Mapping algebras and the Adams spectral sequence
aa r X i v : . [ m a t h . A T ] J u l MAPPING ALGEBRAS AND THE ADAMS SPECTRALSEQUENCE
DAVID BLANC AND SUROJIT GHOSH
Abstract.
For a suitable ring spectrum, such as E = H F p , the E -term of the E -based Adams spectral sequence for a spectrum Y may be described in terms of itscohomology E ∗ Y , together with the action of the primary operations E ∗ E on it.We show how the higher terms of the spectral sequence can be similarly described interms of the higher order truncated E - mapping algebra for Y – that is truncationsof the function spectra Fun( Y , M ) for various E -modules M , equipped with theaction of Fun( M , M ′ ) on them. Introduction
The Adams spectral sequence is an important tool in stable homotopy theory,originally introduced in [A] in order to compute the stable homotopy groups of thesphere (at a prime p ), using the Eilenberg-MacLane spectrum E = H F p . It waslater generalized by Novikov in [N] to more general ring spectra E .The information needed to determine the E -term of the E -based Adams spectralsequence for a spectrum Y are the E -cohomology groups of Y , together with theaction of the primary E -cohomology operations on Y . More generally, we mustconsider the homotopy classes [ Y , M ] for all E -module spectra M , together withthe action of [ M , M ′ ] on them (see [B3, 3.1] to understand why this may be necessaryfor general E ).However, it is not a priori clear what higher order information is needed in orderto determine the E r -terms for r >
2. As we shall see, it turns out that it is sufficientto know the ( r − P r − M E Y h i (see § E - mapping algebra M E Y for Y – that is the function spectra Fun( Y , M ) for various E -modules M , equipped with the action of Fun( M , M ′ ) on them.An explicit computation of was carriedWork of the late Hans Baues and his collaborators shows that the E -term ofthe usual Adams spectral sequence, for Y = S and for E = H F p , mightbe accessible to computation using the “secondary Steenrod algebra”, equivalent tothe first Postnikov section P M E E of the F p -mapping algebra (see [BJ]). Thestructure of the analogous unstable Adams spectral sequence was studied in [BBC](which identifies the E r -terms as certain truncated derived functors) and in [BBS](which describes the differentials as higher cohomology operations).Following [BS], we use a specific version of M E Y to construct a cosimplicialAdams resolution Y → W • , so that the homotopy spectral sequence for Fun( Z , W • )is the E -based Adams spectral sequence for Fun( Z , Y ). Analysis of the differential Mathematics Subject Classification.
Primary: 55T15; secondary: 55P42, 55U35.
Key words and phrases.
Spectral sequence, truncation, differentials, cosimplicial resolution, map-ping algebra. d r − shows that it only depends on the ( r − M E W • , and thusthat the E r -terms are determined by P r − M E Y h i .0.1 . Outline. Section 1 recalls some facts about the category Sp of symmetricspectra and Section 2 defines our main technical tool: spectral functors defined onsmall categories Θ λE of E -modules, for a fixed ring spectrum E , and their truncations.In Section 3 we define mapping algebras – a generalization of the representablespectral functor M λ E Y (defined by M Fun( Y , M )). We use this to constructa monad on spectra, which we analyze in Section 4 in order to overcome certainset-theoretical difficulties. This allows to obtain our first result, in Section 5: Theorem A. If E is a ring spectrum and Y an E -good symmetric spectrum, we canassociate to the representable mapping algebra M λ E Y a cosimplicial spectrum W • such that Tot W • is E -equivalent to Y . See Theorem 5.9 below.In Section 6 we analyze the differentials in the E -based Adams spectral sequencefor Fun( Z , Y ) (in its cosimplicial version), and show: Theorem B.
Given E , Z , and Y as above, for each r ≥ , the d r -differential inthe E -based Adams spectral sequence for Fun( Z , Y ) , and thus its E r +1 -term, canbe calculated from the cosimplicial ( r − -truncated space P r − M λ E Z { W • } . See Theorem 6.1 below.We then use the resolution model category of truncated spectral functors to deduce:
Theorem C. If E = H R for a commutative ring R , Z is a fixed finite spectrum,and Y is a E -good spectrum, then for any r ≥ the E r +2 -term of the E -basedAdams spectral sequence for Fun( Z , Y ) is determined by the r -truncation P r M λ E Y of the E -mapping algebra of Y . See Theorem 6.9 below.0.2.
Notation.
We denote the category of sets by
Set , that of pointed sets by
Set ∗ , that of simplicial sets (called spaces ) by Spaces , and that of pointed simplicialsets by
Spaces ∗ . For X, Y ∈ Spaces ∗ , X ∧ Y := ( X × Y ) / ( X ∨ Y ) is theusual smash product. For any category C and A, B ∈ C , we write C ( A, B ) forHom C ( A, B ) ∈ Set .0.3.
Acknowledgements.
We would like to thank the referee for his or her detailed andpertinent comments. This research was supported by Israel Science Foundation grant770/16. 1.
Symmetric spectra
In this section we recall from [HSS] some basic facts about symmetric spectra. Weprefer this model for the stable homotopy category because it has useful set-theoreticproperties.1.1.
Definition. A symmetric spectrum is a sequence of pointed spaces (simplicialsets) X = ( X n ) n ≥ equipped with:(1) A pointed map σ : S ∧ X n → X n +1 for each n ≥ APPING ALGEBRAS AND THE ADAMS SPECTRAL SEQUENCE 3 (2) A basepoint-preserving left action of the symmetric group Σ n on X n , suchthat the composite σ p = σ ◦ ( S ∧ σ ) ◦ · · · ◦ ( S p − ∧ σ ) : S p ∧ X n → X n + p is Σ p × Σ n -equivariant for p ≥ n ≥ f : X → Y of symmetric spectra is a sequence of Σ n -equivariant maps f n : X n → Y n such that the diagram S ∧ X n Id S ∧ f n (cid:15) (cid:15) σ / / X n +1 f n +1 (cid:15) (cid:15) S ∧ Y n σ / / Y n +1 commutes for all n ≥
0. We denote the category of symmetric spectra by Sp .The smash product of X , Y ∈ Sp is defined to be the symmetric spectrum X ⊗ Y given by ( X ⊗ Y ) n := _ p + q = n Σ n + ∧ Σ p × Σ q X p ∧ Y q . Given X ∈ Sp and a pointed space K , the symmetric spectrum K ⊗ X is definedby ( K ⊗ X ) n := K ∧ X n (see [HSS, § . The model category of symmetric spectra. The (stable) model structureon Sp is defined in [HSS] as follows:A map f : X → Y of symmetric spectra is(i) a stable equivalence if it induces an isomorphism in stable homotopy groups(forgetting the Σ n -actions).(ii) a level trivial fibration if at each level it is a trivial Kan fibration of simplicialsets.(iii) a stable cofibration if it has the left lifting property with respect to level trivialfibrations.(iv) a stable fibration if it has the right lifting property with respect to every stablecofibration which is a stable equivalence.By [HSS, Theorem 3.4.4], the classes of stable equivalences, cofibrations, and fi-brations define a proper, simplicial, symmetric model category structure on Sp ,monoidal with respect to ⊗ . The simplicial enrichment is given bymap Sp ( X , Y ) n = Sp (∆[ n ] + ⊗ X , Y ) . Definition.
Given X , Y ∈ Sp , the function spectrum Fun( X , Y ) is definedto be the symmetric spectrum given by(1.4) Fun( X , Y ) n := map Sp ( X , sh n Y )where sh n Y is the n -shifted symmetric spectrum given by (sh n Y ) k = Y n + k . Theaction of the symmetric group Σ n is induced from the action on sh n Y . Onemay see [HSS, Remark 2.2.12] for the symmetric spectra structures on sh n Y andFun( X , Y ).1.5. Remark.
We have adjoint functors
Spaces ∗ Σ ∞ ⇋ Ω ∞ Sp withΩ ∞ Fun( X , Y ) ≃ map Sp ( X , Y ) . DAVID BLANC AND SUROJIT GHOSH
Moreover, given X , Y , Z ∈ Sp , by [HSS, Theorem 2.2.10] there is a natural adjunc-tion isomorphism(1.6) Sp ( X ⊗ Y , Z ) ∼ = Sp ( X , Fun( Y , Z )) . For any X ∈ Sp , the function spectra Ω X := Fun( S , X ) and P X :=Fun(Σ ∞ ∆[1] + , X ) are called loop and path spectra of X , respectively. Note thatFun( X , − ) commutes with the loop and path constructions.1.7. Definition. A symmetric ring spectrum is a symmetric spectrum R togetherwith spectrum maps m : R ⊗ R → R ( multiplication ) and ι : S → R (the unit map) with m ◦ ( m ⊗ Id) = m ◦ (Id ⊗ m ), such that m ◦ ( ι ⊗ Id) : S ⊗ R → R and m ◦ (Id ⊗ ι ) : R ⊗ S → R are the standard equivalences.An R -module for a symmetric ring spectrum R is a symmetric spectrum M equippedwith a spectrum map µ : R ⊗ M → M with µ ◦ ( m ⊗ Id) = µ ◦ (Id ⊗ µ ) and themap µ ◦ ( ι ⊗ Id) : S ⊗ M → M is the standard equivalence.For any symmetric spectrum Y and symmetric ring spectrum R , the functionspectrum Fun( Y , R ) admits a module structure over R .1.8. Notation.
For any symmetric spectrum X , we write k X k := sup n ∈ N k X n k (thecardinality of the simplicial set). Note that if λ is a limit cardinal and k X k < λ ,then X is λ -small in the usual sense (see [Hi, Definition 10.4.1]).2. Spectral functors
Our main technical tool in this paper is the following:2.1.
Definition.
Given a symmetric ring spectrum E , let E - Mod denote (a skeletonof) the full subcategory of E -module spectra in Sp . We consider spectral functors X : E - Mod → Sp (that is, functors respecting the spectral enrichment), and write X { M } ∈ Sp for the value of X at M ∈ E - Mod . The spectral enrichment – orrather, its truncations – will play a central role in the paper; our main point isthat these provide the data needed to compute the higher differentials in the Adamsspectral sequence. A functor of the form M Fun( Y , M ) for some fixed Y ∈ Sp will be called representable .If λ is some limit cardinal, the corresponding E - spectral theory is the full subcat-egory Θ λE of E - Mod consisting of all E -module spectra which are λ -small. Wedenote by Sp Θ λE the category of all spectral functors from Θ λE . Note that the Θ λE -spectral functor represented by Y ∈ Sp , denoted by M λ E Y , is a homotopyfunctor (that is, it preserves weak equivalences). When Y ∈ Θ λE , we say that M λ E Y is free . Observe that M λ E Y is contravariant in the variable Y .2.2. Lemma. If X is any Θ λE -spectral functor and M λ E M is free (for M ∈ Θ λE ),there is a natural isomorphism Sp Θ λE ( M λ E M , X ) ∼ = Sp ( S , X { M } ) . In particular,if X = M λ E Y is representable, a map S → X { M } corresponds to a map ofspectra Y → M by (1.6) – that is, Sp Θ λE ( M λ E M , M λ E Y ) ∼ = Sp ( S , M λ E Y { M } ) ∼ = Sp ( Y , M ) . Proof.
This follows from the enriched Yoneda Lemma (see [K, Proposition 2.4]). (cid:3)
APPING ALGEBRAS AND THE ADAMS SPECTRAL SEQUENCE 5
Remark.
Any Θ λE -spectral functor X which is a homotopy functor preserveshomotopy pullbacks and pushouts (which are equivalent in Sp ), by [C, Proposition4.1]. In particular, it preserves the path, loop, and suspension operations on spectra,up to weak equivalence. Thus(2.4) π k X { M } ∼ = π Ω k X { M } ∼ = π X { Ω k M } . for any M ∈ Θ λE .2.5 . Truncation of spectral functors. For each n ∈ Z , consider the Postnikovsection functor P n : Sp → Sp (localization with respect to S n +1 → ∗ ), killing allhomotopy groups in dimensions > n , and the ( k − h k i : Sp → Sp (colocalization with respect to ∗ → S k +1 – see [Hi, 1.2 & 5.1]). When n ≥ k ,write P nk for the composite P n ◦ h k i .Note that for any X , Y ∈ Sp we have Fun( X , Y ) = map Sp ( X , Y ) (the simplicialenrichment), by (1.4). For any n ≥
0, we may define P n in Spaces or Spaces ∗ by composing the ( n + 1)-coskeleton functor with a functorial fibrant replacementcommuting with products, so it is monoidal in Spaces ∗ with respect to cartesianproducts (see [Hi, 9.1.14]), with [ P n Fun( X , Y )] ≃ P n map Sp ( X , Y ).We can therefore define a new enrichment on Sp in ( Spaces ∗ , × ) bymap ∗ ( X , Y ) := P n Fun( X , Y ) , and call any X : Θ λE → Sp respecting this enrichment a truncated Θ λE -spectralfunctor, and their category will be denoted by ( Sp Θ λE ) n . In particular, applying P n to any spectral functor X yields such a functor, defined by ( P n X ) { M } := P n (( X { M } ) ), which we call simply the n -truncation of X : explicitly, the action of Θ λE on X , in the form of maps of spectra X { M } ∧ Fun( M , N ) → X { N } , yields a map of simplicial sets ( X { M } ) ∧ Fun( M , N ) → ( X { N } ) , and by precom-posing with the quotient map ( X { M } ) × Fun( M , N ) → ( X { M } ) ∧ Fun( M , N ) and applying our monoidal P n we obtain an action of Θ λE with its new enrichment:(2.6) ( P n X ) { M } × map ∗ ( M , N ) → ( P n X ) { N } . Note that P n X is not itself a spectral functor in the sense of § Sp Θ λE ) n – namely, naturaltransformations inducing weak equivalences for each M ∈ Θ λE .For M ∈ Θ λE , we say that P n M λ E M is a free truncated Θ λE -spectral functor,since we have the following analogue of Lemma 2.2:2.7. Lemma. If X ∈ ( Sp Θ λE ) n and M ∈ Θ λE , there is a natural isomorphism ( Sp Θ λE ) n ( P n M λ E M , X ) ∼ = Sp ( S , X { M } ) . Remark.
Since any Θ λE -spectral homotopy functor X commutes up to weakequivalence with Ω, we have(2.9) P n X { Ω k M } w . e . ≃ P n Ω k X { M } w . e . ≃ Ω k ( P n + kk X ) { M } ) . Therefore, P n X determines P n + kk X up to homotopy for all k ∈ Z . DAVID BLANC AND SUROJIT GHOSH . Model category structures.
By [HSS], Sp has a proper simplicial modelcategory structure, and by [MMSS], it is cofibrantly generated. Since Θ λE is small, by[Hi, Theorems 11.1.6 & 13.1.14] there is a projective proper simplicial model categorystructure on Sp Θ λE , in which the weak equivalences and fibrations are level-wise –in particular, a map f : X → X ′ in Sp Θ λE is a weak equivalence if and only if f ∗ : X { M } → X ′ { M } is a weak equivalence in Sp for each M ∈ Θ λE .We may similarly define a P n - weak equivalence of spectral functors to be a map f : X → Y inducing a weak equivalence after applying P n . Of course, for homotopyspectral functors these are the same those just defined, by (2.9), but in general theyare different. By applying Bousfield (co)localization to the above we obtain the P n - model structure on Sp Θ λE (see [Hi, Ch. 3]).2.11. Proposition.
The P n -model category structure on Sp Θ λE is right proper.Proof. The Postnikov section functor P n is a nullification, so a left Bousfieldlocalization. Hence, by [Hi, Proposition 3.4.4] we have a left proper model structureon the image of P n in Sp Θ λE . The argument of [B3, Theorem 9.9] (which also worksin Sp ) shows that it is also right proper. Since taking connected covers is a rightBousfield localization, by [Hi, loc. cit. ] we see that ( Sp Θ λE ) n is right proper. (cid:3) . Homotopy groups. The homotopy groups π i X { M } are used to defineweak equivalences for a spectral functor X , and we will need to identify the minimalinformation needed to determine them. In fact, by (2.4) we need only the 0-th(stable) homotopy group, if X is a homotopy functor.Since any spectrum B is a homotopy group object, with group operation µ : B × B → B and inverse ν : B → B , for any A ∈ Sp we have µ ∗ : Sp ( A , B ) × Sp ( A , B ) → Sp ( A , B ) and ν ∗ : Sp ( A , B ) → Sp ( A , B ). As by (1.6), Sp ( A , B ) = Sp ( S , Fun( A , B )), we may define a relation ∼ on Sp ( A , B ) by f ∼ g if andonly if there exists F ∈ Sp ( S , P Fun( A , B )) such that µ ∗ ( ν ∗ ( g ) , f ) = p ∗ F , where p : P X → X is the path fibration. We then see:2.13. Lemma. If A ∈ Sp is cofibrant and B ∈ Sp is fibrant, the relation ∼ is anequivalence relation on Sp ( A , B ) which coincides with the (left or right) homotopyrelation on Sp ( A , B ) , which we denote by ≃ . As usual, we write [ A , B ] for Sp ( A , B ) / ∼ . Proof.
The fact that ∼ is an equivalence relation is readily verified. Given twohomotopic maps f ≃ g : A → B , µ ∗ ( ν ∗ ( g ) , f ) is nullhomotopic, so there is F : S → P Fun( A , B ) with µ ∗ ( ν ∗ ( g ) , f ) = p ∗ F . Conversely, given F : S → P Fun( A , B )) with µ ∗ ( ν ∗ ( g ) , f ) = p ∗ F , we see that µ ∗ ( ν ∗ ( g ) , f ) is nullhomotopic,so g ≃ µ ∗ ( g, ∗ ) ≃ µ ∗ ( g, µ ∗ ( ν ∗ ( g ) , f )) ≃ µ ∗ ( µ ∗ ( g, ν ∗ ( g )) , f ) ≃ µ ∗ ( ∗ , f ) ≃ f . (cid:3) Remark.
If we let = (0 →
1) denote the one-arrow category, with a singlenon-identity map, and
Set ∗ the corresponding functor category into pointed sets,we may define a functor b ρ : Sp → Set ∗ by X [ Sp ( S , P X ) p −→ Sp ( S , X )], anddeduce:2.15. Corollary.
For fibrant B ∈ Sp the functor π Fun( − , B ) : Sp cof → Gp (onthe subcategory of cofibrant spectra) factors through b ρ ◦ Fun( − , B ) . APPING ALGEBRAS AND THE ADAMS SPECTRAL SEQUENCE 7 Mapping algebras
We now show that Θ λE -spectral functors X having a certain property (called map-ping algebras ) are representable, up to weak equivalence. To do so, in Section 5 wewill construct a cosimplicial spectrum W • using this structure, and show thatTot W • realizes X , up to weak equivalence.The discussion in § . The arrow set category. For a fixed limit cardinal λ , with Θ λE as above, letΓ Θ λE denote the directed graph associated to the underlying category of Θ λE (see[Ha]). We then define an arrow set A to be a function A : Γ Θ λE → Set ∗ (see § M in Θ λE a map of pointed sets A ( χ M ) : A ( e M ) → A ( b M ),fitting into a commutative square(3.2) A ( e M ) A ( χ M ) (cid:15) (cid:15) A ( e j ) / / A ( e M ′ ) A ( χ M ′ ) (cid:15) (cid:15) A ( b M ) A ( b j ) / / A ( b M ′ )for each map j : M → M ′ in Θ λE .This is equivalent to having a functor from the free category on Γ Θ λE to Set ∗ .We denote the category of such arrow sets by Ξ λ .For each fixed limit cardinal λ we have a functor ρ : Sp Θ λE → Ξ λ , where the arrowset ρ ( X ) assigns to each map j : M → M ′ in Θ λE the commutative square:(3.3) Sp ( S , P X { M } ) p ∗ / / P X { j } (cid:15) (cid:15) Sp ( S , X { M } ) X { j } (cid:15) (cid:15) Sp ( S , P X { M ′ } ) p ∗ / / Sp ( S , X { M ′ } )The map p ∗ is induced by the path fibration p Y : P Y → Y for Y = X { M } (compare § . Maps of arrow sets. Using (3.2), any
A, B ∈ Ξ λ induce the followingdiagram:(3.5) Set ∗ ( A ( e M ) , B ( e M )) B ( χ M ) ∗ (cid:15) (cid:15) B ( e j ) ∗ / / Set ∗ ( A ( e M ) , B ( e M ′ )) B ( χ M ′ ) ∗ (cid:15) (cid:15) Set ∗ ( A ( e M ′ ) , B ( e M ′ )) A ( e j ) ∗ o o B ( χ M ′ ) ∗ (cid:15) (cid:15) Set ∗ ( A ( e M ) , B ( b M )) B ( b j ) ∗ / / Set ∗ ( A ( e M ) , B ( b M ′ )) Set ∗ ( A ( e M ′ ) , B ( b M ′ )) A ( e j ) ∗ o o Set ∗ ( A ( b M ) , B ( b M )) A ( χ M ) ∗ O O B ( b j ) ∗ / / Set ∗ ( A ( b M ) , B ( b M ′ )) A ( χ M ) ∗ O O Set ∗ ( A ( b M ′ ) , B ( b M ′ )) A ( χ M ′ ) ∗ O O A ( b j ) ∗ o o Thus Ξ λ ( A, B ) is a product over all maps j : M → M ′ in Θ λE of the limit ofthe diagrams (3.5).3.6. Remark.
Our goal is to describe the minimal data needed to determine whena map of spectral functors f : X → Y is a weak equivalence ( § f ∗ : π X { M } → π Y { M } isan isomorphism for all M ∈ Θ λE . DAVID BLANC AND SUROJIT GHOSH
By Corollary 2.15, the map of arrow sets ρ f : ρ X → ρ Y suffices for this purpose:in fact, it is enough to consider its values only on the objects of Θ λE (i.e., the verticalarrows in (3.2).The more complicated definition of arrow sets given above is necessary only for thesmallness argument in Section 4 below. However, we do not require that an arrowset be functorial with respect to the compositions in Θ λE , since this is not neededfor our purpose.3.7. Notation.
Let Ξ := S λ Ξ λ (the union taken over all limit cardinals). This isa large category, which we need only in order to be able to discuss all arrow sets atonce.In particular, for each arrow set A ∈ Ξ, let λ be maximal such that A ∈ Ξ λ , andwrite k A k := sup M ∈ Θ λE {| A ( e M ) | , | A ( b M ) |} (where | B | denotes the cardinality ofa set B ). We write L λ E : Sp → Ξ op λ for ρ ◦ M λ E .3.8 . The Stover construction. To describe the right adjoint R λ E : Ξ op λ → Sp to L λ E , we recall a construction due to Stover (see [S] and compare [BS]):We want to have Sp ( Y , R λ E A ) ∼ = Ξ λ ( A, L λ E Y ). By the description of morphismsin Ξ λ (see § λ ( A, L λ E Y ) is theproduct over all M ∈ Θ λE and j : M → M ′ of the limit of the following diagram:(3.9) Q A ( e M ) Sp ( Y , P M ) ( p M ) ∗ (cid:15) (cid:15) Q A ( e M ) ( P j ) ∗ / / Q A ( e M ) Sp ( Y , P M ′ ) ( p M ′ ) ∗ (cid:15) (cid:15) Q A ( e M ′ ) Sp ( Y , P M ′ ) ⊤ A ( e j ) ∗ o o ( p M ′ ) ∗ (cid:15) (cid:15) Q A ( e M ) Sp ( Y , M ) Q A ( e M ) ( j ) ∗ / / Q A ( e M ) Sp ( Y , M ′ ) Q A ( e M ′ ) Sp ( Y , M ′ ) ⊤ A ( e j ) ∗ o o Q A ( b M ) Sp ( Y , M ) A ( χ M ) ∗ O O Q A ( b M ) ( j ) ∗ / / Q A ( b M ) Sp ( Y , M ′ ) A ( χ M ) ∗ O O Q A ( b M ′ ) Sp ( Y , M ′ ) A ( χ M ′ ) ∗ O O A ( b j ) ∗ o o Note that (3.9) splits up as a product of smaller diagrams, indexed by a singlemap φ : Y → M in the left two slots of the bottom row. Moreover, this diagramis really only relevant for nullhomotopic φ .Therefore, given ∗ 6 = φ ∈ A ( b M ) and j : M → M ′ , we define Q ( M ,φ,j ) to bethe limit of the following diagram:(3.10) Q A ( χ M ) − ( φ ) P M Q p M (cid:15) (cid:15) Q P j / / Q A ( χ M ) − ( φ ) P M ′ Q p M ′ (cid:15) (cid:15) Q A ( χ M ′ ) − ( A ( b j )( φ )) P M ′⊤ A ( e j ) ∗ o o Q p M ′ (cid:15) (cid:15) Q A ( χ M ) − ( φ ) M Q j / / Q A ( χ M ) − ( φ ) M ′ Q A ( χ M ′ ) − ( A ( b j )( φ )) M ′⊤ A ( e j ) ∗ o o M diag O O j / / M ′ diag O O M ′ diag O O Note that if φ ∈ Im( A ( χ M )), then A ( b j )( φ ) ∈ Im( A ( χ M ′ )). Thus if A ( b j )( φ ) isnot in the image of A ( χ M ′ ), then all six products the in two top rows of (3.10)have empty indexing sets, so Q ( M ,φ,j ) = M . APPING ALGEBRAS AND THE ADAMS SPECTRAL SEQUENCE 9
Finally, in the special case where φ is actually the zero map ∗ , we set Q ( M ,φ,j ) := Q A ( χ M ) − ( ∗ ) Ω M .3.11. Definition.
For a fixed limit cardinal λ , a mapping algebra is a spectral functor X : Θ λE → Sp preserving all limits of the form (3.10) in Θ λE . In particular,by an appropriate choice of arrow set, we see that such an X preserves loops up tohomotopy. The category of all mapping algebras for λ is denoted by Map Θ λE .Note that any representable spectral functor M λ E Y (see § all limits in E - Mod , and the diagram (3.10) isin fact in Θ λE ⊂ E - Mod (including the path fibrations p ).From the discussion in § Lemma.
For a fixed limit cardinal λ , the right adjoint R λ E : Ξ op λ → Sp of L λ E is given on A ∈ Ξ λ by (3.13) R λ E ( A ) := Y M ∈ Θ λE Y φ ∈ A ( b M ) Y j : M → M ′ Q ( M ,φ,j ) . Remark.
The limits Q ( M ,φ,j ) of (3.10) and the products of (3.13) alwaysexist in E - Mod , but they may or may not be in Θ λE . However, if we let Arr Θ λE denote the set of all morphisms (between any two objects) in Θ λE , with cardinality | Arr Θ λE | , and set κ := max {| Arr Θ λE | , k A k λ } (see § R λ E ( A ) isin Θ ν ( A ) E for ν ( A ) := κ κ , say.3.15. Remark.
Since R λ E is right adjoint to L λ E , we obtain a monad T λ E := R λ E ◦L λ E : Sp → Sp with unit η = d Id L λ E : Id → T λ E and multiplication µ = R λ E ◦ g Id T λ E : T λ E ◦ T λ E → T λ E , as well as a comonad S λ E := L λ E ◦ R λ E on Ξ op λ , with counit ǫ := ] Id R λ E : S λ E → Id and comultiplication δ := L λ E ◦ [ Id R λ E : S λ E → S λ E ◦ S λ E (see [W, § Definition. A coalgebra over the comonad S λ E is an object A ∈ Ξ op λ equippedwith a section ζ A : A → S λ E A of the counit ǫ : S λ E A → A , with S λ E ζ ◦ ζ = δ A ◦ ζ .3.17. Proposition.
Assume given a limit cardinal λ and a Θ λE -mapping algebra X which extends to a Θ κE -mapping algebra for κ = ν ( ρ M λ E R λ E ρ X )) , in the notation of § ρ X has a natural coalgebra structure over S λ E . Remark.
The assumption clearly holds whenever X is representable – but inthis case we already know that the arrow set L λ E Y = ρ M λ E Y has a coalgebrastructure, given by ζ L λ E Y = L λ E ( η ) = L λ E ( \ Id L λ E Y ). Proof.
We want to construct a map ζ ρ X fitting into a commutative diagram(3.19) ρ X ζ ρ X (cid:15) (cid:15) ζ ρ X / / S λ E ( ρ X ) S λ E ζ ρ X (cid:15) (cid:15) S λ E ( ρ X ) ζ S λ E ( ρ X ) / / S λ E S λ E ( ρ X )in Ξ op (so all maps in Ξ are in the opposite direction!) Since S λ E ( ρ X ) = ρ M λ E R λ E ( ρ X ), all objects in (3.19) are in the image of ρ , so itsuffices to produce a map ξ X : V λ E X = M λ E R λ E ρ X → X fitting into a commutativediagram:(3.20) V λ E V λ E X ξ V λ E X (cid:15) (cid:15) V λ E ( ξ X ) / / V λ E X ξ X (cid:15) (cid:15) V λ E X ξ X / / X in Sp Θ λE , and then set ζ ρ X = ( ρξ X ) op . Step 1.
If we let K := R λ E ρ X , by (3.13) we have(3.21) K = Y M ∈ Θ λE Y φ ∈ Sp ( S , X { M } ) Y j : M → M ′ Q ( M ,φ,j ) which is in Θ κE . Thus we have an indexing category I = a M ∈ Θ λE a φ ∈ Sp ( S , X { M } ) a j : M → M ′ I ( M ,φ,j ) (depending on X ), and functors b P ( M ,φ,j ) : I ( M ,φ,j ) → Θ λE such that lim b P ( M ,φ,j ) = Q ( M ,φ,j ) as in (3.10).We can describe the indexing category I ( M ,φ,j ) by:(3.22) ` Φ ∈ χ − ( φ ) (Φ) ` π Φ (cid:15) (cid:15) ` ( γ j ) / / ` Φ ∈ χ − ( φ ) (Φ) ′ ` π ′ Φ (cid:15) (cid:15) ` Ψ ∈ ( χ ′ ) − ( b j ( φ )) (Ψ) ` π ′ Ψ (cid:15) (cid:15) ⊥ ( e ∗ j ) o o ` Φ ∈ χ − ( φ ) (Φ b ) ` ( δ j ) / / ` Φ ∈ χ − ( φ ) (Φ b ) ′ ` Ψ ∈ ( χ ′ ) − ( b j ( φ )) (Ψ b ) ⊥ ( e j ) o o ( b ) diag O O b j / / ( b ′ ) diag O O diag ❧❧❧❧❧❧❧❧❧❧❧ where ` s ∈ S ( s ) is a discrete subcategory with object set S , and diag : ( b ) → ` s ∈ S ( s ) means that there is a single arrow from ( b ) to each ( s ).The notation (Φ) ′ , and so on, is intended to distinguish objects in different discretecategories with the same set of indices χ − ( φ ). The notation ` π Φ for a mapbetween such categories means that each object (Φ) in the upper left corner maps tothe corresponding (Φ b ) beneath it. The reader should keep in mind the motivatingfunctor from (3.22) to Sp , described in (3.9).The somewhat nonstandard notation ⊥ ( e ∗ j ) : a Ψ ∈ ( χ ′ ) − ( b j ( φ )) (Ψ) → a Φ ∈ χ − ( φ ) (Φ) ′ means that if Ψ = e j (Φ) then (Ψ) is sent to (Φ) in the second discretesubcategory.The functor b P = b P ( M ,φ,j ) : I ( M ,φ,j ) → Θ λE is described implicitly by (3.10): thus b P ((Φ)) = P M for each Φ ∈ χ − ( φ ), and so on. The top right left-facing arrow in APPING ALGEBRAS AND THE ADAMS SPECTRAL SEQUENCE 11 (3.10) maps into the copy of P M ′ indexed by Φ (in the top central product) byprojecting the product in the top right onto the factor P M ′ indexed by e j (Ψ).The functors b P ( M ,φ,j ) fit together to define b P : I → Θ λE , with K = lim f ∈I b P ( f ). Step 2.
To define the map ξ X : V λ E X → X , note that since V λ E X = M λ E K , byLemma 2.2 ξ X should correspond to the value of ξ X (Id K ) in Sp ( S , X { K } ).But X { K } = X { lim f ∈I b P ( f ) } = lim f ∈I X { b P ( f ) } , because the mapping algebra X commutes by definition with the limits in (3.21). Thus we may define ξ X (Id K ) tobe the tautological map whose values at X { b P ( f ) } is f itself. Step 3.
A similar calculation shows that L := R λ E ρ V λ E X is a limit of a functor b N = b N V λ E X : J → Θ λE , but in this case the indexing category J can be describedsomewhat more explicitly because V λ E X = ρ M λ E K is also representable. Thus(3.23) L = Y M ∈ Θ λE Y φ : K → M Y j : M → M ′ b N ( M ,φ,j ) which again is in Θ κE . Therefore, J = ` M ∈ Θ λE ` φ : K → M ` j : M → M ′ J ( M ,φ,j ) , where J ( M ,φ,j ) defined analogously to (3.22), and thus the factor b N ( M ,φ,j ) in (3.23)(for nullhomotopic φ : K → M ) is the limit of the diagram:(3.24) Q Φ: φ ∼∗ P M Q p M (cid:15) (cid:15) Q P j / / Q Φ: φ ∼∗ P M ′ Q p M ′ (cid:15) (cid:15) Q Ψ: j ◦ φ ∼∗ P M ′⊤ proj j ◦ Φ o o Q p M ′ (cid:15) (cid:15) Q Φ: φ ∼∗ M Q j / / Q Φ: φ ∼∗ M ′ Q Ψ: j ◦ φ ∼∗ M ′⊤ proj j ◦ Φ o o M diag O O j / / M ′ diag O O diag ♥♥♥♥♥♥♥♥♥♥♥ where we have already taken the limits over the discrete subcategories of J ( M ,φ,j ) .Note that the objects of J ( M ,φ,j ) are actual spectrum maps g from K into thevalue of b N at this object, namely b N ( g ), which is always one of { M , M ′ , P M , P M ′ } .The map of mapping algebras ξ V λ E X = M λ E η K : M λ E L → M λ E K (see Remark 3.18)corresponds under Lemma 2.2 to the tautological map η K : K → lim J b N whichsends K into b N ( g ) by g itself. Step 4.
Similarly, the composite ξ X ◦ ξ V λ E X : V λ E V λ E X → X corresponds underLemma 2.2 to the value of ξ X ◦ ξ V λ E X (Id L ) as a spectrum map S → X { L } , whereagain X { L } = X { lim g ∈J b N ( g ) } = lim g ∈J X { b N ( g ) } . Since we are mappinginto a limit, this is uniquely determined by the map S → X { b N ( g ) } for various g ∈ Obj( J ), given by g ∗ ξ X (Id K ). Step 5.
By definition, the map V λ E ( ξ X ) = M λ E R λ E ρξ X : V λ E V λ E X → V λ E X is inducedby(3.25) R λ E ρξ X = K ϕ −→ T λ E K = L = lim g ∈J b N ( g ) . Again, we are mapping into a limit, so this is uniquely determined by maps ϕ g : K → b N ( g ) for various g ∈ Obj( J ). However K = lim f ∈I b P ( f ), so it hasstructure maps to its constituents and we see that ϕ g is precisely the structure map π f : K → b P ( f ) = b N ( g ) where f = g ∗ ξ X (Id K ) ∈ X { b N ( g ) } . Step 6.
Finally, the map ξ X ◦ V λ E ( ξ X ) : V λ E V λ E X → V λ E X is the composite of thetwo maps given in Steps 2 and 5, respectively. It corresponds under Lemma 2.2 tothe map ψ : S → X { L } which is the image of ξ X (Id K ) : S → X { K } under themap ϕ of (3.25).However, Id K : K → K = lim f ∈I b P ( f ), as a map into a limit, is determined bythe structure maps π f : K → b P ( f ), where ξ X ( π f ) : S → X { b P ( f ) } is given by f itself.Since X { L } = lim g ∈J X { b N ( g ) } is a limit, it is enough to describe the componentof ψ into each constituent X { b N ( g ) } . where it is given by the structure map π f : K → b P ( f ) for f = g ∗ ξ X (Id K ) ∈ X { b N ( g ) } . Thus ψ is determined in this componentby g ∗ ξ X (Id K ) – the same value we got in Step 4.This shows that ξ X ◦ V λ E ( ξ X ) indeed equals ξ X ◦ ξ V λ E X . (cid:3) For the representable mapping algebra X = M λ E Y , Proposition 3.17 and Remark3.18 yield:3.26. Corollary.
The coalgebra map ζ for the arrow set L λ E Y is induced by amap of mapping algebras ζ ′ : M λ E ( R λ E ( ρ M λ E Y )) → M λ E Y , so that ζ = ( ρ ◦ ζ ′ ) op . Small mapping algebras
As noted in Section 3, our goal is to associate to any mapping algebra X a cosim-plicial resolution W • , with Y = Tot W • realizing X : that is, having X weaklyequivalent ( § M E Y .In order to show this, using [B4], Y → W • must be acyclic with respect to any E -module M . However, even if X = M λ E Y to begin with, the modules appearingin Θ λE are of bounded cardinality, so for general E , merely iterating the monad T λ E on Y to produce a coaugmented cosimplicial space Y → W • will not yieldthe required resolution (although for E = H F p , this can be done, as in [BS, § X , there is a cardinal λ such that any map from each spectrum W n to M factors through a module in Θ λE .4.1. Definition.
Given X ∈ Sp and M ∈ E - Mod , any map φ : X → M in Sp is adjoint to an E -module map e φ : E ⊗ X → M with µ M ◦ ( Id E ⊗ φ ) = e φ , where µ M is the module structure map. In symmetric spectra the map e φ has an imageIm( e φ ) inside M , of cardinality ≤ k M k . Since e φ is an E -module map, it fits into acommutative diagram(4.2) E ⊗ ( E ⊗ X ) µ E ⊗ X (cid:15) (cid:15) Id E ⊗ e φ / / E ⊗ Im( e φ ) µ M (cid:15) (cid:15) E ⊗ X e φ / / M APPING ALGEBRAS AND THE ADAMS SPECTRAL SEQUENCE 13
It follows that the image of µ M : E ⊗ Im( e φ ) → M sits inside Im( e φ ), so thelatter has an E -module structure.We say that φ is effectively surjective if M = Im( e φ ), and denote the set of suchmaps by [ Hom( X , M ).If Φ : X → P M is a nullhomotopy of φ : X → M , with p M ◦ Φ = φ (where p M is the path fibration, an E -module map), then (1.6) yields a mapΦ ′ : X ⊗ ∆[1] + → M . Define an E -module map b Φ : E ⊗ X ⊗ ∆[1] + → M bysetting b Φ := µ M ◦ ( Id E ⊗ Φ ′ ). We say that Φ is an effectively surjective nullhomotopy of φ if M = Im( b Φ). The E -module structure on Im( b Φ) is given by (4.2).Note that if φ is effectively surjective, so is Φ. We denote the set of effectivelysurjective nullhomotopies of φ : X → M by [ Hom( X , P M ) φ . If we define Φ : E ⊗ X → P M by Φ( e ⊗ x ) := µ M ( e ⊗ Φ( x )( − )), we see that Φ is a nullhomotopyof e φ .Our goal is now to modify the construction (3.10) used in defining R λ E in termsof effectively surjective maps and nullhomotopies alone, thus obtaining a modifiedversion of T λ E :4.3. Definition.
For any E -module M and effectively surjective φ : X → M , define Q φ to be the pullback in Sp : Q φ / / (cid:15) (cid:15) Y ( j : M → M ′ ) ∈ E - Mod ′ Y [ Hom( X , P M ′ ) jφ P M ′ Q ′ p M ′ (cid:15) (cid:15) M ( j ) / / Y ( j : M → M ′ ) ∈ E - Mod ′ Y [ Hom( X , P M ′ ) jφ M ′ where Q ′ indicates that empty factors are to be omitted from the product, so thatthe limit is in fact taken over a small diagram.Finally, set T E X := Q ′ M ∈ E - Mod Q φ ∈ [ Hom( X , M ) Q φ .4.4. Definition.
For any symmetric spectrum X we define a cardinal λ X := sup M ∈ E - Mod {k Im( e φ ) k : φ : X → M } ∪ {k Im( b Φ) k : Φ : X → P M } . This makes sense since k Im( e φ ) k and k Im( b Φ) k are bounded by k E ⊗ X k and k E ⊗ X ⊗ ∆[1] + k , respectively. Thus for all practical purposes we may simply set λ X := k E ⊗ X ⊗ ∆[1] + k .4.5. Proposition.
For any symmetric spectrum X and κ ≥ λ X we have a canonicalisomorphism T E X ∼ = R κ E L κ E X .Proof. Recall from § T κ E for R κ E L κ E . By the description inLemma 3.12, we know that T κ E X = R κ E ( ρ ◦ Fun( X , − )) is given by(4.6) T κ E X := Y M ∈ Θ λE Y φ : X → M Y j : M → M ′ Q ( M ,φ,j ) . where for nullhomotopic φ : X → M the E -module Q ( M ,φ,j ) is the limit of: Q Φ: φ ∼∗ P M Q P j / / Q jp M $ $ ❏❏❏❏❏❏❏❏❏❏ Q Φ: φ ∼∗ P M ′ Q p M ′ (cid:15) (cid:15) Q Ψ: j ◦ φ ∼∗ P M ′⊤ proj j ◦ Φ o o Q p M ′ (cid:15) (cid:15) Q Φ: φ ∼∗ M ′ Q Ψ: j ◦ φ ∼∗ M ′⊤ proj j ◦ Φ o o M diag ◦ j o o (compare (3.24)).Our goal is to replace this limit by one involving only E -modules in Θ κE , by usingonly effective surjective maps and nullhomotopies.Note that { Ψ : j ◦ φ ∼ ∗} = j ∗ { Φ : φ ∼ ∗} ∐ New , where New is the set ofnullhomotopies of j ◦ φ not induced via j from nullhomotopies of φ .If φ : X → M is effectively surjective, then so is any nullhomotopy Φ : X → P M of φ . So we may replace the index set { Φ : φ ∼ ∗} by [ Hom( X , P M ) φ .If Φ : X → P M ′ is a nullhomotopy of j ◦ φ : X → M ′ which is not effectivelysurjective, we have a commutative diagram X Φ $ $ η X / / Φ ′′ ❋❋❋❋❋❋❋❋❋ E ⊗ X Φ / / (cid:15) (cid:15) P M ′ P M ′′ , P j ′ : : ✉✉✉✉✉✉✉✉✉✉ where M ′′ = Im( b Φ : E ⊗ X ⊗ ∆[1] + → M ′ ) (see § j ′ : M ′′ → M ′ is theinclusion. Thus Φ ′′ is an effectively surjective nullhomotopy.Thus, whenever κ > λ X , we have a cofinal diagram defining T κ E X in whichonly those M ∈ Θ κE appear for which there is either an effective surjection oran effectively surjective nullhomotopy for some map X → M . Therefore, we mayrestrict ourselves to M in Θ λ X E . This shows that the natural map T E X → T κ E X isan isomorphism. (cid:3) Remark.
Proposition 4.5 shows that T E is in fact locally small, in that for every X ∈ Sp , T E X is naturally equivalent to the value of a small functor.In particular, this implies that T E , a posteriori , is a functor, since for any map f : X → Y , we have f ∗ := T E ( f ) = T κ E ( f ) for κ = max { λ X , λ Y } (and similarlyfor composites). However, the reader may find the following explicit description of f ∗ helpful:Let ψ : Y → M an effective surjection and j : M → M ′ a map of E -modulespectra, with Ψ : Y → P M ′ an effectively surjective nullhomotopy of j ◦ ψ . Set M ′′ = Im( ] ψ ◦ f ) and M ′′′ = Im( [ Ψ ◦ f ). By (4.2) it follows that M ′′ and M ′′′ are E -modules. APPING ALGEBRAS AND THE ADAMS SPECTRAL SEQUENCE 15
Note that we have the following commutative diagram X η X (cid:15) (cid:15) f / / Y η Y (cid:15) (cid:15) ψ / / ME ⊗ X ] ψ ◦ f ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ Id E ⊗ f / / E ⊗ Y e ψ rrrrrrrrrr Id E ⊗ ψ / / E ⊗ M µ M O O in Sp . Set φ = ψ ◦ f . By the definitions of M ′′ and M ′′′ we get E -modulemaps j ′′ : M ′′ → M and j ′ : M ′′′ → M ′ fitting into the diagram E ⊗ Y Ψ & & e ψ $ $ ❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏ E ⊗ X Id E ⊗ f o o Φ / / e φ $ $ ❍❍❍❍❍❍❍❍❍ P M ′′′ p M ′′′ (cid:15) (cid:15) P j ′ / / P M ′ p M ′ (cid:15) (cid:15) M ′′ j ′′′ / / j ′′ (cid:15) (cid:15) M ′′′ j ′ $ $ ■■■■■■■■■■ M j / / M ′ The map j ′′′ exists and it is an E -module map because [ Ψ ◦ f ◦ i X = ] j ◦ φ . Here i X is given by the identification of X with X ⊗ { } inside X ⊗ ∆[1] + .The component ϑ of the map f ∗ : T E X → T E Y into the factor Q ψ of T E Y isdefined by projecting from T E X onto Q φ and onto the copy of P M ′′′ indexedby Φ (= Φ ◦ η X ). This then maps by P j ′ to the copy of P M ′ in Q ψ indexedby Ψ.The map f ∗ , restricted to Q φ , is then given by the universal property of thepull-back square as follows: Q φ ϑ + + ❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳ (cid:15) (cid:15) f ∗ (cid:29) (cid:29) ❀❀❀❀❀❀❀❀❀❀ M ′′ j ′′ (cid:26) (cid:26) ✻✻✻✻✻✻✻✻✻✻✻✻✻✻ Q ψ / / (cid:15) (cid:15) Y ( j : M → M ′ ) ∈ E - Mod ′ Y [ Hom( X , P M ′ ) jψ P M ′ Q ′ p M ′ (cid:15) (cid:15) M ( j ) / / Y ( j : M → M ′ ) ∈ E - Mod ′ Y [ Hom( X , P M ′ ) jψ M ′ . Cosimplicial resolutions and the E -based Adams spectral sequence For any limit cardinal λ , the adjoint functors L λ E and R λ E constructed in Section3 define a comonad S λ E = L λ E ◦ R λ E on the category (Ξ λ ) op (see § X ,yields a cosimplicial spectrum W • such that Tot W • realizes X under favorablecircumstances (in particular, when X = M λ E Y for an E -good spectrum Y ). We note that the proper setting for our constructions is the resolution model cate-gory of cosimplicial spectra of [B4, § § . The cosimplicial spectrum W • associated to X . Given a mapping algebra X ∈ Map Θ λE , by iterating the comonad S λ E on the arrow set A = ρ X we obtainas usual an augmented simplicial object ε : e V • → A in Ξ op , with e V k := ( S λ E ) k +1 A ,and face and degeneracy maps induced by the structure maps of the comonad (see[W, 8.6.4]).If we assume that X extends as in Proposition 3.17 – e.g., if it is representable –then A = ρ X has a coalgebra structure ζ A : A → S λ E A = e V over the comonad S λ E , which provides an extra degeneracy for e V • → A . Thus R λ E applied to thisaugmented simplicial object yields a cosimplicial spectrum W • , with W = R λ E ( A ), W = R λ E ( e V ), d = R λ E ( ζ A ), and d = R λ E ( ε ) (see [BS, Prop. 3.27] for a detaileddescription). By applying the functor L λ E to this cosimplicial spectrum we obtaina simplicial object in mapping algebras M λ E W • (by contravariance of R λ E ), whichis augmented to X , yielding a map of simplicial mapping algebras M λ E W • → c ( X ) • .5.2. Definition.
We say that a map f : W • → U • of simplicial spectral functors(e.g., mapping algebras) is an E -equivalence (cf. [J]) if for every M ∈ Θ λE , theinduced map of simplicial abelian groups W • { M } → U • { M } is a weak equivalence(of simplicial sets).5.3. Proposition.
If for X ∈ Map Θ λE and W • as above X is known to be ahomotopy functor, then M λ E W • → c ( X ) • is an E -equivalence. Equivalently, for every M ∈ Θ λE , the augmented simplicial abelian group [ W • , M ] → π ( X { M } ) is acyclic, where [ W • , M ] is the simplicial abelian group obtainedby applying the homotopy functor [ − , M ] (see Lemma 2.13) in each cosimplicialdimension. Proof.
By standard facts about comonads (see [W, Proposition 8.6.10]), the aug-mented simplicial arrow set L λ E W • → ρ X is contractible, so by Corollary 2.15 and(2.4) the augmented simplicial mapping algebra M λ E W • → X is contractible,too. (cid:3) Remark.
Note that each W n is an E -module, and for each 0 ≤ i ≤ n , thecodegeneracy map s in : W n → W n +1 is R λ E ( S λ E ) n − i ǫ ( S λ E ) i A , where ǫ ( S λ E ) i A is thecomonad counit map for ( S λ E ) i A . Thus the codegeneracies are in the image of R λ E and in particular are E -module maps.5.5. Definition.
For any ring spectrum E , G ( E ) := E - Mod is a class of injectivemodels in Sp in the sense of [B4, § G ( E )-localization functor b L G ( E ) : Sp → Sp , with a map η Y : Y → b L G ( E ) Y (see [B4, § Y is called E - good if η Y is an E - equivalence – that is,for each M ∈ G ( E ), the induced map [ b L G ( E ) Y , M ] → [ Y , M ] is an isomorphism(this is called a G ( E )-equivalence in [B4]).5.6. Remark.
By [B1, Theorems 6.5 & 6.6], when E and Y are connective and thecore R of π E is either Z /n or a subring of Q , b L G ( E ) Y is simply the usual R -completion of Y , given by smashing with the Moore spectrum for R (see [B1, § APPING ALGEBRAS AND THE ADAMS SPECTRAL SEQUENCE 17
Notation.
For any Y ∈ Sp , let c λ Y := sup { λ T n E Y } n ∈ N , in the notation of § . The cosimplicial spectrum W • associated to Y. When the mappingalgebra X of § Y , and λ ≥ c λ Y , we can think of thecosimplicial spectrum W • constructed there from X = M λ E Y as having the form W k := T k +1 E Y , with coaugmentation η Y : Y → T E Y .For a cosimplicial spectrum W • the totalization Tot W • as in [B4, Section 2.8]then satisfies5.9. Theorem. If E is a ring spectrum, Y an E -good symmetric spectrum, λ = c λ Y ,and W • is as above, the canonical map Y → Tot W • is an E -equivalence.Proof. By Proposition 4.5, the augmented simplicial group [ W • , M ] → [ Y , M ] isacyclic for all M ∈ G ( E ), using Proposition 5.3. Since Y is E -good, b L G ( E ) Y ≃ Tot W • so Y → Tot W • is an E -equivalence by [B4, § (cid:3) . Cosimplicial Adams resolutions. Recall that an E - Adams resolution for an( E -good) spectrum Y is a sequence of spectra X = X X g o o X g o o · · · o o such that for each s ≥ X s is E -equivalent to Y .(ii) If K s is the cofiber of g s and f s : X s → K s is the structure map, then E ⊗ f s has a retraction.(iii) K s is a retract of E ⊗ K s .(see [R, § E -good spectrum Y with E -mapping algebra M λ E Y , we saw in theprevious section how to construct a cosimplicial spectrum W • such that Tot W • is an E -completion of Y , in the sense of [R, § E -modules given by [SS, Theorem 4.1], andthus an induced Reedy model category E - Mod ∆ of cosimplicial E -modules (see [Hi,Theorem 15.3.4]). We may thus replace the W • of § E - Mod ∆ , (which we also denote by W • , to avoid unnecessary notation).We then have a tower of fibrations(5.11) W = Tot ( W • ) Tot ( W • ) · · · Tot k − ( W • ) h o o Tot k ( W • ) h k o o · · · o o (see [B4, § h k given by Ω k F k , where F k is the fiber ofthe map W k → M k − W • to the matching spectrum of [BK1, X, § X s = Tot s W • and K s := Ω s F s +1 , we see that(5.12) W j (cid:15) (cid:15) Tot ( W • ) · · · j (cid:15) (cid:15) h o o Tot k − ( W • ) o o j k − (cid:15) (cid:15) Tot k ( W • ) j k (cid:15) (cid:15) h k o o · · · o o F Ω F Ω k − F k Ω k F k +1 is an E -Adams resolution for Y .Moreover,(5.13) F k = k − \ j =0 Ker( s j : W k → W k − ) , As noted in § W • are E -module maps, so F k is an E -module.Moreover, the connecting homomorphism δ k : π ∗ F k → π ∗ F k +1 for this tower offibrations is just the differential for the normalized cochains on π ∗ W • – that is,the alternating sum of the coface maps (see [BK1, X, § Z , applying the functor Fun( Z , − ) to W • yieldsa cosimplicial spectrum, whose total spectrum is the E -completion of Fun( Z , Y ),under favorable assumptions. We define the E - based Adams spectral sequence forFun( Z , Y ) to be the homotopy spectral sequence for Fun( Z , − ) applied to (5.11),with(5.14) E k,t = π t − k (Ω k Fun( Z , F k )) ∼ = π (Fun(Σ t − k Z , Ω k F k )) ∼ = π (Fun(Σ t Z , F k )) . (see [BK1, X, § E -based Adams spectral sequencefrom the E -term on (see [R, § Remark.
Note that by Theorem 5.9 W • (and thus our choice for the E -completion of Y ), as well as the E -based Adams spectral sequence for Y , are deter-mined functorially by M λ E Y (in fact, by ρ M λ E Y with its coalgebra structure) andby Z , since the construction of W • in § X .The Reedy model category of cosimplicial E -modules of [SS, Theorem 4.1] also hasfunctorial factorizations, so the same remains true after fibrant replacement of W • .6. Differentials in the Adams spectral sequence
In this section we assume E is a ring spectrum, Y ∈ Sp is E -good, and Z ∈ Sp is finite and λ ≥ c λ Y , c λ Z (in the notation of § X = M λ E Y , with Y → W • constructed from X as in § E -based Adams spectralsequence for Fun( Z , Y ) with the homotopy spectral sequence of the cosimplicialspectrum Fun( Z , W • ). (We do not in fact need Y to be E -good in order for mostof our results to hold, but without some such assumption the spectral sequence neednot converge, so information about it will not be of much use.)We can now state our first main result:6.1. Theorem.
Given E , Z , and Y as above, for each r ≥ , the d r -differential inthe E -based Adams spectral sequence for Fun( Z , Y ) , and thus its E r +1 -term, canbe calculated from the cosimplicial ( r − -truncated space P r − M λ E Z { W • } .Proof. We recall the standard construction of the differentials in the homotopy spec-tral sequence for the Tot tower of fibrations for X • := Fun( Z , W • ), in terms of theinterlocking long exact sequences of Figure 6.2. π k +1 Tot n X • q n (cid:15) (cid:15) δ n / / π k Ω n +1 N n +1 X • j n +1 / / π k Tot n +1 X • q n +1 (cid:15) (cid:15) δ n +1 / / π k − Ω n +2 N n +2 X • π k +1 Tot n − X • δ n − / / π k Ω n N n X • j n / / π k Tot n X • δ n / / π k − Ω n +1 N n +1 X • Figure 6.2.
Exact couple for Tot tower
APPING ALGEBRAS AND THE ADAMS SPECTRAL SEQUENCE 19
Here the normalized chains for X • are given by N n X • = Fun( Z , F n ) (see(5.13)).As we shall see below, the information needed to calculate the differentials at eachstage, consisting of various maps Z → W i , nullhomotopies thereof, and so on:(a) can be expressed in terms of the mapping algebra M λ E Z and the simplicialmapping algebra M λ E W • ;(b) in fact depends only on suitable truncations of these mapping algebras, if weare only calculating differentials up to the r -th stage.For this purpose, we think of the differential d r : E s,tr → E s + r,t + r − r as a “relation”(i.e., partially defined map E s,t → E s + r,t + r − with a certain indeterminacy), inthe spirit of [B2]. Thus a class h γ i ∈ E n,n + kr will be represented by an element γ ∈ E n,n + k such that d ( γ ), · · · d r − ( γ ) all have 0 as a value.In our interpretation, the value [ β ] we compute for the differential d j lies in E n + j,n + k + j − = π (Fun(Σ n + k + j − Z , F n + j )) (see (5.14)), so its vanishing is witnessedby a choice of nullhomotopy. This nullhomotopy takes value in a higher truncation ofthe mapping algebra than the map β , which explains why each successive differentialrequires a higher truncation. Step 1.
Any class γ ∈ E n,n + k is represented in turn by a map ˆ g : Σ k Z → Ω n F n :that is, a map g : Σ k Z → Tot n W • with h n ◦ g = 0 (see (5.12)). Byadjunction this defines a map of cosimplicial spectra G • n : sk n (∆ • ) + ⊗ Σ k Z → W • .The value of the successive differentials d ( γ ) , · · · , d r − ( γ ) serve as the successiveobstructions to lifting G • n to G • n +1 : sk n +1 (∆ • ) + ⊗ Σ k Z → W • , . . . up to G • n + r − :sk n + r − (∆ • ) + ⊗ Σ k Z → W • .The cosimplicial map G • n : sk n (∆ • ) + ⊗ Σ k Z → W • consists of a sequence of mapsof spectra G jn : sk n (∆[ j ]) + ⊗ Σ k Z → W j ( j = 0 , , . . . ). Since W • is Reedyfibrant,(6.3) Ω n F n → Tot n W • h n −→ Tot n − W • is a fibration sequence on the nose, so the fact that ˆ g lands in Ω n F n (and thus G nn lands in F n ) implies that G n = · · · = G n − n = 0. Moreover, sk n ∆[ j ] isdetermined by ∆[ j ] and the coface maps in ∆ • , for j > n , so the maps G jn ( j > n ) are determined by G nn and the coface maps of W • .Note that G nn is adjoint to a map Σ k Z → ( W n ) sk n (∆[ n ]) + , – in other words, itis equivalent to a map e G nn : S → M λ E Σ k Z { M } for M := ( W n ) sk n (∆[ n ]) + ∈ Θ λE , interms of the simplicial structure on E -modules (see [SS]). Step 2.
As noted above, γ represents an element in E if d ( γ ) = 0 in E n +1 ,n + k – that is, if(6.4) φ := n X i =0 ( − i d i ◦ G nn is nullhomotopic in F n +1 ⊆ W n +1 (see Figure 6.2) The differential d ( γ ) thustakes value in π M λ E F n +1 . Step 3.
By (5.13), F n +1 is the (homotopy) limit of the 3 × W n +1 ⊤ s j / / (cid:15) (cid:15) Q nj =0 W n (cid:15) (cid:15) ∗ (cid:15) (cid:15) o o ∗ / / ∗ ∗ o o ∗ / / O O ∗ O O ∗ o o O O and similarly P F n +1 is the (homotopy) limit of the 3 × W n +1 ) ∆[1]ev (cid:15) (cid:15) ⊤ ( s j ) ∆[1] / / Q nj =0 ( W n ) ∆[1] Q ev (cid:15) (cid:15) ∗ ∆[1] = ∗ o o (cid:15) (cid:15) W n +1 ⊤ s j / / Q nj =0 W n ∗ o o ∗ O O / / ∗ O O ∗ o o O O We have a map from (6.6) to (6.5) induced by ev , and by taking limits weobtain the path fibration p : P F n +1 → F n +1 .Thus the path-loop fibration sequence for F n +1 is obtained by taking iteratedpullbacks of diagrams built from W n and W n +1 , first vertically, and then hori-zontally (see [BK1, XI, 4.3]). We therefore see that both the class φ of (6.4) rep-resenting d ( γ ) in π M λ E Σ k Z { F n +1 } , and our choice of a nullhomotopy Φ forit, are determined, according to [M, Theorem 10], by various compatible maps andnullhomotopies into the diagrams (6.6) and (6.5).These maps and nullhomotopies, respectively, correspond to maps and nullhomo-topies, respectively, from S to P M λ E Z { W n +1 } and P M λ E Z { W n } , (composedwith ( s j ) ∗ : M λ E Σ k Z { W n +1 } → M λ E Σ k Z { W n } ) – which can be expressed in termsof the truncated mapping algebra P M λ E Σ k Z and the action on it of the free sim-plicial truncated mapping algebra P M λ E W • (in the sense of (2.6)) – in otherwords, in terms of the 1-truncated cosimplicial space P M λ E Σ k Z { W • } .By a standard argument in the long exact sequence of the fibration (6.3), we canuse Φ to extend G • n to a map G • n +1 : sk n +1 (∆ • ) + ⊗ Σ k Z → W • . Note that this isdetermined by G nn +1 : Σ k Z → ( W n ) ∆[ n ] + , G n +1 n +1 : Σ k Z → ( W n +1 ) ∆[ n +1] + , and themaps between them coming from the coface maps of W • .Because G jn +1 = 0 for j < n , the maps actually land in Ω n W n and Ω n W n +1 ,respectively, so they take value in P M λ E Σ k Z { Ω n W • } . Step 4.
Assume by induction that, for r ≥ γ represents an element in E r , sothe differentials on γ up to d r − vanish, and we have an extension of G • n to G • n + r − : sk n + r − (∆ • ) + ⊗ Σ k Z → W • with G jn + r − = 0 for 0 ≤ j ≤ n − j > n + r − G jn + r − isdetermined by G n + r − n + r − and the coface maps of W • ). As usual, we can extend thisfurther to G • n + r (for some choice of G • n + r − ) if and only if d r ( γ ) vanishes.The map G • n + r − represents a class α r − in π k Tot n + r − X • (as in Figure 6.2).Applying the connecting homomorphism δ n + r − : π k Tot n + r − X • → π k − Ω n + r N n + r X • APPING ALGEBRAS AND THE ADAMS SPECTRAL SEQUENCE 21 to α r − yields a class [ β r − ] ∈ [Σ k − Z , Ω n + r F n + r ], which represents the value of d r ( γ ).Note that β r − (as a map into F n + r ) is represented in turn as in (6.5)above by a map of spectra b b r − : Σ k Z → Ω n + r − W n + r , and thus by b r − ∈ ( P r − M λ E Σ k Z { Ω n W n + r } ) r − (an ( r − P r − ( − ), asin § G • n + r − , . . . , G • n , also come into the picture in the formof (iterated) coface maps of W • applied to earlier simplices β r − , . . . , β . Thisis why we need all of P r − M λ E Σ k Z { Ω n W • } , and not just its ( r − §
5] for an explicit description of the combinatorics in a slightly differentformulation (which is not needed here).We thus see by induction that the choice of G • n + r − , as well as the value of d r ( γ ),may be expressed in terms of P r − M λ E Σ k Z { Ω n W • } .If d r ( γ ) vanishes, for some collection of choices as above, the map β r − isnullhomotopic; as in Step 3, the choice of a nullhomotopy – and thus the lift of G • n + r − to G • n + r and the resulting value of d r +1 ( γ ) – is encoded one simplicialdimension higher – that is, in the cosimplicial space P r M λ E Σ k Z { Ω n W • } .Finally, note that up to homotopy the mapping algebra M λ E Σ k Z is just Ω k M λ E Z ,since it is a homotopy spectral functor, and for the same reason M λ E Σ k Z { Ω n W • } ≃ Ω n M λ E Σ k Z { W • } . (cid:3) . Resolution model categories. Since any spectrum is a homotopy groupobject in Sp , from Lemma 2.2 we see that for all M ∈ Θ λE , the free spectral functor M λ E M is a homotopy cogroup object in Sp Θ λE .Thus by [J, Theorem 2.2.]:(a) There is a resolution model category structure on ( Sp Θ λE ) ∆ op = Sp Θ λE × ∆ op , inwhich the weak equivalences are the E -equivalences (cf. § f : U • → W • of simplicial spectral functors such that for each M ∈ Θ λE the induced map π U • { M } → π W • { M } of simplicial groups is a weakequivalence.(b) Similarly, for each M ∈ Θ λE , any fibrant and cofibrant replacement for M λ E M in the P r -model structure on Sp Θ λE is a homotopy cogroup object there, soby Proposition 2.11, ( Sp Θ λE ) ∆ op also has a P r resolution model categorystructure, with the same E -equivalences.(c) Finally, given a cosimplicial E -module W • , let Θ W denote the simpliciallyenriched category whose objects are W i ( i = 0 , , . . . ) with truncatedsimplicial mapping spaces map ∗ ( W i , W j ) := P r Fun( W i , W j ) as in § Spaces Θ W ∗ of simplicial functors (with respect to map ∗ ) alsohas a proper model category structure (see [BBC, § P r M λ E W i is a cogroup object in Spaces Θ W ∗ , so we get acorresponding resolution model category structure on the simplicial objects( Spaces Θ W ∗ ) ∆ op (see [BBC, § n W • ).We now have: Proposition.
Let W • be constructed from Y as in § U • isany resolution of X = M λ E Y (that is, a cofibrant replacement, in the model categorystructure of § c • ( X ) which is X in eachsimplicial dimension); then P r M λ E W • is E -equivalent to P r U • .Proof. Since π j P r X { M } ∼ = ( π j X { M } for 0 ≤ j ≤ r P r M λ E W • is a resolution of P r X in the model category structure of § (cid:3) From Theorem 6.1 and Proposition 6.8 we deduce:6.9.
Theorem. If E = H R for a commutative ring R , Z is a fixed finite spectrum,and Y is a E -good spectrum, then for any r ≥ the E r +2 -term of the E -basedAdams spectral sequence for Fun( Z , Y ) is determined by the truncated mappingalgebra P r M λ E Y .Proof. Let U • be any resolution of M λ E Y in the model category structure of § P r M λ E Y , up to E -equivalence). By [BBC, Theorem3.21 ff. ], we can construct a cosimplicial resolution U • of Y in the resolution modelcategory structure on Sp ∆ of [B4, § M λ E U • is Reedy equivalent to U • (that is, there is a map of simplicial spectral functors f : M λ E U • → U • witheach f n : M λ E U n → U n a weak equivalence of spectral functors).Thus the truncated cosimplicial space P r M λ E Z { U • } is well defined up to Reedyweak equivalence. Moreover, there is an E -equivalence g : U • → W • (where W • is the cosimplicial spectrum of § E -equivalence of truncatedcosimplicial spaces P r M λ E Z { U • } → P r M λ E Z { W • } , and thus a map of spectralsequences which is an isomorphism form the E -term on. The result then followsfrom Theorem 6.1. (cid:3) This presumably holds for any ring spectrum E , though the results of [BBC, § H R .6.10. Remark.
Our main goal here was to show what sort of general information about E -modules, combined with what specific data on Y and Z , suffice to determine the E r -term of the E -based Adams spectral sequence for Fun( Z , Y ) – modelled onthe way the E -term is a functor of E ∗ Y (under favorable assumptions on E ).As Theorems 6.1 and 6.9 show, the necessary data can be described in the languageof truncated mapping algebras, our main object of study here. For E = H F p , Z = S , and r = 2, this data reduces to the knowledge of H ∗ ( Y ; F p ) as a moduleover the Steenrod algebra, as in [A]. References [A] J.F. Adams, “On the structure and applications of the Steenrod algebra”,
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