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Adams spectral sequence

Contents

Idea

The Adams spectral sequence (Adams 58) is a type of spectral sequences used for computations of stable homotopy groups of spectra in terms of their generalized homology/generalized cohomology. Given a spectrum XX and a ring spectrum EE, then under mild assumptions the Adams spectral sequence converges to the homotopy groups of the EE-nilpotent completion of XX, while under stronger assumptions the latter is the EE-Bousfield localization of spectra. The second page of the spectral sequence is given by the EE-homology of XX as modules over the dual EE-Steenrod operations. The Adams spectral sequence may be seen as a variant of the Serre spectral sequence obtained by replacing a single fibration by an “Adams resolution”.

The original classical Adams spectral sequence is the case where E=H𝔽 pE = H\mathbb{F}_p is ordinary homology mod pp, while the Adams-Novikov spectral sequence (Novikov 67) is the case where E=E = MU is complex cobordism cohomology theory or E=E = BP, Brown-Peterson theory.

Generally, for EE a suitable E-infinity algebra there is a corresponding EE-Adams(-Novikov) spectral sequence whose second page is given by EE-generalized cohomology and which arises as the spectral sequence of a simplicial stable homotopy type of the cosimplicial object which is the Cech nerve/Sweedler coring/Amitsur complex of EE. As such the Adams spectral sequence is an analog in stable homotopy theory of the Bousfield-Kan homotopy spectral sequence in unstable homotopy theory.

Working with the Adams spectral sequence tends to be fairly involved, as is clear from the subtlety of the results it computes (notably stable homotopy groups of spheres) and as witnessed by the fact that one uses further spectral sequences just to compute the low pages of the Adams spectral sequence, e.g. the May spectral sequence and the chromatic spectral sequence.

A clear conceptual picture in higher algebra of what happens in the Adams spectral sequence (Lurie 10) has emerged via the re-formulation in (Miller 81, Hopkins 99). Survey of this perspective includes (Wilson 13).

Here one observes that for EE a ring spectrum, hence an E-∞ ring, the totalization of its Amitsur complex cosimplicial spectrum is really the algebraic dual incarnation of the 1-image factorization of the the terminal morphism

Spec(E) Spec(Tot(E )) p Spec(𝕊) \array{ Spec(E) &\longrightarrow& Spec(Tot(E^{\wedge^\bullet})) \\ \downarrow & \swarrow_{\mathrlap{p}} \\ Spec(\mathbb{S}) }

in E-∞ geometry/spectral geometry.

Moreover, a spectrum XX is equivalently a quasicoherent sheaf on Spec(S)Spec(S) and E XE^{\wedge^\bullet} \wedge X is accordingly the Sweedler coring that expresses the descent property of XX pullled back along the cover pp, dually the EE-localization of XX. The Adams spectral sequence may then be seen to be the computation of the homotopy groups of the EE-localization of XX in terms of its restriction to that cover.

In general, notably for E=H𝔽 pE = H \mathbb{F}_p, the 1-image of Spec(E)Spec(𝕊)Spec(E) \to Spec(\mathbb{S}) is smaller than Spec(𝕊)Spec(\mathbb{S}) and therefore this process computes not all of XX, but just the restriction to that one image (for instance just the pp-local component). Examples of ring spectra which are “complete” with respect to the sphere spectrum in that the above 1-image coincides with Spec(S)Spec(S) notably includes the complex cobordism cohomology spectrum E=E = MU (Hopkins 99, p. 70).

That explains the relevance of the Adams-Novikov spectral sequence (noticing that the wedge summands of MU (p)MU_{(p)} are the BP-spectra) and the close interplay between the ANSS and chromatic homotopy theory.

Details

  1. Via injective resolutions

  2. As derived descent in higher algebra

The EE-Adams spectral sequence

We here discuss Adams spectral sequences for computation of EE-localization of mapping spectra [Y,X][Y,X] for by EE a general commutative ring spectrum which is flat in a certain sense (def. 8 below).

The classical Adams spectral sequence is the special case with Y=X=𝕊Y = X = \mathbb{S} and E=E = A𝔽 p\mathbb{F}_p, discussed below.

The Adams-Novikov spectral sequence is the special case with Y=X=𝕊Y = X = \mathbb{S} and E=E = MU, discussed below.

EE-Adams resolutions

A streamlined discussion of EE-Adams resolutions in close analogy to injective resolutions in homological algebra was given in (Miller 81), advertized in (Hopkins 99) and worked out in more detail in (Aramian).

Notice that the standard concept of exact sequences and injective objects makes sense in abelian categories, but not in the stable homotopy category of spectra, as the latter is only an additive category. Of course this is because the stable homotopy theoretic analog of what are exact sequences in abelian categories are homotopy fiber sequences of spectra. But for computational purposes it turns out useful to consider a blend between these two concepts (due to Miller 81), where a sequence of spectra X X_\bullet is regarded as exact if the homotopical functor to the abelian category of abelian groups that it represents takes values in exact sequences. With respect to this hybrid concept, EE-Adams resolutions in the stable homotopy category are the direct analog of injective resolutions in an abelian category.

Definition

For XX a spectrum, we say that the homotopical functor that it co-represents is the functor of stable homotopy groups of the mapping spectrum-construction out of XX, regarded as a functor on the stable homotopy category:

π [X,]:Ho(Spectra)Ab . \pi_\bullet[X, -] \colon Ho(Spectra) \longrightarrow Ab^{\mathbb{Z}} \,.

(Regarded as an (∞,1)-functor on the stable (∞,1)-category of spectra this is also called the homological functor co-represented by XX.)

Example

For X=𝕊X = \mathbb{S} the sphere spectrum then the homotopical functor that it co-represents according to def. 1

π [𝕊,]π () \pi_\bullet[\mathbb{S},- ]\simeq \pi_\bullet(-)

is the stable homotopy group-functor.

Throughout,EE is a ring spectrum.

Definition

Say that

  1. a sequence of spectra

    A 1A 2A n A_1 \longrightarrow A_2 \longrightarrow \cdots \longrightarrow A_n

    is

    1. a (long) exact sequence if the induced sequence of homotopy functors according to def. 1, is a long exact sequence in [HoSpectra,Ab ][HoSpectra,Ab^{\mathbb{Z}}];

    2. (for n=2n = 2) a short exact sequence if

      0A 1A 2A 30 0 \longrightarrow A_1 \longrightarrow A_2 \longrightarrow A_3 \longrightarrow 0

      is (long) exact in the above sense;

  2. a morphism ABA \longrightarrow B is

    1. a monomorphism if 0AB0 \longrightarrow A \longrightarrow B is an exact sequence in the above sense;

    2. an epimorphism if AB0A \longrightarrow B \longrightarrow 0 is an exact sequence in the above sense.

For EE a ring spectrum, then a sequence of spectra is called (long/short) EE-exact and a morphism is epi/mono, respectively, if becomes long/short exact or epi/mono, respectively, after taking smash product with EE.

Example

Every homotopy cofiber sequence of spectra is exact in the sense of def. 2.

Example

The suspension functor Σ:Ho(Spectra)Ho(Spectra)\Sigma \colon Ho(Spectra) \to Ho(Spectra) preserves exact sequences in the sense of def. 2.

Proof

By the suspension/looping adjunction-isomorphism [ΣA ,][A ,Ω()][\Sigma A_\bullet, -]\simeq [A_\bullet, \Omega(-)] (prop.) and so the statement follows from the assumption that A A_\bullet is long exact.

Example

If a morphism, s:ABs \colon A \to B has a retraction r:BAr \colon B \to A in Ho(Spectra) then it is a monomorphism in the sense of def. 2.

Proof

We need to check that for every XX the morphism i *:[B,X][A,X]i^\ast \colon [B, X]\to [A,X] is surjective. By retraction, given f:AXf \colon A \to X, then rf:BrAfXr \circ f \colon B \stackrel{r}{\to} A \stackrel{f}{\to} X is a preimage.

Example

For any spectrum XX the morphism

X𝕊XeidEX X \simeq \mathbb{S} \wedge X \stackrel{e \wedge id}{\longrightarrow} E \wedge X

is an EE-monomorphism in the sense of def. 2.

Proof

We need to check that EXideidEEXE \wedge X \stackrel{id \wedge e \wedge id}{\longrightarrow} E \wedge E \wedge X is a monomorphism in the sense of def. 2. Observe that this morphism has a retraction given by μid\mu \wedge id. Therefore it is a monomorphism by example 4.

Remark/Warning

Consecutive morphisms in an EE-exact sequence according to def. 2 in general need not compose up to homotopy, to the zero morphism. But this does become true (lemma 3 below) for sequences of EE-injective objects, defined below in def. 3.

Lemma
  1. If f:BAf \colon B\longrightarrow A is a monomorphism in the sense of def. 2, then there exists a morphism g:CAg \colon C \longrightarrow A such that the wedge sum morphism is a weak homotopy equivalence

    fg:BCA. f \vee g \;\colon\; B \wedge C \stackrel{\simeq}{\longrightarrow} A \,.

    In particular, every morphism in Ho(Spectra) has an extension along a monomorphism in this sense.

  2. If f:ABf \colon A \longrightarrow B is an epimorpimsm in the sense of def. 2, then there exists a homotopy section s:BAs \colon B\to A, i.e. fsIdf\circ s\simeq Id, together with a morphism g:CAg \colon C \to A such that the wedge sum morphism is a weak homotopy equivalence

    sf:BCA. s \vee f \colon B\vee C \stackrel{\simeq}{\longrightarrow} A \,.
Proof

Given a monomorphism f:ABf \colon A \longrightarrow B, consider the correspondiing homotopy cofiber sequence

AfBrCδΣAΣfΣB. A\stackrel{f}{\longrightarrow} B \stackrel{r}{\longrightarrow} C \stackrel{\delta}{\longrightarrow} \Sigma A \stackrel{-\Sigma f}{\longrightarrow} \Sigma B \,.

We first observe that the connecting homomorphism is equivalent to the zero morphism δ0\delta \simeq 0. This follows because by example 2 the sequence

[C,X]δ *[ΣA,X](Σf) *[ΣB,X] [C,X] \stackrel{\delta^\ast_\bullet}{\longleftarrow} [\Sigma A, X] \stackrel{(-\Sigma f)^\ast_\bullet}{\longleftarrow} [\Sigma B, X]

is an exact sequence (of homotopy groups) for every XX, while by example 3 the morphism on the right is epi, so that δ *=0\delta^\ast_\bullet = 0.

Now since BrCδ0B \stackrel{r}{\longrightarrow} C \stackrel{\delta \simeq 0}{\longrightarrow} is also a homotopy fiber sequence, the pasting law identifies BC×ACAB \simeq C \times A \simeq C \vee A:

BC×A A 0 C 0 ΣA. \array{ B \simeq C \times A &\longrightarrow& A &\longrightarrow& 0 \\ \downarrow && \downarrow && \downarrow \\ C &\longrightarrow& 0 &\longrightarrow& \Sigma A } \,.
Definition

For EE a ring spectrum, say that a spectrum SS is EE-injective if for each morphism ASA \longrightarrow S and each EE-monomorphism f:ASf \colon A \longrightarrow S in the sense of def. 2, there is a diagram in HoSpectra of the form

A S B. \array{ A &\longrightarrow & S \\ \downarrow & \nearrow_{\mathrlap{\exists}} \\ B } \,.
Lemma

A spectrum is EE-injective in the sense of def. 3, precisely if it is a retract in HoSpectra of a free EE-modules, hence of EXE \wedge X for some spectrum XX.

Proof

In one direction, assume that SS is EE-injective and consider the diagram

S id S eid ES. \array{ S &\stackrel{id}{\longrightarrow}& S \\ {}^{\mathllap{e \wedge id}}\downarrow \\ E \wedge S } \,.

By example 5 here the vertical morphism is an EE-monomorphism, and so by assumption there is a lift

S id S ES \array{ S &\stackrel{id}{\longrightarrow}& S \\ \downarrow & \nearrow \\ E \wedge S }

which exhibits SS as a retract of ESE \wedge S.

In the other direction, given a retraction SsrEXS \stackrel{\overset{r}{\longleftarrow}}{\underset{s}{\longrightarrow}} E \wedge X we show that there exist extensions in

A g S f B \array{ A &\stackrel{g}{\longrightarrow} & S \\ {}^{\mathllap{f}}\downarrow \\ B }

whenever the vertical morphism is an EE-monomorphism. To see this, complete the extension problem to the following commuting diagram

A eid EA f g μididsg idf S r EX B eid EB. \array{ A &&& \stackrel{e \wedge id}{\longrightarrow} &&& E \wedge A \\ {}^{\mathllap{f}}\downarrow &\searrow^{\mathrlap{g}} & && & {}^{\mathllap{{\mu \wedge id} \atop {\circ id \wedge s g }}}\swarrow & \downarrow^{\mathrlap{id \wedge f}} \\ && S &\stackrel{r}{\longleftarrow}& E \wedge X \\ \downarrow && && && \downarrow \\ B &&& \stackrel{e \wedge id}{\longrightarrow} &&& E \wedge B } \,.

Now, since ff is assumed to be an EE-monomorphism, the morphism EidfEid\wedge f on the right is a monomorphism in the sense of def. 2, and so by lemma 1 there exists an extension hh in

A eid EA f g μididsg idf S r EX h B eid EB. \array{ A &&& \stackrel{e \wedge id}{\longrightarrow} &&& E \wedge A \\ {}^{\mathllap{f}}\downarrow &\searrow^{\mathrlap{g}} & && & {}^{\mathllap{{\mu \wedge id} \atop {\circ id \wedge s g }}}\swarrow & \downarrow^{\mathrlap{id \wedge f}} \\ && S &\stackrel{r}{\longleftarrow}& E \wedge X \\ \downarrow && && &\nwarrow^{\mathrlap{h}}& \downarrow \\ B &&& \stackrel{e \wedge id}{\longrightarrow} &&& E \wedge B } \,.

By composition and commutativity, this gives the required extension of gg along ff.

Definition

For EE a ring spectrum, then an EE-Adams resolution of an spectrum SS is a long exact sequence, in the sense of def. 2, of the form

0SI 0I 1I 2 0 \longrightarrow S \longrightarrow I_0 \longrightarrow I_1 \longrightarrow I_2 \longrightarrow \cdots

such that each I jI_j is EE-injective, def. 3.

Lemma

Any two consecutive maps in an EE-Adams resolution, def. 4, compose to the zero morphism.

The following lemma says that EE-Adams resolutions may be extended along morphisms.

Lemma

For XX X \to X_\bullet an EE-Adams resolution, def. 4, and for XYX \longrightarrow Y any morphism, then there exists an EE-Adams resolution YJ Y \to J_\bullet and a commuting diagram

X I f g Y J . \array{ X &\longrightarrow& I_\bullet \\ \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{g_\bullet}} \\ Y &\longrightarrow& J_\bullet } \,.

There are two EE-Adams resolutions that we will consider. Following (Hopkins 99) we call them the “normalized resolution” and the “standard resolution”. But beware that what all the traditional literature (Adams 74, Bousfield 79, Ravenel 86, …) considers (and somewhat implicitly) is the “normalized” resolution, not the standard resolution. The standard resolution is standard only from the more recent perspective of E-∞ geometry: it is the Amitsur complex of the 𝕊\mathbb{S}-algebra EE, exhibiting the formal dual of the Cech nerve of Spec(E)Spec(𝕊)Spec(E) \to Spec(\mathbb{S}).

Example

(normalized EE-Adams resolution)

Let E¯\overline{E} denote the homotopy fiber of the unit of the ring spectrum EE, fitting into a homotopy fiber sequence

E¯𝕊eEΣE¯. \overline{E} \longrightarrow \mathbb{S} \overset{e}{\longrightarrow} E \longrightarrow \Sigma \overline{E} \,.

For XX a spectrum, its normalized EE-Adams resolution is the top row of

X (e,id) EX E(ΣE¯)X E(ΣE¯)(ΣE¯)X (e,id) (e,id) (ΣE¯)X (ΣE¯)E¯X \array{ X &\overset{(e,id)}{\longrightarrow}& E \wedge X && \longrightarrow && E \wedge (\Sigma \overline{E}) \wedge X && \longrightarrow && E \wedge (\Sigma \overline{E}) \wedge (\Sigma \overline{E}) \wedge X && \longrightarrow && \cdots \\ && & \searrow && \nearrow_{\mathrlap{(e,id)}} && \searrow && \nearrow_{\mathrlap{(e,id)}} \\ && && (\Sigma \overline{E})\wedge X &&&& (\Sigma \overline{E}) \wedge \overline{E} \wedge X }

(e.g. Hopkins 99, corollary 5.3).

Remark

The notation for E¯\overline{E} in def. 6 follows (Bousfield 79, section 5). In (Hopkins 99) the same notation is used not for the homotopy fiber but for the homotopy cofiber. While our notation makes plenty of “Σ\Sigma”s appear in the above resolution, the advantage is that in the induced inverse sequence of a normalized resolution below in example 8 these all drop out and we are left with the original form of the expressions as considered by (Adams 74) and followed in most of the literature.

Example

(standard EE-Adams resolution)

Any ring spectrum EE gives rise to an augmented cosimplicial spectrum (its bar construction)

𝕊EEEEEE \mathbb{S} \longrightarrow E \stackrel{\longrightarrow}{\longrightarrow} E \wedge E \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} E \wedge E \wedge E \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}} \cdots

whose

  • coface maps are given by inserting the unit 𝕊eE\mathbb{S} \stackrel{e}{\to} E

    d i:E nE i𝕊E niideidE n+1; d^i \colon E^{\wedge^n} \simeq E^{\wedge^i} \wedge \mathbb{S} \wedge E^{\wedge^{n-i}} \stackrel{id \wedge e \wedge id}{\longrightarrow} E^{\wedge^{n+1}} \,;
  • codegeneracy maps are given by the product operation μ:EEE\mu \colon E \wedge E \to E

    s j:E n+1E iEEE n1iidμidE n. s^j \colon E^{\wedge^{n+1}} \simeq E^{\wedge^{i}} \wedge E \wedge E \wedge E^{\wedge n-1-i} \stackrel{id \wedge \mu \wedge id}{\longrightarrow} E^n \,.

The corresponding Amitsur complex is given by forming alternating sums of the coface maps

𝕊eEeidideEEeidideid+ideEEE. \mathbb{S} \stackrel{e}{\longrightarrow} E \stackrel{e\wedge id - id \wedge e}{\longrightarrow} E \wedge E \stackrel{e \wedge id - id\wedge e \wedge id + id \wedge e}{\longrightarrow} E \wedge E \wedge E \stackrel{}{\longrightarrow} \cdots \,.

Given any spectrum XX, then forming the smash product ()X(-)\wedge X with this sequence yields a sequence of the form

XEXEEXEEEX. X \stackrel{}{\longrightarrow} E \wedge X \stackrel{}{\longrightarrow} E \wedge E \wedge X \stackrel{}{\longrightarrow} E \wedge E \wedge E \wedge X \stackrel{}{\longrightarrow} \cdots \,.

This is called the standard EE-Adams resolution of XX.

(e.g. Hopkins 99, def. 5.4).

Proposition

The standard resolution of example 7 is indeed an EE-Adams resolution of XX in the sense of def. 4.

Proof

As generally for bar resolutions, one checks that the alternating sum of the codegeneracy maps constitute homotopy operators that give contracting homotopies when commuted with the alternating sum of the coface maps. This gives that the sequence is EE-exact. Moreover, the terms in the sequence are all EE-injective by lemma 2.

EE-Adams towers
Definition

An EE-Adams tower of a spectrum XX is a commuting diagram in the stable homotopy category of the form

p 2 X 2 κ 2 Ω 2I 3 p 1 X 1 κ 1 ΩI 2 p 0 X X 0=I 0 κ 0 I 1 \array{ && \vdots \\ && \downarrow^{\mathrlap{p_2}} \\ && X_2 &\stackrel{\kappa_2}{\longrightarrow}& \Omega^2 I_3 \\ &\nearrow& \downarrow^{\mathrlap{p_1}} \\ && X_1 &\stackrel{\kappa_1}{\longrightarrow}& \Omega I_2 \\ &\nearrow& \downarrow^{\mathrlap{p_0}} \\ X &\underset{}{\longrightarrow}& X_0 = I_0 &\stackrel{\kappa_0}{\longrightarrow}& I_1 }

such that

  1. each hook is a homotopy fiber sequence;

  2. the composition of the (ΣΩ)(\Sigma \dashv \Omega)-adjuncts of Σ p n1\Sigma_{p_{n-1}} with Σ nκ n\Sigma^n \kappa_n

    i n+1:I nΣp n1˜Σ nX nΣ nκ nI n+1 i_{n+1} \;\colon\; I_n \stackrel{\widetilde {\Sigma p_{n-1}}}{\longrightarrow} \Sigma^n X_n \stackrel{\Sigma^{n}\kappa_n}{\longrightarrow} I_{n+1}

    constitute an EE-Adams resolution of XX, def. 4:

    0Xi 0I 0i 1I 1. 0 \to X \stackrel{i_0}{\to} I_0 \stackrel{i_1}{\to} I_1 \stackrel{}{\to} \cdots \,.

Call this the associated EE-Adams resolution of the EE-Adams tower.

(Hopkins 99, def. 4.10)

The following is the main statement of the above little theory of EE-injective spectra.

Proposition

Every EE-Adams resolution XI X \to I_\bullet (def. 4) induces an EE-Adams tower, def. 5 of which it is the associated EE-Adams resolution.

Proof idea

Given an EE-Adams resolution

Xi 0I 0i 1I 1 X \overset{i_0}{\longrightarrow} I_0 \overset{i_1}{\longrightarrow} I_1 \longrightarrow \cdots

consider the induced diagram

C 1 C 3 ρ 1 σ 1 ρ 3 I 0 i 1 I 1 i 2 I 2 i 3 σ 0i 0 ρ 2 σ 2 C 0X C 2 \array{ && && C_1 && && && && C_3 \\ && & {}^{\mathllap{\rho_1}}\nearrow && \searrow^{\mathrlap{\sigma_1}} && && && {}^{\mathllap{\rho_3}}\nearrow \\ && I_0 && \underset{i_1}{\longrightarrow} && I_1 && \overset{i_2}{\longrightarrow} && I_2 && \underset{i_3}{\longrightarrow} & \cdots \\ & {}^{\mathllap{\sigma_0 \coloneqq i_0}}\nearrow && && && {}_{\mathllap{\rho_2}}\searrow && \nearrow_{\mathrlap{\sigma_2}} \\ C_0 \coloneqq X && && && && C_2 }

constructed inductively as follows:

To start with, ρ 1\rho_1 is the homotopy cofiber of i 0i_0, and σ 1\sigma_1 is the morphism universally induced from this by the fact that i 1i 00i_1 \circ i_0 \simeq 0, by lemma 3. Observe that σ 1\sigma_1 is an EE-monomorphism and ρ 1\rho_1 is an EE-epimorphism in the sense of def. 2.

Then assume that an EE-epi/mono factorization

i n:I n 1ρ nC nσ nI n i_n \colon I_{n_1} \overset{\rho_n}{\longrightarrow} C_n \overset{\sigma_n}{\to} I_n

has been constructed. Let now ρ n+1\rho_{n+1} be its homotopy cofiber. Since ρ n\rho_{n} is EE-epi, the equivalence 0i n+1i n=i n+1σ nρ n0 \simeq i_{n+1} \circ i_n = i_{n+1}\circ \sigma_n \circ \rho_n from lemma 3 implies that already i n+1σ n0i_{n+1} \circ \sigma_n \simeq 0. With this, the universal property of the homotopy cofiber induces a morphism σ n+1:C n+1I n+1\sigma_{n+1}\colon C_{n+1}\to I_{n+1}. As before, ρ n+1\rho_{n+1} is EE-epi and σ n+1\sigma_{n+1} is EE-mono, and so the induction proceeds.

Using this, we now construct an EE-Adams tower as follows (…).

There is another tower associated with an EE-Adams resolutions:

Definition

Given an EE-Adams resolutions XI X \to I_\bullet (def. 4), its associated inverse sequence is

X=C 0 γ 0 Σ 1C 1 γ 1 Σ 2C 2 I 0 Σ 1I 1 Σ2I 2 \array{ X = C_0 &\stackrel{\gamma_0}{\longleftarrow}& \Sigma^{-1} C_1 &\stackrel{\gamma_1}{\longleftarrow}& \Sigma^{-2} C_2 &\longleftarrow& \cdots \\ \downarrow && \downarrow && \downarrow \\ I_0 && \Sigma^{-1} I_1 && ^\Sigma^{-2} I_2 }

with the C iC_i as in the proof of prop. 2 and γ nΣ hofib(σ n)\gamma_n \coloneqq \Sigma^{-} hofib(\sigma_n).

Example

Let XI =(E(ΣE¯) 1E)X \to I_\bullet = (E \wedge (\Sigma \overline{E})^{\wedge^{\bullet-1}}\wedge E) be a normalized EE-Adams resolution according to example 6. Then its associated inverse sequence according to def. 6 is

X γ 0 E¯X γ 1 E¯E¯X EX Σ 1(E(ΣE¯)X) Σ 2(E(ΣE¯)(ΣE¯)X \array{ X &\stackrel{\gamma_0}{\longleftarrow}& \overline{E} \wedge X &\stackrel{\gamma_1}{\longleftarrow}& \overline{E} \wedge \overline{E} \wedge X &\longleftarrow& \cdots \\ \downarrow && \downarrow && \downarrow \\ E \wedge X && \Sigma^{-1}(E \wedge (\Sigma \overline{E}) \wedge X) && \Sigma^{-2}(E \wedge (\Sigma\overline{E}) \wedge (\Sigma \overline{E}) \wedge X }

hence

X γ 0 E¯X γ 1 E¯E¯X EX EE¯X EE¯E¯X. \array{ X &\stackrel{\gamma_0}{\longleftarrow}& \overline{E} \wedge X &\stackrel{\gamma_1}{\longleftarrow}& \overline{E} \wedge \overline{E} \wedge X &\longleftarrow& \cdots \\ \downarrow && \downarrow && \downarrow \\ E \wedge X && E \wedge \overline{E} \wedge X && E \wedge \overline{E} \wedge \overline{E} \wedge X } \,.

This is the tower of spectra considered in the original texts (Adams 74, p. 318) and (Bousfield 79, p. 271).

Remark

In (Ravenel 84, p. 356) it is the associated inverse sequence as in example 8 that is called the “Adams tower”, while in (Ravenel 86, def. 2.21) this is called an “EE-Adams resolution”. We instead follow (Hopkins 99) in using “EE-Adams resoltuon” for “EE-injective resolution” as in def. 4, “EE-Adams tower” for def. 5 and follow (Aramian) in saying “associated inverse sequence” for the above.

EE-Adams spectral sequences
Definition

Given spectra XX and YY, and given an EE-Adams resolution of XX, def. 4, or equivalently (by prop. 2) an EE-Adams tower over XX, def. 5,

p 2 X 2 κ 2 Ω 2I 3 p 1 X 1 κ 1 ΩI 2 p 0 X X 0=I 0 κ 0 I 1 \array{ && \vdots \\ && \downarrow^{\mathrlap{p_2}} \\ && X_2 &\stackrel{\kappa_2}{\longrightarrow}& \Omega^2 I_3 \\ &\nearrow& \downarrow^{\mathrlap{p_1}} \\ && X_1 &\stackrel{\kappa_1}{\longrightarrow}& \Omega I_2 \\ &\nearrow& \downarrow^{\mathrlap{p_0}} \\ X &\underset{}{\longrightarrow}& X_0 = I_0 &\stackrel{\kappa_0}{\longrightarrow}& I_1 }

then the corresponding EE-Adams spectral sequence for the mapping spectrum [Y,X][Y,X] is the associated spectral sequence of a tower of fibrations of the image of that tower of fibrations under the mapping spectrum operation [Y,][Y,-]:

[Y,p 2] [Y,X 2] [Y,κ 2] [Y,Ω 2I 3] [Y,p 1] [Y,X 1] [Y,κ 1] [Y,ΩI 2] [Y,p 0] [Y,X] [Y,X 0]=[Y,I 0] [Y,κ 0] [Y,I 1]. \array{ && \vdots \\ && \downarrow^{\mathrlap{[Y,p_2]}} \\ && [Y,X_2] &\stackrel{[Y,\kappa_2]}{\longrightarrow}& [Y,\Omega^2 I_3] \\ &\nearrow& \downarrow^{\mathrlap{[Y,p_1]}} \\ && [Y,X_1] &\stackrel{[Y,\kappa_1]}{\longrightarrow}& [Y,\Omega I_2] \\ &\nearrow& \downarrow^{\mathrlap{[Y,p_0]}} \\ [Y,X] &\underset{}{\longrightarrow}& [Y,X_0] = [Y,I_0] &\stackrel{[Y,\kappa_0]}{\longrightarrow}& [Y,I_1] } \,.

More in detail, the associated exact couple of the tower is

𝒟 p 𝒟 κ \array{ \mathcal{D} && \stackrel{p}{\longrightarrow} && \mathcal{D} \\ & {}_{\mathllap{\partial}}\nwarrow && \swarrow_{\mathrlap{\kappa}} \\ && \mathcal{E} }

with

𝒟 s,t𝒟 s,t s,tπ ts([Y,X s]) \mathcal{D} \coloneqq \oplus_{s,t} \mathcal{D}^{s,t} \coloneqq \oplus_{s,t} \pi_{t-s}([Y,X_s])
s,t s+1,t s,tπ ts([Y,Ω sI s+1]) \mathcal{E} \coloneqq \oplus_{s,t} \mathcal{E}^{s+1,t} \coloneqq \oplus_{s,t} \pi_{t-s}([Y,\Omega^s I_{s+1}])

and

p:π ts([Y,X s+1])π ts([Y,p s])X ts([Y,X s]) p \colon \pi_{t-s}([Y,X_{s+1}]) \stackrel{\pi_{t-s}([Y,p_s])}{\longrightarrow} X_{t-s}([Y,X_s])
κ:π ts([Y,X s])π ts([Y,κ s])π ts([Y,Ω sI s+1]) \kappa \colon \pi_{t-s}([Y,X_s]) \stackrel{\pi_{t-s}([Y,\kappa_s])}{\longrightarrow} \pi_{t-s}([Y,\Omega^s I_{s+1}])
:π ts([Y,Ω sI s+1])π ts([Y, s])π ts([Y,ΣX s+1]). \partial \colon \pi_{t-s}([Y,\Omega^s I_{s+1}]) \stackrel{\pi_{t-s}([Y,\partial_s])}{\longrightarrow} \pi_{t-s}([Y,\Sigma X_{s+1}]) \,.

The EE-Adams spectral sequence of the EE-Adams tower is the spectral sequence induced by this exact couple.

Proposition

Given two EE-Adams towers, def. 5, for some XX, then the corresponding two EE-Adams spectral sequences, def. 7, are isomorphic from the 2\mathcal{E}_2-page on.

Proposition

Given an EE-Adams resolution (def. 4), there is an isomorphism of spectral sequences between

  1. the tower spectral sequence of its associated EE-Adams tower (def. 5), i.e. the spectral sequence of def. 7;

  2. the tower spectral sequence of its associated inverse sequence (def. 6).

Remark

Hence both of these construction are to be called the EE-Adams spectral sequence. It is in fact the second construction – for the case of the normalized resolution as in example 8 – that is considered in the original sources (Adams 74, p. 318, Bousfield 79, p. 271). But it is the first construction that relates to the totalization tower of the cosimplicial spectrum E XE^{\wedge^\bullet} \wedge X.

The first page

Due to prop. 3, for understanding the 2\mathcal{E}_2-page of any EE-Adams spectral sequence, def. 7, it is sufficient to understand the 1\mathcal{E}_1-page of the EE-Adams spectral sequence that is induced by the standard EE-resolution of example 7. By construction, that page is

1 s,π ([Y,E (s+1)X]) \mathcal{E}_1^{s,\bullet} \simeq \pi_\bullet(\;[Y,E^{\wedge (s+1)}\wedge X] \;)

with the differentials being the image under π \pi_\bullet of the alternating sum of the morphisms that insert unit elements.

We discuss now how, under favorable conditions, these homotopy groups of mapping spectra of the form [Y,E (s+1)X][Y,E^{\wedge (s+1)}\wedge X] may alternatively be computed as morphisms of EE-homology equipped with suitable comodule structure over a Hopf algebroid structure on the dual EE-Steenrod operations E (E)E_\bullet(E). Then below we discuss that, as a result, the d 1d_1-homology of the 1\mathcal{E}_1-page is seen to compute the Ext-groups from the EE-homology of YY to the EE-homology of XX, regarded as E (E)E_\bullet(E)-comodules. This re-formulation of the 2\mathcal{E}_2-page is the one that makes it be useful for computations.

The first condition needed for this to work is the following.

Definition

Call the commutative ring spectrum EE flat if one, equivalently both, of the morphisms

η Lπ (eid):E E (E) \eta_L \coloneqq \pi_\bullet(e \wedge id) \;\colon\; E_\bullet \longrightarrow E_\bullet(E)
η rπ (ide):E E (E) \eta_r \;\coloneqq\; \pi_\bullet(id \wedge e) \colon E_\bullet \longrightarrow E_\bullet(E)

is a flat morphism.

Example

Examples of commutative ring spectra that are flat according to def. 8 include E=E =

Example

Examples of ring spectra that are not flat in the sense of def. 8 include HZ, and MSUM S U.

The key consequence of the assumption that EE is flat in the sense of def. 8 is the following.

Proposition

If EE is flat, def. 8, then for all spectra XX there is a natural isomorphisms

E (E) π (E)E (X)π (EEX) E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(X) \stackrel{}{\longrightarrow} \pi_\bullet(E \wedge E \wedge X)

and hence for all nn \in \mathbb{N} there are isomorphisms

π (E (n+2)X)E (E) π (E) π (E)E (E)n+1factors π (E)E (X). \pi_\bullet(E^{\wedge^{(n+2)}}\wedge X ) \simeq \underset{n+1\,factors}{ \underbrace{E_\bullet(E) \otimes_{\pi_\bullet(E)} \cdots \otimes_{\pi_\bullet(E)} E_\bullet(E) }} \otimes_{\pi_\bullet(E)} E_\bullet(X) \,.

(e.g. Adams 74, part III, lemma 12.5, Schwede 12, prop. 6.20)

Proof

The desired natural homomorphism

E (E) π (E)E (X)π (EEX) E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(X) \longrightarrow \pi_\bullet(E \wedge E \wedge X)

is given on [α]π (EE)[\alpha] \in \pi_\bullet(E \wedge E) and [β]π (EX)[\beta] \in \pi_\bullet(E \wedge X) by ([α,β])[(idμid)(αβ)]([\alpha, \beta])\mapsto [(id \wedge \mu \wedge id) \circ (\alpha \wedge \beta)].

To see that this is an isomorphism, observe that by flatness of EE, the assignment XE (E) π (E)E ()X \mapsto E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(-) is a generalized homology functor, hence represented by some spectrum. The above morphism, natural in XX, thus constitutes a homomorphism of generalized homology theories. By the Whitehead theorem for generalized homology for this to be an isomorphism it is sufficient to check that it induces isomorphisms on the point. This is manifestly the case.

Finally we get the claimed isomorphisms for all nn by induction:

π (E n+2X) π (EEE nX)) E (E) π (E)E (E nX) =E (E) π (E)π (E n+1X). \begin{aligned} \pi_\bullet(E^{\wedge^{n+2}} \wedge X) & \simeq \pi_\bullet(E \wedge E \wedge E^{\wedge^n} \wedge X)) \\ &\simeq E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet( E^{\wedge^n} \wedge X ) \\ & = E_\bullet(E) \otimes_{\pi_\bullet(E)} \pi_\bullet( E^{\wedge^{n+1}} \wedge X ) \end{aligned} \,.
Commutative Hopf algebroids
Definition

A commutative Hopf algebroid is an internal groupoid in the opposite category CRing op{}^{op} of commutative rings, regarded with its cartesian monoidal category structure.

(e.g. Ravenel 86, def. A1.1.1)

Remark

We unwind def. 9. For RCRingR \in CRing, write Spec(R)Spec(R) for same same object, but regarded as an object in CRing opCRing^{op}.

An internal category in CRing opCRing^{op} is a diagram in CRing opCRing^{op} of the form

Spec(Γ)×Spec(A)Spec(Γ) Spec(Γ) s i t Spec(A), \array{ Spec(\Gamma) \underset{Spec(A)}{\times} Spec(\Gamma) \\ \downarrow^{\mathrlap{\circ}} \\ Spec(\Gamma) \\ {}^{\mathllap{s}}\downarrow \; \uparrow^{\mathrlap{i}} \downarrow^{\mathrlap{t}} \\ Spec(A) } \,,

(where the fiber product at the top is over ss on the left and tt on the right) such that the pairing \circ defines an associative composition over Spec(A)Spec(A), unital with respect to ii. This is an internal groupoid if it is furthemore equipped with a morphism

inv:Spec(Γ)Spec(Γ) inv \;\colon\; Spec(\Gamma) \longrightarrow Spec(\Gamma)

acting as assigning inverses with respect to \circ.

The key basic fact to use now is that tensor product of commutative rings exhibits the cartesian monoidal category structure on CRing opCRing^{op}, see at CRing – Properties – Cocartesian comonoidal structure:

Spec(R 1)×Spec(R 3)Spec(R 2)Spec(R 1 R 3R 2). Spec(R_1) \underset{Spec(R_3)}{\times} Spec(R_2) \simeq Spec(R_1 \otimes_{R_3} R_2) \,.

This means that the above is equivalently a diagram in CRing of the form

ΓAΓ Ψ Γ η L ϵ η R A \array{ \Gamma \underset{A}{\otimes} \Gamma \\ \uparrow^{\mathrlap{\Psi}} \\ \Gamma \\ {}^{\mathllap{\eta_L}}\uparrow \downarrow^{\mathrlap{\epsilon}} \; \uparrow^{\mathrlap{\eta_R}} \\ A }

as well as

c:ΓΓ c \; \colon \; \Gamma \longrightarrow \Gamma

and satisfying formally dual conditions, spelled out as def. 10 below. Here

  • η L,etaR\eta_L, \etaR are called the left and right unit maps;

  • ϵ\epsilon is called the co-unit;

  • Ψ\Psi is called the comultiplication;

  • cc is called the antipode or conjugation

Remark

Generally, in a commutative Hopf algebroid, def. 9, the two morphisms η L,η R:AΓ\eta_L, \eta_R\colon A \to \Gamma from remark 5 need not coincide, they make Γ\Gamma genuinely into a bimodule over AA, and it is the tensor product of bimodules that appears in remark 5. But it may happen that they coincide:

An internal groupoid 𝒢 1ts\mathcal{G}_1 \stackrel{\overset{s}{\longrightarrow}}{\underset{t}{\longrightarrow}} for which the domain and codomain morphisms coincide, s=ts = t, is euqivalently a group object in the slice category over 𝒢 0\mathcal{G}_0.

Dually, a commutative Hopf algebroid Γη Rη LA\Gamma \stackrel{\overset{\eta_L}{\longleftarrow}}{\underset{\eta_R}{\longleftarrow}} A for which η L\eta_L and η R\eta_R happen to coincide is equivalently a commutative Hopf algebra Γ\Gamma over AA.

Writing out the formally dual axioms of an internal groupoid as in remark 5 yields the following equivalent but maybe more explicit definition of commutative Hopf algebroids, def. 9

Definition

A commutative Hopf algebroid is

  1. two commutative rings, AA and Γ\Gamma;

  2. ring homomorphisms

    1. (left/right unit)

      η L,η R:AΓ\eta_L,\eta_R \colon A \longrightarrow \Gamma;

    2. (comultiplication)

      Ψ:ΓΓAΓ\Psi \colon \Gamma \longrightarrow \Gamma \underset{A}{\otimes} \Gamma;

    3. (counit)

      ϵ:ΓA\epsilon \colon \Gamma \longrightarrow A;

    4. (conjugation)

      c:ΓΓc \colon \Gamma \longrightarrow \Gamma

such that

  1. (co-unitality)

    1. ϵη L=ϵη R=id A\epsilon \circ \eta_L = \epsilon \circ \eta_R = id_A;

    2. (id Γ Aϵ)Δ=(ϵ Aid Γ)Δ=id Γ(id_\Gamma\otimes_A\epsilon) \circ \Delta = (\epsilon \otimes_A id_\Gamma) \circ \Delta = id_\Gamma;

  2. (co-associativity) (id Γ AΨ)Ψ=(Ψ Aid Γ)Ψ(id_\Gamma \otimes_A \Psi) \circ \Psi = (\Psi \otimes_A id_\Gamma) \circ \Psi;

  3. (inverses)

    1. cc=id Γc \circ c = id_\Gamma;

    2. cη L=η Rc\circ \eta_L = \eta_R; cη R=η Lc \circ \eta_R = \eta_L;

    3. the universally induced c:Γ AΓΓ\nabla_c \colon \Gamma \otimes_A \Gamma \longrightarrow \Gamma satifies

      cΨ=ϵη L=ϵη R\nabla_c \circ \Psi = \epsilon \circ \eta_L = \epsilon \circ \eta_R.

Comodules and cotensor product
Definition

Given a commutative Hopf algebroid Γ\Gamma over AA as in def. 10, hence an internal groupoid in CRing opCRing^{op}, then a comodule ring over it is an action in CRing opCRing^{op} of that internal groupoid.

In the same spirit, a comodule over a commutative Hopf algebroid (not necessarily a comodule ring) is a quasicoherent sheaf on the corresponding internal groupoid (regarded as a (algebraic) stack) (e.g. Hopkins 99, prop. 11.6). Explicitly in components:

Definition

Given a commutative Hopf algebroid Γ\Gamma over AA, def. 10, then a left comodule over Γ\Gamma is

  1. an AA-module NN;

  2. an AA-module homomorphism (co-action)

    Ψ N:NΓ AN\Psi_N \;\colon\; N \longrightarrow \Gamma \otimes_A N;

such that

  1. (unitality)

    (ϵ Aid N)Ψ N=id N(\epsilon \otimes_A id_N) \circ \Psi_N = id_N;

  2. (associativity)

    (Ψ Aid N)Ψ N=(id Γ AΨ N)Ψ N(\Psi \otimes_A id_N) \circ \Psi_N = (id_\Gamma \otimes_A \Psi_N)\circ \Psi_N.

A homomorphism between comodules N 1N 2N_1 \to N_2 is a homomorphism of underlying AA-modules making commuting diagrams with the co-action morphism. Write

ΓCoMod \Gamma CoMod

for the resulting category of left comodules over Γ\Gamma. Analogously for right comodules.

Example

Given a commutative Hopf algebroid Γ\Gamma over AA, def. 10, then AA itself becomes a left Γ\Gamma-comodule (def. 12) with coaction given by

Ψ A:Aη LΓΓ AA \Psi_A \;\colon\; A \overset{\eta_L}{\longrightarrow} \Gamma \simeq \Gamma \otimes_A A

and a right Γ\Gamma-comodule with coaction given by

Ψ A:Aη RΓΓ AA. \Psi_A \;\colon\; A \overset{\eta_R}{\longrightarrow} \Gamma \simeq \Gamma \otimes_A A \,.
Proposition

Given a commutative Hopf algebroid Γ\Gamma over AA, there is a free-forgetful adjunction

ΓCoModBotforgetcofreeAMod \Gamma CoMod \underoverset {\underset{forget}{\longrightarrow}} {\overset{co-free}{\longleftarrow}} {\Bot} A Mod

between the category of Γ\Gamma-comodules, def. 12 and the category of modules over AA, where the cofree functor is right adjoint.

The co-free Γ\Gamma-comodule on an AA-module NN is Γ AN\Gamma \otimes_A N equipped with the coaction induced by the comultiplication Ψ\Psi in Γ\Gamma.

Proposition

Consider a commutative Hopf algebroid Γ\Gamma over AA, def. 10. Any left comodule NN over Γ\Gamma (def. 12) becomes a right comodule via the coaction

NΨΓ ANN AΓid AcN AΓ, N \overset{\Psi}{\longrightarrow} \Gamma \otimes_A N \overset{\simeq}{\longrightarrow} N \otimes_A \Gamma \overset{id \otimes_A c}{\longrightarrow} N \otimes_A \Gamma \,,

where the isomorphism in the middle the is braiding in AModA Mod and where cc is the conjugation map of Γ\Gamma.

Dually, a right comodule NN becoomes a left comodule with the coaction

NΨN AΓΓ ANc AidΓ AN. N \overset{\Psi}{\longrightarrow} N \otimes_A \Gamma \overset{\simeq}{\longrightarrow} \Gamma \otimes_A N \overset{c \otimes_A id}{\longrightarrow} \Gamma \otimes_A N \,.
Definition

Given a commutative Hopf algebroid Γ\Gamma over AA, def. 10, and given N 1N_1 a right Γ\Gamma-comodule and N 2N_2 a left comodule (def. 12), then their cotensor product N 1 ΓN 2N_1 \Box_\Gamma N_2 is the kernel of the difference of the two coaction morphisms:

N 1 ΓN 2ker(N 1 AN 2Ψ N 1 Aidid AΨ N 2). N_1 \Box_\Gamma N_2 \;\coloneqq\; ker \left( N_1 \otimes_A N_2 \overset{\Psi_{N_1}\otimes_{A} id - id \otimes_A \Psi_{N_2} }{\longrightarrow} \right) \,.

If both N 1N_1 and N 2N_2 are left comodules, then their cotensor product is the cotensor product of N 2N_2 with N 1N_1 regarded as a right comodule via prop. 7.

e.g. (Ravenel 86, def. A1.1.4).

Example

Given a commutative Hopf algebroid Γ\Gamma over AA, (def.), and given NN a left Γ\Gamma-comodule (def.). Regard AA itself canonically as a right Γ\Gamma-comodule via example 11. Then the cotensor product

Prim(N)A ΓN Prim(N) \coloneqq A \Box_\Gamma N

is called the primitive elements of NN:

Prim(N)={nN|Ψ N(n)=1n}. Prim(N) = \{ n \in N \;\vert\; \Psi_N(n) = 1 \otimes n \} \,.
Proposition

Given a commutative Hopf algebroid Γ\Gamma over AA, def. 10, and given N 1,N 2N_1, N_2 two left Γ\Gamma-comodules (def. 12), then their cotensor product (def. 13) is commutative, in that there is an isomorphism

N 1N 2N 2N 1. N_1 \Box N_2 \;\simeq\; N_2 \Box N_1 \,.

(e.g. Ravenel 86, prop. A1.1.5)

Lemma

Given a commutative Hopf algebroid Γ\Gamma over AA, def. 10, and given N 1,N 2N_1, N_2 two left Γ\Gamma-comodules (def. 12), such that N 1N_1 is projective as an AA-module, then

  1. The morphism

    Hom A(N 1,A)f(id Af)Ψ N 1Hom A(N 1,Γ AA)Hom A(N 1,Γ)Hom A(N 1,A) AΓ Hom_A(N_1, A) \overset{f \mapsto (id \otimes_A f) \circ \Psi_{N_1}}{\longrightarrow} Hom_A(N_1, \Gamma \otimes_A A) \simeq Hom_A(N_1, \Gamma) \simeq Hom_A(N_1, A) \otimes_A \Gamma

    gives Hom A(N 1,A)Hom_A(N_1,A) the structure of a right Γ\Gamma-comodule;

  2. The cotensor product (def. 13) with respect to this right comodule structure is isomorphic to the hom of Γ\Gamma-comodules:

    Hom A(N 1,A) ΓN 2Hom Γ(N 1,N 2). Hom_A(N_1, A) \Box_\Gamma N_2 \simeq Hom_\Gamma(N_1, N_2) \,.

    Hence in particular

    A ΓN 2Hom Γ(A,N 2) A \Box_\Gamma N_2 \;\simeq\; Hom_\Gamma(A,N_2)

(e.g. Ravenel 86, lemma A1.1.6)

Remark

In computing the second page of EE-Adams spectral sequences, the second statement in lemma 5 is the key translation that makes the comodule Ext-groups on the page be equivalent to a Cotor-groups. The latter lend themselves to computation, for instance via Lambda-algebra or via the May spectral sequence.

The Hopf algebroid of dual EE-Steenrod operations

Now we identify the commutative Hopf algebroids arising in the EE-Adams spectral sequence:

Proposition

If EE is flat according to def. 8, then, via the isomorphism of proposition 5, the cosimplicial spectrum E XE^{\wedge^\bullet} \wedge X (the EE-standard resolution of XX from example 7) exhibits:

  1. for X=EX = E: Hopf algebroid-structure, def. 9, remark 5, on E (E)E_\bullet(E) over π (E)\pi_\bullet(E) – called the dual EE-Steenrod algebra;

  2. for general XX: comodule-structure on E (X)E_\bullet(X) over the dual EE-Steenrod algebra.

(e.g. Baker-Lazarev 01, theorem 1.1)

Proof

Via prop. 5, the image under π ()\pi_\bullet(-) of the cosimplicial spectrum E (E)E^{\wedge^\bullet}(E) is identified as on the right of the following diagram

π (EEE) E (E) π (E)E (E) π (ideid) Ψ π (EE) = E (E) π (eid) π (μ) π (ide) η L ϵ η R π (E) = π (E). \array{ \pi_\bullet(E\wedge E \wedge E) &\simeq& E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(E) \\ \uparrow^{\mathrlap{\pi_\bullet(id \wedge e \wedge id)}} && \uparrow^{\mathrlap{\Psi}} \\ \pi_\bullet(E \wedge E) &=& E_\bullet(E) \\ {}^{\mathllap{\pi_\bullet(e \wedge id)}}\uparrow \downarrow^{\mathrlap{\pi_\bullet(\mu)}} \;\;\;\;\;\; \uparrow^{\mathrlap{\pi_\bullet(id \wedge e)}} && {}^{\mathllap{\eta_L}}\uparrow \downarrow^{\mathrlap{\epsilon}} \uparrow^{\mathrlap{\eta_R}} \\ \pi_\bullet(E) &=& \pi_\bullet(E) } \,.

Analogously the coaction is induced as on the right of the following diagram

π (EEX) E (E) π (E)E (X) π (ideid) Ψ X π (EX) = E (X). \array{ \pi_\bullet(E\wedge E \wedge X) &\simeq& E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(X) \\ \uparrow^{\mathrlap{\pi_\bullet(id \wedge e \wedge id)}} && \uparrow^{\mathrlap{\Psi_X}} \\ \pi_\bullet(E \wedge X) &=& E_\bullet(X) } \,.
Example

Examples of commutative ring spectra EE for which the dual EE-Steenrod algebra E (E)E_\bullet(E) over π (E)\pi_\bullet(E) of corollary 9 happens to be a commutative Hopf algebra over π (E)\pi_\bullet(E) instead of a more general commutative Hopf algebroid, according to remark 6, includes the cases

E=E =

  • H𝔽 p\mathbb{F}_p,

The key use of the Hopf coalgebroid structure of prop. 9 for the present purpose is that it is extra structure inherited from maps of spectra under smashing with EE:

Example

For Y,NY,N any two spectra, the morphism (of \mathbb{Z}-graded abelian groups) given by smash product with EE

π (E):π ([Y,N])Hom Ab (E (Y),E (N)) \pi_\bullet(E \wedge -) \;\colon\; \pi_\bullet([Y,N]) \longrightarrow Hom^\bullet_{Ab}(E_\bullet(Y), E_\bullet(N))

factors through E (E)E_\bullet(E)-comodule homomorphisms over the dual EE-Steenrod algebra:

π (E):π ([Y,N])Hom E (E) (E (Y),E (N))Hom Ab (E (Y),E (N)). \pi_\bullet(E \wedge -) \;\colon\; \pi_\bullet([Y,N]) \longrightarrow Hom^\bullet_{E_\bullet(E)}(E_\bullet(Y), E_\bullet(N)) \longrightarrow Hom^\bullet_{Ab}(E_\bullet(Y), E_\bullet(N)) \,.

In order to put all this together, we need to invoke a universal coefficient theorem in the following form.

Proposition

If EE is among the examples S, HR for R=𝔽 pR = \mathbb{F}_p, MO, MU, MSp, KO, KU, then for all EE-module spectra NN with action ρ:ENN\rho \colon E\wedge N \to N the morphism of \mathbb{Z}-graded abelian groups

π [Y,N]ϕρ(idϕ)Hom π (E) (E (Y),π N) \pi_\bullet[Y,N] \stackrel{\phi \mapsto \rho \circ (id\wedge \phi)}{\longrightarrow} Hom_{\pi_\bullet(E)}^\bullet(E_\bullet(Y), \pi_\bullet N)_\bullet

(from the stable homotopy group of the mapping spectrum to the hom groups of π (E)\pi_\bullet(E)-modules)

is an isomorphism.

This is the universal coefficient theorem of (Adams 74, chapter III, prop. 13.5), see also (Schwede 12, chapter II, prop. 6.20), and see at Kronecker pairing – Universal coefficient theorem.

With this we finally get the following statement, which serves to identity maps of certain spectra with their induced maps on EE-homology:

Proposition

If the assumptions of prop. 10 hold, then for X,NX,N any two spectra, the morphism of \mathbb{Z}-graded abelian groups from example 14 of the form

π (E()):π [Y,EN]Hom E (E) (E (Y),E (Y))) \pi_\bullet(E\wedge (-)) \;\colon\; \pi_\bullet[Y, E\wedge N] \stackrel{}{\longrightarrow} Hom_{E_\bullet(E)}^\bullet(E_\bullet(Y), E_\bullet(Y)))

is an isomorphism.

(Adams 74, part III, page 323)

Proof

By the general formula for expressing adjuncts, the morphism fits into the following commuting diagram

[Y,EN] π (E()) Hom E (E)(E (Y),E (EN)) ϕμ(idϕ) Hom π (E)(E (Y),E (N)) Hom E (E)(E (Y),E (E) π (E)E (E)), \array{ [Y, E \wedge N] &\stackrel{\pi_\bullet(E\wedge(-))}{\longrightarrow}& Hom_{E_\bullet(E)}( E_\bullet(Y), E_\bullet(E \wedge N) ) \\ {}^{\mathllap{{\phi \mapsto} \atop {\mu \circ (id \wedge \phi)}}} \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ Hom_{\pi_\bullet(E)}(E_\bullet(Y), E_\bullet(N)) &\stackrel{\simeq}{\longleftarrow}& Hom_{E_\bullet(E)}( E_\bullet(Y), E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(E) ) } \,,

where

  1. the right vertical map comes from the isomorphism of prop. 5;

  2. the bottom isomorphism is the cofree/forgetful adjunction isomorphism of prop. 6;

  3. the the left vertical morphism is an isomorphism by prop. 10.

Therefore also the top morphism is an iso.

In conclusion:

Proposition

For X,YX, Y spectra, and for EE a commutative ring spectrum from the list in example 9, then the 1\mathcal{E}_1-page of the EE-Adams spectral sequence, def. 7, for [Y,X][Y,X], induced by the standard EE-Adams resolution for XX from example 7, is of the form

0Hom E (E) (E (Y),E (X))d 1Hom E (E) (E (Y),E (E) π (E)E (X))d 1Hom E (E) (E (Y),E (E) π (E)E (E) π (E)E (X))d 1. 0 \to Hom_{E_\bullet(E)}^\bullet(E_\bullet(Y),E_\bullet(X)) \stackrel{d_1}{\longrightarrow} Hom_{E_\bullet(E)}^\bullet( E_\bullet(Y), E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(X) ) \stackrel{d_1}{\longrightarrow} Hom_{E_\bullet(E)}^\bullet( E_\bullet(Y), E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(X) ) \stackrel{d_1}{\longrightarrow} \cdots \,.

The next step is to identify the chain homology of this d 1d_1 with the comodule Ext-groups.

The second page

Theorem

If EE is flat, def. 8, and satisfies the conditions of prop. 10, and E (Y)E_\bullet(Y) a projective module over π (E)\pi_\bullet(E), then the entries of the 2\mathcal{E}_2-page of any EE-Adams spectral sequence, def. 7, for [Y,X][Y,X] are the Ext-groups of commutative Hopf algebroid-comodules for the commutative Hopf algebroid structure on EE-operations E (E)E_\bullet(E) from prop. 9:

2 s,tExt E (E) s,t(E (Y),E (X)). \mathcal{E}^{s,t}_2 \simeq Ext^{s,t}_{E_\bullet(E)}(E_\bullet(Y), E_\bullet(X)) \,.

In the special case that Y=𝕊Y = \mathbb{S}, then (by prop. 5) these are equivalently Cotor-groups

2 s,tCotor E (E) s,t(π (E),E (X)). \mathcal{E}^{s,t}_2 \simeq Cotor^{s,t}_{E_\bullet(E)}(\pi_\bullet(E), E_\bullet(X)) \,.
Proof

By prop. 3 it is sufficient to show this for the standard EE-Adams resolution of prop. 1. For that case the 1\mathcal{E}_1 page is given by prop. 12, and so by the standard theory of derived functors in homological algebra (see the section Via acyclic resolutions), it is now sufficient to see that:

  1. the category E (E)CoModE_\bullet(E) CoMod is an abelian category;

  2. the graded chain complex of prop. 12 is the image under the hom-functor FHom E (E)(E (Y),)F \coloneqq Hom_{E_\bullet(E)}(E_\bullet(Y),-) of an FF-acyclic resolution of E (X)E_\bullet(X).

These two statements are prop. 13 and prop. 15 below.

Homological co-algebra

We now discuss the relevant general aspects of homological algebra in categories of comodules over commutative Hopf algebroids needed for the proof of theorem 1 from prop. 12.

Proposition

If a commutative Hopf algebroid Γ\Gamma over AA, def. 9, 10 is such that η L,η R:AΓ\eta_L, \eta_R \colon A \longrightarrow \Gamma is a flat morphism, then the category ΓCoMod\Gamma CoMod of comodules over Γ\Gamma, def. 12, is an abelian category.

(e.g. Ravenel 86, theorem A1.1.3)

Proof

It is clear that, without any condition the Hopf algebroid, ΓCoMod\Gamma CoMod is an additive category.

We need to show that with the assumption that Γ\Gamma is flat over AA, then this is also a pre-abelian category in that kernels and cokernels exist. Let f:(N 1,Ψ N 1)(N 2,Ψ N 2)f \colon (N_1,\Psi_{N_1}) \longrightarrow (N_2,\Psi_{N_2}) be a morphism of comodules, hence a commuting diagram in AAMod of the form

N 1 f N 2 Ψ N 1 Ψ N 2 Γ AN 1 id Γ Af Γ AN 2. \array{ N_1 &\stackrel{f}{\longrightarrow}& N_2 \\ \downarrow^{\mathrlap{\Psi_{N_1}}} && \downarrow^{\mathrlap{\Psi_{N_2}}} \\ \Gamma \otimes_A N_1 &\stackrel{id_\Gamma \otimes_A f}{\longrightarrow}& \Gamma \otimes_A N_2 } \,.

Consider the kernel ker(f)ker(f) of ff in AAMod and its image under Γ A()\Gamma \otimes_A (-)

ker(f) N 1 f N 2 Ψ N 1 Ψ N 2 Γ Aker(f) Γ AN 1 id Γ Af Γ AN 2. \array{ ker(f) &\longrightarrow& N_1 &\stackrel{f}{\longrightarrow}& N_2 \\ \downarrow && \downarrow^{\mathrlap{\Psi_{N_1}}} && \downarrow^{\mathrlap{\Psi_{N_2}}} \\ \Gamma \otimes_A ker(f) &\longrightarrow& \Gamma \otimes_A N_1 &\stackrel{id_\Gamma \otimes_A f}{\longrightarrow}& \Gamma \otimes_A N_2 } \,.

By the assumption that Γ\Gamma is a flat module over AA, also Γ Aker(f)ker(Γ Af)\Gamma \otimes_A ker(f) \simeq ker(\Gamma \otimes_A f) is a kernel. By its universal property this induces uniquely a morphism as shown on the left, making the above diagram commute. This means that the AA-module ker(f)ker(f) uniquely inherits the structure of a Γ\Gamma-comodule such as to make ker(f)N 1ker(f) \to N_1 a comodule homomorphism. By the same universal property it follows that ker(f)ker(f) with this comodule structure is in fact the kernel of ff in ΓCoMod\Gamma CoMod.

The argument for the existence of cokernels proceeds formally dually. Therefore it follows that the comparison morphism

coker(ker(f))ker(coker(f)) coker(ker(f)) \longrightarrow ker(coker(f))

formed in ΓCoMod\Gamma CoMod has underlying it the corresponding comparison morphism in AModA Mod. There this is an isomorphism, hence it is an isomorphism also in ΓCoMod\Gamma CoMod, and so the latter is not just a pre-abelian category but in fact an abelian category itself.

Proposition

If a commutative Hopf algebroid Γ\Gamma over AA, def. 9, 10 is such that η L,η R:AΓ\eta_L, \eta_R \colon A \longrightarrow \Gamma is a flat morphism, then

  1. every co-free Γ\Gamma-comodule, def. 6, on an injective module over AA is an injective object in ΓCoMod\Gamma CoMod;

  2. ΓCoMod\Gamma CoMod has enough injectives (if the axiom of choice holds in the ambient set theory).

(e.g. Ravenel 86, lemma A1.2.2)

Proof

First of all, assuming the axiom of choice, then the category of modules AModA Mod has enough injectives (see this proposition). Now by prop. 6 we have the adjunction

ΓCoModforgetcofreeAMod. \Gamma CoMod \underoverset {\underset{forget}{\longrightarrow}} {\overset{co-free}{\longleftarrow}} {\bot} A Mod \,.

Observe that the left adjoint is a faithful functor (being a forgetful functor) and that, by the proof of prop. 13, it is an exact functor. With this a standard lemma applies (here) which says that

  1. with IAModI \in A Mod an injective module, then the co-free comodule Γ AI\Gamma \otimes_A I is an injective object in ΓCoMod\Gamma CoMod;

  2. for NΓCoModN \in \Gamma CoMod any object, and for i:U(N)Ii \colon U(N) \hookrightarrow I a monomorphism of AA-modules into an injective AA-module, then the adjunct i˜:NΓ AI\tilde i \colon N \hookrightarrow \Gamma\otimes_A I is a monomorphism in ΓCoMod\Gamma CoMod (and into an injective comodule).

Proposition

Let Γ\Gamma be a commutative Hopf algebroid over AA, def. 9, 10, such that η L,η R:AΓ\eta_L, \eta_R \colon A \longrightarrow \Gamma is a flat morphism, Let NΓCoModN \in \Gamma CoMod be a Hopf comodule, def. 12, such that the underlying AA-module is a projective module (a projective object in AAMod).

Then (assuming the axiom of choice) every co-free comodule, prop. 6, is an FF-acyclic object for FF the hom functor Hom ΓCoMod(N,)Hom_{\Gamma CoMod}(N,-).

Proof

We need to show that the derived functors R Hom Γ(N,)R^{\bullet} Hom_{\Gamma}(N,-) vanish in positive degree on all co-free comodules, hence on Γ AK\Gamma \otimes_A K, for KAModK \in A Mod.

To that end, let I I^\bullet be an injective resolution of KK in AModA Mod. By prop. 14 then Γ AI \Gamma \otimes_A I^\bullet is a sequence of injective objects in ΓCoMod\Gamma CoMod and by the assumption that Γ\Gamma is flat over AA it is an injective resolution of Γ AK\Gamma \otimes_A K in ΓCoMod\Gamma CoMod. Therefore the derived functor in question is given by

R 1Hom Γ(N,Γ AK) H 1(Hom Γ(N,Γ AI )) H 1(Hom A(N,I )) 0. \begin{aligned} R^{\bullet \geq 1} Hom_\Gamma(N, \Gamma \otimes_A K) & \simeq H_{\bullet \geq 1}( Hom_\Gamma( N, \Gamma \otimes_A I^\bullet ) ) \\ & \simeq H_{\bullet \geq 1}( Hom_A(N, I^\bullet) ) \\ & \simeq 0 \end{aligned} \,.

Here the second equivalence is the cofree/forgetful adjunction isomorphism of prop. 6, while the last equality then follows from the assumption that the AA-module underlying NN is a projective module (since hom functors out of projective objects are exact functors (here) and since derived functors of exact functors vanish in positive degree (here)).

With prop. 15 the proof of theorem 1 is completed.

Convergence

Definition

For XX a spectrum and EE a ring spectrum, consider the inverse sequence

E¯ X=(XE¯XE¯E¯X) \overline{E}^{\wedge^\bullet} \wedge X = \left( X \leftarrow \overline{E}\wedge X \leftarrow \overline{E} \wedge \overline{E} \wedge X \leftarrow \cdots \right)

associated to the normalized EE-Adams resolution of XX, as in example 8. The E-nilpotent completion E ^XE^{\hat{}} X of XX is the homotopy limit over the sequence of homotopy cofibers of this tower:

E^Xlim(cofib(E¯E¯X)cofib(E¯XX)0). E\hat{}X \coloneqq \underset{\longleftarrow}{\lim} \left( \cdots \to cofib(\overline{E}\wedge \overline{E}\wedge X) \to cofib(\overline{E}X \to X) \to 0 \right) \,.

This exists and comes with a canonical morphism XE^XX \to E\hat{}X.

(Bousfield 79, prop. 5.5, recalled as Ravenel 84, theorem 1.13)

(Ravenel 84, example 1.16

Remark

There is a canonical map

L EXlim n(E S n+1 SX) L_E X \stackrel{}{\longrightarrow} \underset{\leftarrow}{\lim}_n (E^{\wedge^{n+1}_S}\wedge_S X)

from the EE-Bousfield localization of spectra of XX into the totalization.

We consider now conditions for this morphism to be an equivalence.

Definition

For RR a ring, its core cRc R is the equalizer in

cRRRR. c R \longrightarrow R \stackrel{\longrightarrow}{\longrightarrow} R \otimes R \,.
Proposition

Let EE be a connective ring spectrum such that the core of π 0(E)\pi_0(E), def. 20, is either of

  • the localization of the integers at a set JJ of primes, cπ 0(E)[J 1]c \pi_0(E) \simeq \mathbb{Z}[J^{-1}];

  • n\mathbb{Z}_n for n2n \geq 2.

Then the map in remark 11 is an equivalence

L EXlim n(E S n+1 SX). L_E X \stackrel{\simeq}{\longrightarrow} \underset{\leftarrow}{\lim}_n (E^{\wedge^{n+1}_S}\wedge_S X) \,.

(Bousfield 79).

Example
  • E=H𝔽 pE = H \mathbb{F}_p the Eilenberg-MacLane spectrum of a prime field. For XX a connective spectrum, its H𝔽 pH \mathbb{F}_p-nilpotent completion is its p-completion

    (H𝔽 p)^XX p ^lim nXM(/p n) (H\mathbb{F}_p)\hat{}X \simeq X^{\hat{}}_p \coloneqq \underset{\longleftarrow}{\lim}_{n \in \mathbb{N}} X \wedge M(\mathbb{Z}/p^n)

    (where MAM A denotes the Moore spectrum of the abelian group AA).

  • E=E = MU. Every spectrum is alreay MUMU-nilpotently complete

    MU^XX. MU\hat{}X \simeq X \,.
  • E=E = BP at prime pp. For every spectrum XX its BPBP-nilpotent completion is its p-localization

    BP^XX (p)XM (p) BP\hat{}X \simeq X_{(p)} \coloneqq X \wedge M \mathbb{Z}_{(p)}

    (where (p)\mathbb{Z}_{(p)}\subset \mathbb{Q} is the result of inverting all primes different from pp).

For more discussion of E-infinity (derived) formal completions via totalizations of Amitsur complexes, see (Carlsson 07).

Theorem

For XX a spectrum and EE a ring spectrum, consider the EE-Adams spectral sequence { r ,,d r}\{\mathcal{E}_r^{\bullet,\bullet}, d_r\} of XX (def. 7, prop. 3, prop. 4). If for each s,ts,t there is rr such that

r s,t s,t \mathcal{E}_r^{s,t} \simeq \mathcal{E}_\infty^{s,t}

then the EE-Adams spectral sequence converges strongly (def.) to the stable homotopy groups of the E-nilpotent completion of XX (def. 14):

1 s,tπ (E^X). \mathcal{E}_1^{s,t} \;\Rightarrow\; \pi_\bullet(E\hat{}X) \,.

(Bousfield 79, recalled as Ravenel 84, theorem 1.15)

Examples

Examples
  • For X=𝕊X = \mathbb{S} and E=H𝔽 pE = H\mathbb{F}_p, then theorem 1 and theorem 2 with example 1 gives a spectral sequence

    Ext 𝒜 p *(𝔽 p,𝔽 p)π (𝕊)Z p . Ext_{\mathcal{A}^\ast_p}(\mathbb{F}_p, \mathbb{F}_p) \;\Rightarrow\; \pi_\bullet(\mathbb{S})\otimes Z^\wedge_p \,.

    This is the classical Adams spectral sequence.

  • For X=𝕊X = \mathbb{S} and E=E = MU, then theorem 1 and theorem 2 with example 1 gives a spectral sequence

    Ext MU *(MU)(MU *,MU *)π (𝕊). Ext_{MU_\ast(MU)}(MU_\ast, MU_\ast) \;\Rightarrow\; \pi_\bullet(\mathbb{S}) \,.

    This is the Adams-Novikov spectral sequence.

As derived descent in higher algebra

We discuss the general definition of EE-Adams-Novikov spectral sequences for suitable E-∞ rings EE expressed in higher algebra, as in (Lurie, Higher Algebra). We follow Lurie 10, a nice exposition is in (Wilson 13).

First we recall

for the general case of filtered objects in suitable stable (∞,1)-categories. Then we consider the specialization of that to the

Finally we consider specifically the examples of such given by

In conclusion this yields for each suitable E-∞ algebra EE over SS and SS-∞-module XX a spectral sequence converging to the homotopy groups of the EE-localization of XX, and this is

The quick idea is this: Given an SS -module XX, regard it as a quasicoherent sheaf on Spec(S)Spec(S). Choose a map Spec(E)Spec(S)Spec(E) \to Spec(S). This is a cover of its 1-image Spec(E)Spec(S) Spec(E) Spec(E) \to Spec(S)^\wedge_{Spec(E)}, which is the derived formal completion of Spec(S)Spec(S) at the image of Spec(E)Spec(E). Restrict attention then to the restriction of XX to that formal completion X Spec(E) X^\wedge_{Spec(E)}. (So if Spec(E)Spec(S)Spec(E) \to Spec(S) was already an atlas, hence was already complete, we stick with the original XX). Then pull back SS to the Cech nerve of the cover Spec(E)Spec(S) Spec(E) wedgeSpec(E) \to Spec(S)^wedge_{Spec(E)}. The realization of this Cech nerve reproduces the completed image, and hence the canonical filtration on the Cech nerve gives a filtration spectral sequence for X Spec(E) X^\wedge_{Spec(E)}.

Spectral sequences computing homotopy groups of filtered objects

Let thoughout 𝒞\mathcal{C} be a stable (∞,1)-category equipped with a t-structure such that its heart is an abelian category.

Example

For instance

Definition

A generalized filtered object in 𝒞\mathcal{C} is simply a sequential diagram X:(,<)𝒞X \colon (\mathbb{Z}, \lt) \to \mathcal{C}

X n+1X nX n1. \cdots X_{n+1} \to X_n \to X_{n-1} \to \cdots \,.

Or rather, the object being filtered is the homotopy limit

Xlim nX n X \coloneqq \underset{\leftarrow}{\lim}_n X_n

and the sequential diagram exhibits the filtering.

This appears as (Higher Algebra, def. 1.2.2.9).

Definition

For a generalized filtered object X X_\bullet, def. 16, write

F nfib(X nX n+1) F_n \coloneqq fib(X_n \to X_{n+1})

for the homotopy fiber of the nnth structure map, for all nn \in \mathbb{Z}, and define an exact couple

π (F ) π (X ) π (X ) \array{ && \pi_\bullet(F_\bullet) \\ & \swarrow && \nwarrow \\ \pi_\bullet(X_\bullet) && \stackrel{}{\longrightarrow} && \pi_\bullet(X_\bullet) }

where the maps are given by the long exact sequences of homotopy groups

π (X n+1)π (F n)π (X n)π (X n+1)π +1(F n) \cdots \to \pi_\bullet(X_{n+1}) \to \pi_\bullet(F_n) \to \pi_\bullet(X_n) \to \pi_\bullet(X_{n+1}) \to \pi_{\bullet+1}(F_n) \to \cdots

We now have the spectral sequence of a filtered stable homotopy type.

Proposition

Let 𝒞\mathcal{C} be a stable (∞,1)-category equipped with a t-structure such that its heart is an abelian category.

If 𝒞\mathcal{C} has sequential limits and if X n0X_n \simeq 0 for all n>n 0n \gt n_0 then the spectral sequence induced by the exact couple of def. 17 converges to the homotopy groups of the homotopy limit lim nX n\underset{\leftarrow}{\lim}_n X_n of the generalized filtered object:

E 1 p,q=π p+qF p1π p+q(limX ) E^{p,q}_1 = \pi_{p+q} F_{p-1} \Rightarrow \pi_{p+q} (\underset{\leftarrow}{\lim} X_\bullet)

This is due to (Higher Algebra, prop. 1.2.2.14). Review is in (Wilson 13, theorem 1.2.1).

For the traditional statement in the category of chain complexes see at spectral sequence of a filtered complex.

Homotopy groups of cosimplicial totalizations filtered by coskeleta

Definition

Given an cosimplicial object

Y:Δ𝒞 Y \;\colon\; \Delta \longrightarrow \mathcal{C}

its totalization TotYlim nY nTot Y \simeq \underset{\leftarrow}{\lim}_n Y_n is filtered, def. 16, by the totalizations of its coskeleta

TotYTot(cosk 2Y)Tot(cosk 1Y)Tot(cosk 0Y)0. Tot Y \to \cdots \to Tot (cosk_2 Y) \to Tot (cosk_1 Y) \to Tot (cosk_0 Y) \to 0 \,.
Proposition

The filtration spectral sequence, prop. 17, applied to the filtration of a totalization by coskeleta as in def. 18, has as E 2E_2-term the cohomology groups of the Moore complex associated with the cosimplicial object

E 2 p,q=H p(π q(Tot(cosk (Y))))π pqTot(Y). E_2^{p,q} = H^p(\pi_q(Tot (cosk_\bullet(Y)))) \Rightarrow \pi_{p-q} Tot(Y) \,.

This is (Higher Algebra, remark 1.2.4.4). Review is around (Wilson 13, theorem 1.2.4).

Canonical cosimplicial resolution of E E_\infty-algebras

We discuss now the special case of coskeletally filtered totalizations coming from the canonical cosimplicial objects induced from E-∞ algebras (“Amitsur complexes”, “Sweedler corings”).

In this form this appears as (Lurie 10, theorem 2). A review is in (Wilson 13, 1.3). For the analog of this in the traditional formulation see (Ravenel 86, ch. 3, prop. 3.1.2).

Definition

Let SS be an E-∞ ring and let EE be an E-∞ algebra over SS, hence an E-∞ ring equipped with a homomorphism

SE. S \longrightarrow E \,.

The canonical cosimplicial object associated to this (the “\infty-Sweedler coring” or “Amitsur complex”) is that given by the iterated smash product/tensor product over SS:

E S +1:Δ𝒞. E^{\wedge^{\bullet+1}_S} \;\colon\; \Delta \to \mathcal{C} \,.

More generally, for XX an SS-∞-module, the canonical cosimplicial object is

E S +1 SX:Δ𝒞. E^{\wedge^{\bullet+1}_S}\wedge_S X \;\colon\; \Delta \to \mathcal{C} \,.
Proposition

If EE is such that the self-generalized homology E (E)π (E SE)E_\bullet(E) \coloneqq \pi_\bullet(E \wedge_S E) (the dual EE-Steenrod operations) is such that as a module over E π (E)E_\bullet \coloneqq \pi_\bullet(E) it is a flat module, then there is a natural equivalence

π (E S n+1 SX)E (E S n) E E (X). \pi_\bullet \left( E^{\wedge^{n+1}_S} \wedge_S X \right) \simeq E_\bullet(E^{\wedge^n_S}) \otimes_{E_\bullet} E_\bullet(X) \,.
Remark

This makes (E ,E (E))(E_\bullet, E_\bullet(E)) be the Hopf algebroid formed by the EE-Steenrod algebra. See there for more on this.

Example

The condition in prop. 19 is satisfied for

It is NOT satisfied for

Remark

Under good conditions (…), π \pi_\bullet of the canonical cosimplicial object provides a resolution of comodule tensor product and hence computes the Ext-groups over the Hopf algebroid:

H p(π q(Tot(cosk (E S +1 SX))))Ext E (E) p(Σ qE ,E (X)). H^p(\pi_q(Tot(cosk_\bullet(E^{\wedge^{\bullet+1}_S } \wedge_S X)))) \simeq Ext^p_{E_\bullet(E)}(\Sigma^q E_\bullet, E_\bullet(X)) \,.

(…)

Here the homotopy groups are expressed by Ext-groups using a universal coefficient theorem for generalized cohomology (Adams 74, III.13).

(e.g. Wilson 13, theorem 1.3.5, based on Bousfield 79)

Remark

There is a canonical map

L EXlim n(E S n+1 SX) L_E X \stackrel{}{\longrightarrow} \underset{\leftarrow}{\lim}_n (E^{\wedge^{n+1}_S}\wedge_S X)

from the EE-Bousfield localization of spectra of XX into the totalization.

We consider now conditions for this morphism to be an equivalence.

Definition

For RR a ring, its core cRc R is the equalizer in

cRRRR. c R \longrightarrow R \stackrel{\longrightarrow}{\longrightarrow} R \otimes R \,.
Proposition

Let EE be a connective E-∞ ring such that the core of π 0(E)\pi_0(E), def. 20, is either of

  • the localization of the integers at a set JJ of primes, cπ 0(E)[J 1]c \pi_0(E) \simeq \mathbb{Z}[J^{-1}];

  • n\mathbb{Z}_n for n2n \geq 2.

Then the map in remark 11 is an equivalence

L EXlim n(E S n+1 SX). L_E X \stackrel{\simeq}{\longrightarrow} \underset{\leftarrow}{\lim}_n (E^{\wedge^{n+1}_S}\wedge_S X) \,.

(Bousfield 79).

The EE-Adams-Novikov spectral sequence

Summing this up yields the general EE-Adams(-Novikov) spectral sequence

Corollary

Let EE a connective E-∞ ring that satisfies the conditions of prop. 20. Then by prop. 17 and prop. 20 there is a strongly convergent multiplicative spectral sequence

E p,qπ qpL cπ 0EX E^{p,q}_\bullet \Rightarrow \pi_{q-p} L_{c \pi_0 E} X

converging to the homotopy groups of the cπ 0(E) c \pi_0(E)-localization of XX. If moreover the dual EE-Steenrod algebra E (E)E_\bullet(E) is flat as a module over E E_\bullet, then, by prop. 18 and remark 10, the E 2E_2-term of this spectral sequence is given by the Ext-groups over the EE-Steenrod Hopf algebroid.

E p,q=Ext E (E) p(Σ qE ,E X). E^{p,q}_\bullet = Ext^p_{E_\bullet(E)}(\Sigma^q E_\bullet, E_\bullet X) \,.
tower diagram/filteringspectral sequence of a filtered stable homotopy type
filtered chain complexspectral sequence of a filtered complex
Postnikov towerAtiyah-Hirzebruch spectral sequence
chromatic towerchromatic spectral sequence
skeleta of simplicial objectspectral sequence of a simplicial stable homotopy type
skeleta of Sweedler coring of E-∞ algebraAdams spectral sequence
filtration by support
slice filtrationslice spectral sequence

References

General

The original sources are

Convergence was notably discussed in

  • Aldridge Bousfield, The localization of spectra with respect to homology, Topology 18 (1979), no. 4, 257–281. (pdf)

  • Douglas Ravenel, Localization with respect to certain periodic homology theories, American Journal of Mathematics, Vol. 106, No. 2, (Apr., 1984), pp. 351-414 (pdf)

The spectrum-level discussion of the ASS goes back to around

  • R. M. F. Moss, On the composition pairing of Adams spectral sequences, Proceedings of the London Mathematical Society 3.1 (1968): 179-192.

A streamlined presentation of this which is close in spirit to constructions in homological algebra was given in

  • Haynes Miller, On relations between Adams spectral sequences, with an application to the stable homotopy of a Moore space, J. Pure Appl. Algebra 20 (1981) (pdf)

further highlighted in

and worked out in some more detail in

For full details of some of the steps involved see also (Schwede 12).

Reviews include

The modern point of view of higher algebra is in

  • Jacob Lurie, Chromatic Homotopy Theory (2010)

    • lecture 8, The Adams Spectral Sequence (pdf)

    • lecture 9, The Adams Spectral Sequence for MUMU (pdf)

    • lecture 10, The proof of Quillen’s theorem (pdf)

    • lecture 30, Localizations and the Adams-Novikov Spectral Sequence (pdf)

based on

and nicely surveyed in

Hopf algebroid ExtExt-structure on E 2E^2

Further examples with more general coefficients

For tmf

  • Mark Behrens, The Adams spectral sequence for tmftmf (pdf)

  • Michael Hill, The 3-local tmftmf-homology of BΣ 3B \Sigma_3, Proceedings of the AMS, 2007 (pdf)

Revised on May 20, 2016 08:54:33 by Urs Schreiber (131.220.184.222)