C → ˜ U ∗ ( B ⊗ C ).By these remarks, it follows that ˜ U ∗ E ∗ is a commutative monoid object in Mod Z ∗ Γ ∗ , andthat ˜ U ∗ B is tautologically a ˜ U ∗ E ∗ -module object for all B in Alg ∗ T . Let Mod Z ∗ ˜ U ∗ E ∗ denote thecategory of ˜ U ∗ E ∗ -modules; it is a symmetric monoidal category, with tensor product definedby M ⊗ ˜ U ∗ E ∗ N def = cok (cid:2) M ⊗ ˜ U ∗ E ∗ ⊗ N ⇒ M ⊗ N (cid:3) . There is an evident forgetful functor Mod Z ∗ ˜ U ∗ E ∗ → Mod Z ∗ E ∗ to the Z -graded category of E ∗ -modules, and this forgetful functor is strongly symmetric monoidal: the underlying Z -graded E ∗ -module of M ⊗ ˜ U ∗ E ∗ N is just the usual Z -graded tensor product of the underlying E ∗ -modules of M and N .Thus, the functor ˜ U ∗ : Alg ∗ T → Mod Z ∗ Γ ∗ lifts tautologically to a functor˜ U ∗ : Alg ∗ T → Mod Z ∗ ˜ U ∗ E ∗ . Furthermore, it is straightforward to check (by looking at what happens on the underlying E ∗ -modules) that the lifted ˜ U ∗ is a strong symmetric monoidal functor; i.e., ˜ U ∗ B ⊗ ˜ U ∗ E ∗ ˜ U ∗ C → ˜ U ∗ ( B ⊗ C ) is an isomorphism.The rest of this section is devoted to giving a more elementary description of Mod Z ∗ ˜ U ∗ E ∗ .7.10. Periodicity. Observe that the underlying E -module of ˜ U i E ∗ is canonically isomor-phic to π − i E .7.11. Proposition. The functor Mod Γ i → Mod Γ i +2 defined by M M ⊗ ˜ U E ∗ is anequivalence of categories, and there are natural isomorphisms ˜ U i ( M ) ⊗ ˜ U E ∗ ≈ ˜ U i +2 ( M ) offunctors Alg ∗ T → Mod Γ i +2 . Proof. The isomorphism ˜ U ( E ) ⊗ ˜ U − ( E ) ≈ ˜ U ( E ) ≈ E produces an inverse up toisomorphism for this functor. (cid:3) As a consequence, we note the following.7.12. Lemma. Let V ′ : Mod Γ × Mod Γ → Mod Z ∗ ˜ U ∗ E ∗ be the functor which on objects sends ( M , M ) to N i , where N i = M ⊗ ˜ U i E ∗ and N i +1 = M ⊗ ˜ U i E ∗ , and N is given a ˜ U ∗ E ∗ -module structure in the evident way. Then V ′ is an equivalence of categories. Suspension. Define functions e q : Γ q + i → Γ i for all q ≥ i ∈ Z so that e q ( f )is the composite ω i/ ≈ ω − q/ ⊗ ω ( q + i ) / ⊗ f −−−→ ω − q/ ⊗ T ( ω ( q + i ) / ) E q −→ T ( ω − q/ ⊗ ω ( q + i ) / ) ≈ T ( ω i/ ) , where E q is was defined in § Proposition. The maps e q : Γ i + q → Γ i are homomorphisms of associative rings under E .Proof. This is a lengthy but straightforward diagram chase, which depends essentially on(4.27). (cid:3) Proposition. The diagram Γ i + j + a + b ∆ i + a,j + b / / e i + j (cid:15) (cid:15) Γ i + a ⊗ Γ j + be i ⊗ e j (cid:15) (cid:15) Γ a + b ∆ a,b / / Γ a ⊗ Γ b commutes.Proof. This is a lengthy but straightforward diagram chase, which depends essentially on(4.28). (cid:3) Let e ∗ q : Mod Γ i → Mod Γ i + q denote the functor obtained by restricting along the ringhomomorphism e q .7.16. Proposition. There are natural and coherent isomorphisms e ∗ a ( M ) ⊗ e ∗ b ( N ) → e ∗ a + b ( M ⊗ N ) of functors Mod Γ i × Mod Γ j → Mod Γ i + j + a + b . Proposition. The suspension map e : Γ i → Γ i is an isomorphism for all i ∈ Z .Thus, the functor e ∗ : Mod Γ i → Mod Γ i is an equivalence of categories.Proof. By periodicity, this amounts to the observation that e : Γ − → Γ − is an isomor-phism. We read this off of the fact that the suspension map E B Σ ρp k → E B Σ ρp k is anisomorphism on primitives, as proved in Strickland and discussed in (7.5). (cid:3) HE CONGRUENCE CRITERION FOR POWER OPERATIONS IN MORAVA E -THEORY 33 Z -graded Γ ∗ -modules from spheres. For q ≥ 0, define a Z -graded ˜ U ∗ E ∗ -module η ( q ) by η ( q ) def = Ker (cid:2) ˜ U ∗ ( E ∗ S q ) → ˜ U ∗ ( E ∗ (pt)) (cid:3) . Lemma. There are natural and coherent isomorphisms η ( q ) ⊗ ˜ U ∗ E ∗ η ( q ′ ) ≈ η ( q + q ′ ) .Proof. Read this off using the K¨unneth isomorphism E ∗ ( S q × S q ′ ) ≈ E ∗ S q ⊗ E ∗ S q ′ . (cid:3) Clearly, η (0) ≈ ˜ U ∗ E ∗ . Observe that η (2) is a symmetric object of Mod Z ∗ ˜ U ∗ E ∗ (in the senseof § η (1) is an odd square-root of η (2).7.20. Proposition. There is an isomorphism η ( q ) ≈ e ∗ q ( ˜ U ∗ E ∗ ) of Z -graded Γ ∗ -modules.Proof. This is a straightforward consequence of (3.29). (cid:3) Proposition. There are natural isomorphisms e ∗ q M ≈ η ( q ) ⊗ ˜ U ∗ E ∗ M of Z -graded ˜ U ∗ E ∗ -modules.Proof. Clear from (7.20) and (7.21). (cid:3) Z / -graded Γ -modules. Let Mod ∗ Γ denote the category of Z / § 2, where we twist by thesymmetric object ω ∈ Alg Γ described above.Let V : Mod Γ → Mod Z ∗ ˜ U ∗ E ∗ denote the functor which on objects sends M to N , where N i = M ⊗ ˜ U i E ∗ and N i +1 = 0.It is clear that V is a strong symmetric monoidal functor. Furthermore, it is clear from thedefinitions that V ( ω ) ≈ η (2).Using the method of (2.6), we obtain a symmetric monoidal functor V ∗ : Mod ∗ Γ → Mod Z ∗ ˜ U ∗ E ∗ which extends V and sends ω / to η (1).7.23. Proposition. The symmetric monoidal functor V ∗ : Mod ∗ Γ → Mod Z ∗ ˜ U ∗ E ∗ is an equiva-lence of categories.Proof. Explicitly, the functor V ∗ sends an object M = { M , M } of Mod ∗ Γ to the object V ( M ) ⊕ V ( M ) ⊗ η (1). By (7.21) this functor is naturally isomorphic to the M V ( M ) ⊕ e ∗ V ( M ). Since e ∗ : Mod Γ → Mod Γ is an equivalence of categories by (7.17), the resultfollows using (7.12). (cid:3) The forgetful functor U . Since V ∗ is an equivalence of tensor categories, there is afunctor ˜ U : Alg ∗ T → Mod ∗ Γ such that there is a monoidal isomorphism V ∗ ˜ U ≈ ˜ U ∗ of monoidalfunctors.7.25. Proposition. The functor ˜ U : Alg ∗ T → Mod ∗ Γ lifts to a plethyistic functor U : Alg ∗ T → Alg ∗ Γ . Proof. In the sequence of forgetful functorsAlg ∗ T U −→ Alg ∗ Γ U ′ −→ Alg E ∗ , the functor U ′ is clearly plethyistic, and the composite U ′ U is plethyistic by (4.23). Theresult follows. (cid:3) Rational algebras We say that an object of Alg ∗ T (resp. an E ∗ -algebra) is rational if the underlying E ∗ -module is a (graded) rational vector space. In this section, we prove that if B is an T -algebra,then Q ⊗ B is also an T -algebra in a canonical way; and also that a rational T -algebra is thesame thing as a rational Γ-algebra. Our argument follows that of Knutson [Knu73]. Recallthat U : Alg ∗ T → Alg E ∗ admits a right adjoint G ( § Lemma. If A ∈ Alg E ∗ is rational, then so is U G ( A ) .Proof. Let i = 0 or 1, and let X ( A ) denote the i th graded piece of U G ( A ), regarded as anabelian group. We have that X ( A ) ≈ Mod E ∗ ( ω i/ , U G ( A )) ≈ Alg E ∗ ( T ( ω i/ ) , A ) , where the abelian group structure on X ( A ) is determined by the coproduct on T ( ω i/ ).We need to show that X ( A ) is rational, and our proof amounts to observing that X is a nilpotent abelian group scheme. More precisely, let X q ( A ) denote the quotient of the set X ( A ) under the following equivalence relation; we say that f and g in X ( A ) are equivalentif their restrictions to L qk =0 T k ( ω i/ ) are equal. Because the coproduct map ∆ + : T ( ω i/ ) → T ( ω i/ ) ⊗ T ( ω i/ ) restricts to a defined on this summand, X q ( A ) is in fact a quotient abeliangroup of X ( A ). We have that X ( A ) = 0, and X ( A ) = lim q X q ( A ) , so that it suffices to show that each X q ( A ) is rational. Furthermore, we haveker[ X q ( A ) → X q − ( A )] = Mod E ∗ ( M q , A ) , where M q = cok[ L
Let B ∈ Alg ∗ T . There exists a morphism B → B ′ ∈ Alg ∗ T which isinitial among morphisms from B to rational objects of Alg ∗ T . Furthermore, the evident map Q ⊗ U B → U B ′ is an isomorphism of rational E ∗ -algebras.Proof. Let ψ : U B → ( U G )( U B ) denote the unit of the adjunction. We claim that thereis a unique map ψ ′ : Q ⊗ U B → ( U G )( Q ⊗ U B ) making Q ⊗ U B into a coalgebra for thecomonad U G such that U B → Q ⊗ U B is a map of coalgebras. Consider the diagram U B ψ / / (cid:15) (cid:15) ( U G )( U B ) (cid:15) (cid:15) Q ⊗ U B / / ( U G )( Q ⊗ U B ) HE CONGRUENCE CRITERION FOR POWER OPERATIONS IN MORAVA E -THEORY 35 Since ( U G )( Q ⊗ U B ) is a rational E ∗ -algebra by the above, it is clear that there is a uniquedotted arrow making the diagram commute. (cid:3) Let Alg ∗ T , Q ⊂ Alg ∗ T and Alg ∗ Γ , Q ⊂ Alg ∗ Γ denote the full subcategories of rational objects.8.3. Proposition. Let B be a rational object of Alg ∗ Γ . Then there is a unique T -algebrastructure on B compatible with the given Γ -algebra structure.Proof. It is enough to check this for S ⊗ Q , where S ≈ Sym E (Γ ⊗ ω q/ ) is the free Γ-algebraon one generator in some degree q/ 2; observe that S ⊗ Q is clearly an object of Alg ∗ Γ . Let P ≈ T ( ω q/ ) denote the free T -algebra on one generator, and consider the tautological map f : S → P . By (8.2), we see that P ⊗ Q is a T -algebra. The result follows from the observationthat f ⊗ Q : S ⊗ Q → P ⊗ Q is an isomorphism (7.4). (cid:3) Corollary. The forgetful functor U : Alg ∗ T → Alg ∗ Γ induces an equivalence Alg ∗ T , Q → Alg ∗ Γ , Q . Critical weights Let U : D → A be a plethyistic functor, where A is the category of commutative monoidobjects in an abelian tensor category C . Let T : C → C be the monad associated to thissituation. Recall from § η : I → T and µ : T T → T defining the monad structure, and natural maps ι : k → T ( M ) and δ : T ( M ) ⊗ T ( M ) → T ( M )defining the commutative ring structure on T ( M ). We will assume that T is equipped witha weight decomposition, as in § { f j : A j → B } in a category called an epimorphicfamily if for any two morphisms g, h : B → C , gf j = hf j for all f j implies g = h . We saythat m ≥ regular weight of T if the family of maps in C (9.1) δ : T i ( M ) ⊗ T m − i ( M ) → T m ( M ) , µ : T d T m/d ( M ) → T m ( M )is an epimorphic family, where i ranges over integers such that 0 < i < m and d rangesover divisors of m such that 1 < d < m . In addition, we say that 0 is a regular weight if ι : k → T ( M ) is an epimorphism, and that 1 is a regular weight if η : M → T ( M ) is anepimorphism. We say that m ≥ critical weight of T if it is not a regular weight.The idea of the following proposition is that T is in some sense “generated” by phenomenain critical weights.9.2. Proposition. Let ψ : T ( M ) → M be a T -algebra structure on M ∈ C . Let N ⊆ M bea subobject of M in A . If ψ ( T m ( N )) ⊆ N for each critical weight m , then there is a unique T -algebra structure on N such that the inclusion N → M is a morphism of T -algebras.Proof. It suffices to show that the image of the composite T ( N ) → T ( M ) ψ −→ M is containedin N . We will show by induction on m ≥ α m : T m ( N ) → T m ( M ) ψ −→ M factorsthrough N . When m is a critical weight this is true by hypothesis. If m = 0 is not critical,then ι : k → T ( M ) is epimorphic, and thus α factors through N since N is a subobject.If m = 1 and is not critical, then η : M → T ( M ) is epimorphic, and thus α must factor through N since ψ ◦ η = id M . If m ≥ T i ( N ) ⊗ T m − i ( N ) δ / / ψ ⊗ ψ (cid:15) (cid:15) T m ( N ) / / (cid:15) (cid:15) T m ( M ) ψ (cid:15) (cid:15) N ⊗ N product / / N / / / / M and T d T m/d ( M ) µ / / T d ( ψ ) (cid:15) (cid:15) T m ( N ) / / (cid:15) (cid:15) T m ( M ) ψ (cid:15) (cid:15) T d ( N ) ψ / / N / / / / M commute. (cid:3) Proof of the congruence criterion In this section we prove Theorem A, using the results of § § The only critical weight of T is p . The proof of the following proposition owessomething to McClure’s discussion in [BMMS86, Ch. IX].10.2. Proposition. With respect to the standard weight decompositions, the only criticalweight for the algebraic approximation functor T is p .Proof. We will see later in this section exactly how p is a critical weight. Right now we provethat all other weights are regular.It is clear that 0 and 1 are regular weights for T , since ι : E ∗ → T ( M ∗ ) and η : M → T ( M ∗ ) are isomorphisms for all E ∗ -modules.To show that m is a regular weight, it suffices to show that the collection of maps of (9.1)is epimorphic when M ∗ is a finitely generated and free E ∗ -module, since T commutes withfiltered colimits and reflexive coequalizers. Suppose that M ∗ is a finite free E ∗ -module. For m ≥ m = p , there are two cases. Case of m relatively prime to p . In this case, let r ≥ p r < m . I claim that δ : T p r ( M ∗ ) ⊗ T m − p r ( M ∗ ) → T m ( M ∗ )is surjective. Case of p | m and m = p . In this case, let d = m/p > 1. I claim that µ : T d T p ( M ∗ ) → T m ( M ∗ )is surjective.In either case, the claim proves regularity. To prove these claims, choose a finite free E -module N such that π ∗ N ≈ M ∗ . Then the claims amount to showing that L P Σ pr × Σ m − pr N → L P m N and L P Σ d ≀ Σ p N → L P m N HE CONGRUENCE CRITERION FOR POWER OPERATIONS IN MORAVA E -THEORY 37 are surjective on homotopy groups; in fact, each of these maps admits a section by (3.18),since Σ p r × Σ m − p r ⊂ Σ m (in the first case) and Σ d ≀ Σ p ⊂ Σ m (in the second case) have indexprime to p . (cid:3) The Frobenius class. Let I ⊂ E B Σ p k denote the kernel of the augmentation map i ∗ : E B Σ p k → E , and let J ⊂ E B Σ p k denote the ideal generated by the images of transfermaps from subgroups Σ i × Σ p k − i ⊂ Σ p k , for 0 < i < p k .10.4. Lemma. For k > , the image i ∗ ( J ) of J under the augmentation map is pE , and E B Σ p k / ( I + J ) ∼ −→ E / ( p ) .Proof. We compute the projection of the transfer ideal along the augmentation map. Let m ≥ 1, and consider the fiber squareΣ m / (Σ i × Σ m − i ) / / j (cid:15) (cid:15) B (Σ i × Σ m − i ) (cid:15) (cid:15) ∗ i / / B Σ m . Applying the double-coset formula in E -cohomology shows that the image of E B (Σ i × Σ m − i ) transfer −−−−→ E B Σ m i ∗ −→ E is precisely the ideal generated by the binomial coefficient (cid:0) mi (cid:1) . The result follows from the observation that gcd { (cid:0) p k i (cid:1) | < i < p k } = p for k > m prime to p , E B Σ m /J ≈ (cid:3) Proposition. The augmentation map i ∗ : E B Σ p → E restricts induces an isomor-phism J → pE , and induces a homomorphism σ ∗ : E B Σ p /J → E / ( p ) on quotients.Furthermore, the commutative square E B Σ p π ∗ / / i ∗ (cid:15) (cid:15) E B Σ p /J σ ∗ (cid:15) (cid:15) E π ∗ / / E / ( p ) is a pullback square of E -modules.Proof. Since k = 1, the ideal J is equal to the image in cohomology of the stable transfermap t : Σ ∞ + B Σ p → Σ ∞ + B { e } associated to the inclusion of the trivial subgroup { e } ⊂ Σ p .Thus J is a free E -module on the one generator t ∗ (1) ∈ ˜ E B Σ p , and therefore the surjectivemap J → pE must be an isomorphism. The existence of σ ∗ is immediate from (10.4), andthe square is clearly a pullback. (cid:3) For any E ∗ -module M ∗ , there are maps u : M ∗ ⊗ E ∗ Γ[1] → T p ( M ∗ ) , v : Sym pE ∗ ( M ∗ ) → T p ( M ∗ ) . The map u is induced by the standard Γ-module structure T M ∗ ⊗ Γ → T M ∗ , which carries T M ∗ into T p M ∗ . The map v is induced by the tautological commutative E ∗ -algebra map Sym ∗ E ( M ∗ ) → T ( M ∗ ). We will be particularly be interested in the case when M ∗ = E ∗ , sothat all objects are concentrated in even degree and the maps have the form u : Γ[1] → T p ( E ) , v : E → T p ( E ) . In this case, observe that hom E ( E B Σ p , E ) ≈ T p ( E ), and that the inclusion u : Γ[1] → T p ( E ) is dual to the projection E B Σ p → E B Σ p /I . Likewise, observe that v is dual tothe augmentation map i ∗ : E B Σ p → E .Let ¯ σ ∈ Γ[1] ⊗ E E /pE denote the element corresponding via duality to the homomor-phism σ ∗ : E B Σ p /I → E /pE of (10.5). We will call ¯ σ the Frobenius class . Pick arepresentative σ ∈ Γ[1] of this congruence class.10.6. Proposition. There is a pushout square in E -modules of the form E · p / / ( σ, − id) (cid:15) (cid:15) E β (cid:15) (cid:15) Γ[1] ⊕ E u,v ) / / T p ( E ) Proof. Consider the diagram0 / / E B Σ p ( π ∗ ,j ∗ ) / / β ∗ (cid:15) (cid:15) E B Σ p /I ⊕ E σ ∗ , − π ∗ ) / / ( σ ∗ , − id) (cid:15) (cid:15) E /pE / / / / E · p / / E / / E /pE / / β ∗ is the unique mapmaking the diagram commute. The long exact sequence associated to Ext ∗ E ( − , E ) givesexact sequences0 / / E · p / / ( σ, − id) (cid:15) (cid:15) E / / β (cid:15) (cid:15) Ext E ( E /pE , E ) / / / / Γ[1] ⊕ E u,v ) / / T p ( E ) / / Ext E ( E /pE , E ) / / (cid:3) The corresponding odd degree result is simpler.10.7. Proposition. The inclusion Γ[1] ⊗ ω q/ → T p ( ω q/ ) is an isomorphism when q is odd.Proof. By periodicity, we may assume that q = 1. In this case, this amounts to the claimthat E ∗ B Σ ρp → E ∗ B Σ ρp /I is an isomorphism, where I is the image of the transfer map, which is immediate from(3.21). (cid:3) HE CONGRUENCE CRITERION FOR POWER OPERATIONS IN MORAVA E -THEORY 39 The case of weight p . Proposition. Let M ∗ be an E ∗ -module. The image of the map T p ( M ∗ ) → T p ( M ∗ ) ⊗ Q is generated by the images of u , v , and by elements of the form ( σx − x p ) /p as x ranges overelements of M .Proof. The functors T p and Sym p commute with filtered colimits and reflexive coequalizers,so it suffices to check the result when M ∗ is a finite free E ∗ -module. The “binomial formula” T p ( M ∗ ⊕ N ∗ ) ≈ M i + j = p T i ( M ∗ ) ⊗ T j ( N ∗ ) , together with the fact that u : Sym i ( M ∗ ) → T i ( M ∗ ) is an isomorphism for i ≤ p , allows usto reduce to the case that M ∗ is free on one generator in some degree, which amounts to(10.6) for even degree and (10.7) for odd degree. (cid:3) Proof of Theorem A. Say an object of Alg ∗ T (resp. Alg ∗ Γ ) is torsion free itsunderlying E ∗ -module is torsion free as an abelian group. Say that an object B of Alg ∗ Γ satisfies the congruence condition if σx ≡ x p mod pB for all x ∈ B .Let U : Alg ∗ T → Alg ∗ Γ denote the forgetful functor. If B ∈ Alg ∗ T , then U ( B ) satisfies thecongruence condition; this is the content of (10.6). In fact, if x ∈ B , represented by a map f : T ( E ) → B , then σx − x p ∈ B is represented by the composite E σ, − id) −−−−−→ Γ[1] ⊕ E u,v ) −−−→ T p ( E ) f −→ B, which factors through multiplication by p by (10.6). Proof of Theorem A. By the above remarks, it suffices to show that an object B of Alg ∗ Γ which is torsion free and which satisfies the congruence condition admits a unique structureof T -algebra.Let B ′ = B ⊗ Q ; this is an object of Alg ∗ Γ , Q by (8.2), and thus there is a unique T -algebrastructure on B ′ compatible with the Γ-algebra structure. By (9.2) and (10.2), it suffices toshow that the dotted arrow exists in T p ( B ) / / (cid:15) (cid:15) T p ( B ′ ) ψ (cid:15) (cid:15) B / / / / B ′ Because B is a Γ-algebra, we know that ψ ( u ( B ⊗ Γ[1])) ⊆ B and ψ ( v (Sym p ( B ))) ⊆ B . Themap T p ( B ) → B ′ factors uniquely through a map T p ( B ) ⊗ Q → B ′ , and thus by (10.9) itsuffices to check that ψ (( σx − x p ) /p ) ∈ B for all x ∈ B ; this is precisely the congruencecondition. (cid:3) Sheaves on categories of deformations Fix a prime p and an integer n ≥ 1. Let k be a perfect field of characteristic p , and let G be a (one-dimensional, commutative) formal group of height n defined over k .11.1. Deformations of G . Let b R denote the category whose objects are complete localrings whose residue field has characteristic p , and whose morphisms are continuous localhomomorphisms. If R ∈ b R , we write m ⊂ R for the maximal ideal, and π : R → R/ m forthe quotient map.Let R ∈ b R . A deformation of G to R is a triple ( G, i, α ), consisting of(a) a formal group G defined over R ,(b) a homomorphism i : k → R/ m , and(c) an isomorphism α : i ∗ G → π ∗ G of formal groups over R/ m .11.2. The Frobenius isogeny. Suppose that R ∈ b R has characteristic p . We write φ : R → R for the p th power homomorphism ( φ ( x ) = x p ). For each formal group G over such a ring R , the Frobenius isogeny Frob : G → φ ∗ G is the homomorphism of formal groups over R induced by the relative Frobenius map on rings. We write Frob r : G → ( φ r ) ∗ G for thehomomorphism inductively defined by Frob r = φ ∗ (Frob r − ) ◦ Frob.11.3. The “deformations of Frobenius” category. Let ( G, i, α ) and ( G ′ , i ′ , α ′ ) be twodeformations of G to R . We say that a homomorphism f : G → G ′ of formal groups over R is a deformation of Frob r if(i) i ◦ φ r = i ′ , so that ( φ r ) ∗ i ∗ G ≈ ( i ′ ) ∗ G , and(ii) the square i ∗ G r / / α (cid:15) (cid:15) ( φ r ) ∗ i ∗ G α ′ (cid:15) (cid:15) π ∗ G f / / π ∗ G ′ of homomorphisms of formal groups over R/ m commutes.We let Def R denote the category whose objects are deformations of G to R , and whosemorphisms are homomorphisms which are a deformation of Frob r for some (unique) r ≥ Remark. Say that a morphism in Def R has a height r (where r ≥ r . The height of a composition of morphisms is the sum of the heights.For G , G ′ objects of Def R , let Def rR ( G, G ′ ) ⊂ Def R ( G, G ′ ) denote the subset of the hom-setconsisting of morphisms of height r . Note that Def R ( G, G ′ ) are precisely the isomorphismsfrom G to G ′ in Def, i.e., the isomorphisms G → G ′ of formal groups which are deformationsof the identity map of G .Observe that if R is an F p -algebra, then for every object ( G, i, α ) in Def R the Frobeniusisogeny defines a morphism Frob : ( G, i, α ) → ( φ ∗ G, φi, φ ∗ α ) in this category.Given f : R → R ′ ∈ b R , we define a functor f ∗ : Def R ′ → Def R by base change. The basechange functors are coherent, in the sense that if R f −→ R ′ g −→ R ′′ , then there are naturalisomorphisms g ∗ f ∗ ≈ ( gf ) ∗ satisfying the evident coherence property. HE CONGRUENCE CRITERION FOR POWER OPERATIONS IN MORAVA E -THEORY 41 Remark. We can summarize the above using 2-categorical language: Def is a pseudo-functor from b R to the 2-category grCat of graded categories.11.6. Lubin-Tate theory. According to the deformation theory of Lubin-Tate, there existsa ring L ∈ b R , and a deformation ( G univ , id , α univ ) which is a deformation of G to L , suchthat for each G ∈ Def R there exists a unique map f : L → R ∈ b R for which there is anisomorphism u : G → f ∗ G univ in Def R , and that furthermore the isomorphism u is itselfunique. The ring L has the form L ≈ W k [[ u , . . . , u n − ]]. As we hope the reader is alreadyaware, L is canonically identified with π E for the Morava E -theory spectrum associated to G .Say that a groupoid D is thin if between any two objects of D there is at most oneisomorphism. As a consequence of the Lubin-Tate theorem, the groupoids Def R are thin.Given a category C and a subcategory D ⊆ C which is a thin groupoid and which containsall the objects of C , we can define a quotient category C/D as follows. The object setob( C/D ) is defined to be ob C/ ∼ , where we say that X ∼ Y if there exists f : X → Y ∈ D ;morphisms in C/D are defined by ( C/D )([ X ] , [ Y ]) = C ( X, Y ). That this becomes a categoryuses the fact that D is thin. There is an evident quotient functor C → C/D , which is anequivalence of categories.11.7. Subgroups of formal groups. Let Sub R def = Def R / Def R . Thus the quotient functorsDef R → Sub R are equivalences of categories for all R . The assignment R Sub R is a functor b R → Cat, (i.e., not merely a pseudofunctor.)11.8. Proposition. There is a one-to-one correspondence obSub R ≈ b R ( L, R ) .Proof. This is Lubin-Tate deformation theory. (cid:3) For objects G, G ′ ∈ Sub R , let Sub rR ( G, G ′ ) ⊂ Sub R ( G, G ′ ) denote the image ofDef rR ( G, G ′ ).11.9. Proposition. The assignment f ker f is a one-to-one correspondence between ` [ G ′ ] ∈ obSub R Sub rR ( G, G ′ ) and the set of finite subgroup schemes of G which have rank p r . Thus any pair ([ G ] , K ) consisting of an object [ G ] of Sub R and a subgroup K of G of rank p r determines an object [ G ′ ] of Sub R , which is represented by any deformation of Frobenius f : G → G ′ with ker f = K ; we’ll write [ G/K ] for the object [ G ′ ].11.10. Proposition. For each r ≥ there is a complete local ring L [ r ] such that b R ( L [ r ] , R ) is in one to one correspondence with the set of pairs ([ G ] , K ) , where [ G ] is an object of Sub R and K is a subgroup of G of degree p r . Under the continuous homomorphism L → L [ r ] which classifies ([ G ] , K ) [ G ] , the ring L [ r ] is finite and free as an L -module.Proof. See [Str97]. (cid:3) Corollary. Let Sub rR = ` [ G ] , [ G ′ ] ∈ obSub R Sub rR ( G, G ′ ) . Then Sub rR ≈ b R ( L [ r ] , R ) . The formal graded category scheme Sub . Let can ∗ : L [ r ] → E / ( p ) be the mapwhich classifies the kernel of Frob r : π ∗ G univ → ( φ r ) ∗ π ∗ G univ , where π : E → E / ( p ). Ob-serve that if a deformation g of Frob r is represented by some ring map f : L [ r ] → R , then f factors through can ∗ if and only if g is equal to Frob r .Let can ∗ : L [ r ] → k denote the composite of can ∗ with E / ( p ) → k ; it classifies the kernelof Frob r : G → ( φ r ) ∗ G . The maps can ∗ and can ∗ are surjective, and the kernel of can ∗ isprecisely the maximal ideal of L [ r ]. Let s ∗ , t ∗ : L [0] → L [ r ] be the maps in b R which classify([ G ] , K ) [ G ] and ([ G ] , K ) [ G/K ] respectively. It is clear that the diagram L [0] s ∗ / / can ∗ (cid:15) (cid:15) L [ r ] can ∗ (cid:15) (cid:15) L [0] t ∗ o o can ∗ (cid:15) (cid:15) k id / / k k φ r o o commutes. Since k is perfect, we see that the tensor product L [ r ] s ⊗ E t L [ r ′ ] is also acomplete local ring with residue field k . There are maps c ∗ : L [ r + r ′ ] → L [ r ] s ⊗ E t L [ r ′ ]and i ∗ : L [0] → E in b R , corresponding to composition and identity maps in Sub; fromwhat we have already shown, it follows that c ∗ and i ∗ are local homomorphisms, inducingisomorphisms on the residue fields. In fact, i ∗ is an isomorphism.Thus, Sub is what we might call a formal affine category scheme ; it is represented bydata L = ( E , L [ r ] , s ∗ , t ∗ , i ∗ , c ∗ ), which are objects and morphisms in b R .11.13. Sheaves on the deformations category. For R ∈ b R , let Mod R and Alg R denotethe category of R -modules and commutative R -algebras respectively. If f : R → R ′ ∈ b R ,let f ∗ : Mod R → Mod R ′ (or f ∗ : Alg R → Alg R ′ ) denote the evident basechange functors,defined in either case by f ∗ ( M ) ≈ R ′ ⊗ R M . These functors are coherent, in the sense thatif R f −→ R ′ g −→ R ′′ then there are natural isomorphisms g ∗ f ∗ ≈ ( gf ) ∗ satisfying the evidentcoherence property.A quasi-coherent sheaf of O -modules on Def consists of data M = ( { M R } , { M f } ),where for each R ∈ b R we have a functor M R : Def op R → Mod R , and for each f : R → R ′ ∈ b R we are given natural isomorphisms M f : f ∗ M R ∼ −→ M R ′ f ∗ , such that for all R f −→ R ′ g −→ R ′′ ,both ways of constructing a natural isomorphism g ∗ f ∗ M R → M R ′ g ∗ f ∗ coincide, and forid R : R → R the natural map M id R : id ∗ R M R → M R id ∗ R is the identity transformation.A morphism γ : M → N of quasi-coherent sheaves modules consists of data { γ R } , where γ R : M R → N R for each R ∈ b R is a natural transformation compatible with base change, inthe sense that if f : R → R ′ in b R , then N f ( f ∗ γ R ) = ( γ R f ∗ ) M f .Let Sh(Def , Mod) denote the category of quasi-coherent sheaves of O -modules. It isa tensor category, with tensor product defined “objectwise”, so that ( M ⊗ N ) R ( G ) = M R ( G ) ⊗ R N R ( G ). The unit object is O , defined so that O R : Def R → Mod R is the constantfunctor sending every object to R .In a similar way, we define a category Sh(Def , Alg) of quasi-coherent sheaves of O -algebras , by replacing “Mod R ” in the above definitions with “Alg R ”. The categorySh(Def , Alg) is plainly isomorphic to the category of commutative monoid objects inSh(Def , Mod). HE CONGRUENCE CRITERION FOR POWER OPERATIONS IN MORAVA E -THEORY 43 It is clear from the definition that Sh(Def , Alg) and Sh(Def , Mod) have all small colimitsand finite products.11.14. Remark. The above is easily summarized using 2-categorical language: Def, Mod,and Alg are pseudofunctors b R → Cat, and Sh(Def , Mod) and Sh(Def , Alg) are defined to bethe categories of pseudonatural transformations and modifications between pseudofunctorsDef → Mod and Def → Alg respectively.11.15. Sheaves are comodules. Recall that for a graded affine category scheme such asdefined by the data L , there is an associated category of comodules Comod L (5.7).11.16. Proposition. There is an equivalence of tensor categories between Sh(Def , Mod) and Comod L .Proof. This is straightforward. Given an L -comodule ( M, ψ : M → V M ), we produce asheaf by the following procedure. Let˜ ψ r : L [ r ] t ⊗ E M → L [ r ] s ⊗ E M denote the extension of ψ r : M → V M → L [ r ] s ⊗ E M to a map of L [ r ]-modules.On objects h : E → R of Def R , the functor M R : Def R → Mod R is given on objects ofDef R by M R ( h ) = R h ⊗ E M. On morphisms g : L [ r ] → R of Pow R , the functor M R is given by M R ( g : L [ r ] → R ) = ( R t ∗ g ⊗ E M ≈ R g ⊗ L [ r ] L [ r ] t ⊗ E M ˜id ⊗ ψ r −−−−→ R g ⊗ L [ r ] L [ r ] s ⊗ E M ≈ R s ∗ g ⊗ E M ) . Likewise, given an object M of Sh(Def , Mod), we obtain an L -comodule by evaluating M at the the universal deformation of G , and at the universal examples of deformations ofFrob r . (cid:3) (Note that although L is really a formal graded category scheme, formality does not playa role here.)11.17. The sheaves F , ω , and Z / -graded sheaves. If G is a formal group over a localring R , let F G denote the ring of functions on G ; it is an R -algebra non-canonically isomorphicto R [[ x ]]. We define F ∈ Sh(Def , Alg) by F R (( G, i, α )) def = F G . The evident algebra maps O → F e −→ O induce a direct sum splitting F ≈ O ⊕ I as modules.Let ω def = I / I as an object of Sh(Def , Mod); observe that the R -modules ω R (( G, i, α )) are non-canonicallyisomorphic to R , so that ω is a symmetric object in the sense of § , Mod) ∗ denote the Z / ω -twisted tensor category ( § 2) associated toSh(Def , Mod). Let Sh(Def , Alg) ∗ denote the category of commutative monoid objects inSh(Def , Mod) ∗ . Frobenius congruence. Suppose that R ∈ b R has characteristic p . The p th powermap φ defines a functor φ ∗ : Alg R → Alg R by base change, together with a natural transfor-mation Frob : φ ∗ → id, where Frob : φ ∗ A → A is the relative Frobenius defined as the uniquemap making the diagram R φ / / (cid:15) (cid:15) R (cid:15) (cid:15) R (cid:15) (cid:15) A / / φ φ ∗ A Frob / / A commute.Given an object B ∈ Sh(Def , Alg), we say it satisfies the Frobenius congruence if forall R ∈ b R of characteristic p , and for all G ∈ Def R , the diagram φ ∗ B R ( G ) B φ ∼ / / Frob & & MMMMMMMMMM B R ( φ ∗ G ) B R (Frob) (cid:15) (cid:15) B R ( G )commutes. Roughly speaking, the Frobenius congruence condition says that B carries therelative Frobenius on formal groups to the relative Frobenius on algebras.11.19. Example. Both O and F satisfy the Frobenius congruence.Let Sh(Def , Alg) cong ⊂ Sh(Def , Alg) denote the full subcategory consisting of sheaveswhich satisfy the Frobenius congruence. Let Sh(Def , Alg) ∗ cong ⊂ Sh(Def , Alg) ∗ denote thefull subcategory consisting of sheaves whose even degree part is in Sh(Def , Alg) cong .11.20. Proposition. Let A be an object of Sh(Def , Alg) , and let ( B, ψ : B → CB ) be thecorresponding commutative monoid object in Comod A . Then A satisfies the Frobenius con-gruence if and only if the composite map B ψ −→ L [1] s ⊗ E B can ∗ ⊗ id −−−−−→ ( E / ( p )) s ⊗ E B ≈ B/pB is equal to x x p .Proof. This amounts to checking the congruence condition on the universal example. (cid:3) Bialgebra rings as sheaves on the deformation category We retain the notation of the previous section.12.1. The formal graded category scheme Pow . The discussion of § L -bialgebra Γ of power operations defined in § 6. Thus, let A = ( E , A [ r ] , s ∗ , t ∗ , i ∗ , c ∗ )be the data obtained by “dualizing” Γ, so that A [ r ] ≈ H Γ[ r ] ( E ) ≈ E B Σ p r / (transfer), andin particular A [0] ≈ E . Our first order of business is to show that, like L , the data A consists of objects and morphisms of b R , and thus defines a formal graded category scheme(11.12), which we will call Pow. HE CONGRUENCE CRITERION FOR POWER OPERATIONS IN MORAVA E -THEORY 45 For r > 0, let A [ r ] denote the quotient ring of A [ r ] defined by A [ r ] def = E B Σ p r / ((transfer) + (aug)) , where (aug) denotes the kernel of the augmentation map E B Σ p r → E induced by inclu-sion of the base point ∗ → B Σ p r . Additionally, we set A [0] def = E /pE . By (10.4), theaugmentation quotient induces an isomorphism A [ r ] → E / ( p ).12.2. Lemma. The rings A [ r ] are local rings, with maximal ideal given by the kernel of thecomposite A [ r ] → A [ r ] ≈ E /pE → k .Proof. The ring A [ r ] is finite as an E -module (using s ∗ ), and thus every maximal ideal of A [ r ] must contract to the maximal ideal of E by the “going up theorem”. Thus it sufficesto show that A [ r ] ⊗ E k is a local ring. Since A [ r ] is a quotient of E B Σ p r , it suffices toshow that E B Σ p r ⊗ E k is a local ring. Let K denote the generalized cohomology theoryobtained by killing the maximal ideal of E , so that E B Σ p r ⊗ E k ≈ K B Σ p r ; the resultfollows from the observation that the augmentation ideal of K B Σ p r is nilpotent, since B Σ p r is connected of finite type. (cid:3) Lemma. Let B be an object in Alg T . Then the composite B → Hom E ∗ (Γ[ r ] , B ) ≈ B ⊗ E s A [ r ] → B ⊗ E s A [ r ] ≈ B/pB sends x x p r .Proof. Immediate using (4.21) and (10.4). (cid:3) Let can ∗ : A [ r ] → A [ r ] ≈ k denote the evident projection. The diagram A [0] s ∗ / / can ∗ (cid:15) (cid:15) A [ r ] can ∗ (cid:15) (cid:15) A [0] t ∗ o o can ∗ (cid:15) (cid:15) k id / / k k φ r o o commutes, because s ∗ is induced by the “standard” inclusion E → E B Σ p k , while t ∗ isinduced by the structure map E → Hom E (Γ[ r ] , E ) which defines the canonical Γ-modulestructure on the ground ring E , and thus is the p r th power map on quotients by (12.3). Thusthe rings A [ r ] s ⊗ A [0] A [ r ′ ] are complete local with residue field k , and the homomorphisms s ∗ , t ∗ , c ∗ , i ∗ are all local homomorphisms.Thus, Pow is a formal graded category scheme. Observe that Pow R is a thin groupoid forall R ∈ b R , and that Pow : b R → grCat is a functor (not just a pseudofunctor).12.4. Isomorphism between Sub and Pow . We are going to construct a pseudonaturaltransformation F : Pow → Def of pseudofunctors b R → grCat, and we will show, by applyinga theorem of Strickland, that F R : Pow R → Def R is an equivalence of graded categories foreach R ∈ b R . This immediately implies the following.12.5. Proposition. The natural transformation F ′ : Pow F −→ Def → Sub of functors b R → grCat induces an isomorphism Pow R → Sub R of graded categories for all R in b R . Observe that an object M ∈ Mod Γ determines functors M R : Pow R → Mod R , as follows.Let ψ : M → H Γ[ r ] ( M ) ≈ A [ r ] s ⊗ E M denote the structure map of the Γ-module M ; it isa map of E -modules, using the module structure on A [ r ] given by the ring homomorphism t ∗ : E → A [ r ]. If X is a space, then E X is an object of Alg Γ , and thus we get a functor( E X ) R : Pow R → Alg ( R ).The next observation is that if we take X = CP ∞ , then we get a functor from Pow R toDef R . More precisely, for each f : E → R , the projective system { R ⊗ E E CP m } determinesa formal scheme over R , and the maps { R ⊗ E E CP i + j → R ⊗ E E ( CP i × CP j ) } give itthe structure of a formal group over R , which we will call f ∗ G univ . Let f : k → R/ m bethe homomorphism obtained from f by passing to residue fields. Then the triple F ( f ) =( f ∗ G univ , f , id : f ∗ G → π ∗ f ∗ G univ ) is a deformation of G to R .A map g : A [ r ] → R determines a homomorphism of formal groups ( s ∗ g ) ∗ G univ → ( t ∗ g ) ∗ G univ over R .12.6. Lemma. This homomorphism is a deformation of Frob r ; i.e., it determines a mor-phism F ( s ∗ g ) → F ( t ∗ g ) in Def rR .Proof. It suffices to check the universal example, corresponding to the identity map of A [ r ].In this case, it is enough to note that the composite E CP ∞ ψ −→ H Γ[ k ] ( E CP ∞ ) → E CP ∞ ⊗ E E / ( p ) → E CP ∞ ⊗ E k is the map x x p r , and that this composite is the same as E CP ∞ → A [ r ] s ⊗ E E CP ∞ → A [ r ] s ⊗ E E CP ∞ . (cid:3) Thus we have obtained functors F : Pow R → Def R , which are clearly natural in R , andthus a functor F ′ : Pow R → Sub R . By the Yoneda lemma, the functor F ′ determines and isdetermined by a collection of homomorphisms F ∗ : L [ r ] → A [ r ] in b R .It remains to show that F ′ is an isomorphism of graded categories. It is clear that F ′ is a bijection on objects. To show that F is a bijection on morphism, it suffices to checkthat F ∗ : L [ r ] → A [ r ] is an isomorphism for each r ≥ 0. By the definition of L [ r ], this mapclassifies a pair ([ G ] , K ), where G = F ( s ∗ : E → A [ r ]) and K is the kernel of the isogeny F (id A [ r ] ) : F ( s ∗ : E → A [ r ]) → F ( t ∗ : E → A [ r ]). The kernel K is a closed subscheme of G ; its function ring O K is the pushout of E -algebras E CP ∞ P pr / / i ∗ (cid:15) (cid:15) E ( CP ∞ × B Σ p r ) / (tr) (cid:15) (cid:15) E / / O K Lemma. Let ρ C p r denote the complex vector bundle over B Σ p r associated to the permu-tation representation, and let π : P ( ρ p r ⊗ C ) → B Σ p r denote the projective bundle associated HE CONGRUENCE CRITERION FOR POWER OPERATIONS IN MORAVA E -THEORY 47 to this vector bundle, with tautological line bundle L classified by ℓ : P ( ρ C p r ) → CP ∞ . Thenthere is a pushout square of E -algebras of the form E CP ∞ P pr / / i ∗ (cid:15) (cid:15) E ( CP ∞ × B Σ p r ) / ( tr ) ( ℓ,π ) ∗ (cid:15) (cid:15) E / / E P ( ρ C p r ) / ( tr ) Proof. Examine the diagram(12.8) ˜ E ( CP ∞ ) L P m / / z ∗ (cid:15) (cid:15) ˜ E ( CP ∞ × B Σ m ) L ⊠ ρ C m / / z ∗ (cid:15) (cid:15) ˜ E ( CP ∞ × B Σ m ) L ⊠ ρ C m /I ′ (cid:15) (cid:15) E CP ∞ P m / / π ∗ (cid:15) (cid:15) E CP ∞ × B Σ m / / π ∗ (cid:15) (cid:15) E CP ∞ × B Σ m /I ¯ π ∗ (cid:15) (cid:15) E S ( L ) (cid:15) (cid:15) E S ( L ⊠ ρ C m ) / / (cid:15) (cid:15) E S ( L ⊠ ρ C m ) /I ′′ (cid:15) (cid:15) D ( V ) , S ( V )) associated to the Thom space X V ; they are exact, sincethe Euler class of any complex bundle of the form L ⊠ V → CP ∞ × X is a non-zero divisor.The right-hand vertical column is obtained by quotienting the middle column by transferideals, denoted I , I ′ , and I ′′ . We claim that the right-hand column of (12.8) is also exact,which amounts to showing that π ∗ I = I ′′ , which in turns amounts to the observation thatthe diagrams E ( CP ∞ × B Σ i × B Σ m − i ) transfer / / π ∗ i (cid:15) (cid:15) E CP ∞ × B Σ mπ ∗ (cid:15) (cid:15) E S ( L ⊠ ρ C i ⊠ ρ C m − i ) transfer / / E S ( L ⊠ ρ C m )commute, and that the map marked π ∗ i is also surjective. (Note that if m is not a p th power,the right-hand column of (12.8) is identically 0.)The two long horizontal composites in (12.8) are ring homomorphisms, which we denote P m .Choose any complex orientation for E , let u ∈ ˜ E ( CP ∞ ) L denote the Thom class of L , andlet x = z ∗ ( u ) ∈ E CP ∞ denote the associated Euler class, so that E S ( L ) ≈ E CP ∞ / ( x ). We have that i ∗ P m ( x ) = i ∗ z ∗ P m ( u ) = 0, and therefore there is a ring homomorphismindicated by the dotted arrow in (12.8) making the square E CP ∞ P m / / π ∗ (cid:15) (cid:15) E CP ∞ × B Σ m /I ¯ π ∗ (cid:15) (cid:15) E S ( L ) / / E S ( L ⊠ ρ C m ) /I ′′ commute. Since the class P m ( u ) is a Thom class for L ⊠ ρ C m , the kernel of π ∗ is a cyclic idealgenerated by P m ( x ) = z ∗ P m ( u ), and thus the kernel of ¯ π ∗ is also a cyclic ideal generated by P m ( x ). Therefore we conclude that the square is a pushout square in rings; since S ( L ) ≈ ∗ and S ( L ⊠ ρ C m ) ≈ P ( ρ C m ), we obtain the desired result. (cid:3) Theorem (Strickland, [Str98]) . The map L [ r ] → A [ r ] ≈ E B Σ p r / ( tr ) classifying thesubgroup K of G described above is an isomorphism. As an immediate corollary, the maps F ∗ : L [ r ] → A [ r ] are isomorphisms as desired.12.10. Equivalence of Sh(Def , Mod) and Mod Γ , and proof of Theorem B. 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