Context

Stable Homotopy theory

stable homotopy theory

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 $X$ and a ring spectrum $E$, then under mild assumptions the Adams spectral sequence converges to the homotopy groups of the $E$-nilpotent completion of $X$, while under stronger assumptions the latter is the $E$-Bousfield localization of spectra. The second page of the spectral sequence is given by the $E$-homology of $X$ as modules over the dual $E$-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 $E = H\mathbb{F}_p$ is ordinary homology mod $p$, while the Adams-Novikov spectral sequence (Novikov 67) is the case where $E =$ MU is complex cobordism cohomology theory or $E =$ BP, Brown-Peterson theory.

Generally, for $E$ a suitable E-infinity algebra there is a corresponding $E$-Adams(-Novikov) spectral sequence whose second page is given by $E$-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 $E$. 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 $E$ 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

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

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

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

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

Details

The $E$-Adams spectral sequence

We here discuss Adams spectral sequences for computation of $E$-localization of mapping spectra $[Y,X]$ for by $E$ 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 $Y = X = \mathbb{S}$ and $E =$ A$\mathbb{F}_p$, discussed below.

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

$E$-Adams resolutions

A streamlined discussion of $E$-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 $X_\bullet$ 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, $E$-Adams resolutions in the stable homotopy category are the direct analog of injective resolutions in an abelian category.

Definition

For $X$ 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 $X$, regarded as a functor on the stable homotopy category:

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

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

Example

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

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

is the stable homotopy group-functor.

Throughout,$E$ is a ring spectrum.

Definition

Say that

1. a sequence of spectra

$A_1 \longrightarrow A_2 \longrightarrow \cdots \longrightarrow A_n$

is

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

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

$0 \longrightarrow A_1 \longrightarrow A_2 \longrightarrow A_3 \longrightarrow 0$

is (long) exact in the above sense;

2. a morphism $A \longrightarrow B$ is

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

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

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

Example

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

Example

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

Proof

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

Example

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

Proof

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

Example

For any spectrum $X$ the morphism

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

is an $E$-monomorphism in the sense of def. 2.

Proof

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

Remark/Warning

Consecutive morphisms in an $E$-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 $E$-injective objects, defined below in def. 3.

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

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

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

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

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

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

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

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

$[C,X] \stackrel{\delta^\ast_\bullet}{\longleftarrow} [\Sigma A, X] \stackrel{(-\Sigma f)^\ast_\bullet}{\longleftarrow} [\Sigma B, X]$

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

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

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

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

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

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

Proof

In one direction, assume that $S$ is $E$-injective and consider the diagram

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

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

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

which exhibits $S$ as a retract of $E \wedge S$.

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

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

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

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

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

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

By composition and commutativity, this gives the required extension of $g$ along $f$.

Definition

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

$0 \longrightarrow S \longrightarrow I_0 \longrightarrow I_1 \longrightarrow I_2 \longrightarrow \cdots$

such that each $I_j$ is $E$-injective, def. 3.

Lemma

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

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

Lemma

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

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

There are two $E$-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 $\mathbb{S}$-algebra $E$, exhibiting the formal dual of the Cech nerve of $Spec(E) \to Spec(\mathbb{S})$.

Example

(normalized $E$-Adams resolution)

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

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

For $X$ a spectrum, its normalized $E$-Adams resolution is the top row of

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

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

Remark

The notation for $\overline{E}$ 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 “$\Sigma$”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 $E$-Adams resolution)

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

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

whose

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

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

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

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

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

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

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

This is called the standard $E$-Adams resolution of $X$.

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

Proposition

The standard resolution of example 7 is indeed an $E$-Adams resolution of $X$ 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 $E$-exact. Moreover, the terms in the sequence are all $E$-injective by lemma 2.

$E$-Adams towers
Definition

An $E$-Adams tower of a spectrum $X$ is a commuting diagram in the stable homotopy category of the form

$\array{ && \vdots \\ && \downarrow^{\mathrlap{p_2}} \\ && X_2 &\stackrel{\kappa_2}{\longrightarrow}& \Omega^2 I_3 \\ &\nearrow& \downarrow^{\mathrlap{p_1}} \\ && X_1 &\stackrel{\kappa_1}{\longrightarrow}& \Omega I_2 \\ &\nearrow& \downarrow^{\mathrlap{p_0}} \\ X &\underset{}{\longrightarrow}& X_0 = I_0 &\stackrel{\kappa_0}{\longrightarrow}& I_1 }$

such that

1. each hook is a homotopy fiber sequence;

2. the composition of the $(\Sigma \dashv \Omega)$-adjuncts of $\Sigma_{p_{n-1}}$ with $\Sigma^n \kappa_n$

$i_{n+1} \;\colon\; I_n \stackrel{\widetilde {\Sigma p_{n-1}}}{\longrightarrow} \Sigma^n X_n \stackrel{\Sigma^{n}\kappa_n}{\longrightarrow} I_{n+1}$

constitute an $E$-Adams resolution of $X$, def. 4:

$0 \to X \stackrel{i_0}{\to} I_0 \stackrel{i_1}{\to} I_1 \stackrel{}{\to} \cdots \,.$

Call this the associated $E$-Adams resolution of the $E$-Adams tower.

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

Proposition

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

Proof idea

Given an $E$-Adams resolution

$X \overset{i_0}{\longrightarrow} I_0 \overset{i_1}{\longrightarrow} I_1 \longrightarrow \cdots$

consider the induced diagram

$\array{ && && C_1 && && && && C_3 \\ && & {}^{\mathllap{\rho_1}}\nearrow && \searrow^{\mathrlap{\sigma_1}} && && && {}^{\mathllap{\rho_3}}\nearrow \\ && I_0 && \underset{i_1}{\longrightarrow} && I_1 && \overset{i_2}{\longrightarrow} && I_2 && \underset{i_3}{\longrightarrow} & \cdots \\ & {}^{\mathllap{\sigma_0 \coloneqq i_0}}\nearrow && && && {}_{\mathllap{\rho_2}}\searrow && \nearrow_{\mathrlap{\sigma_2}} \\ C_0 \coloneqq X && && && && C_2 }$

constructed inductively as follows:

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

Then assume that an $E$-epi/mono factorization

$i_n \colon I_{n_1} \overset{\rho_n}{\longrightarrow} C_n \overset{\sigma_n}{\to} I_n$

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

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

There is another tower associated with an $E$-Adams resolutions:

Definition

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

$\array{ X = C_0 &\stackrel{\gamma_0}{\longleftarrow}& \Sigma^{-1} C_1 &\stackrel{\gamma_1}{\longleftarrow}& \Sigma^{-2} C_2 &\longleftarrow& \cdots \\ \downarrow && \downarrow && \downarrow \\ I_0 && \Sigma^{-1} I_1 && ^\Sigma^{-2} I_2 }$

with the $C_i$ as in the proof of prop. 2 and $\gamma_n \coloneqq \Sigma^{-} hofib(\sigma_n)$.

Example

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

$\array{ X &\stackrel{\gamma_0}{\longleftarrow}& \overline{E} \wedge X &\stackrel{\gamma_1}{\longleftarrow}& \overline{E} \wedge \overline{E} \wedge X &\longleftarrow& \cdots \\ \downarrow && \downarrow && \downarrow \\ E \wedge X && \Sigma^{-1}(E \wedge (\Sigma \overline{E}) \wedge X) && \Sigma^{-2}(E \wedge (\Sigma\overline{E}) \wedge (\Sigma \overline{E}) \wedge X }$

hence

$\array{ X &\stackrel{\gamma_0}{\longleftarrow}& \overline{E} \wedge X &\stackrel{\gamma_1}{\longleftarrow}& \overline{E} \wedge \overline{E} \wedge X &\longleftarrow& \cdots \\ \downarrow && \downarrow && \downarrow \\ E \wedge X && E \wedge \overline{E} \wedge X && E \wedge \overline{E} \wedge \overline{E} \wedge X } \,.$

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 “$E$-Adams resolution”. We instead follow (Hopkins 99) in using “$E$-Adams resoltuon” for “$E$-injective resolution” as in def. 4, “$E$-Adams tower” for def. 5 and follow (Aramian) in saying “associated inverse sequence” for the above.

$E$-Adams spectral sequences
Definition

Given spectra $X$ and $Y$, and given an $E$-Adams resolution of $X$, def. 4, or equivalently (by prop. 2) an $E$-Adams tower over $X$, def. 5,

$\array{ && \vdots \\ && \downarrow^{\mathrlap{p_2}} \\ && X_2 &\stackrel{\kappa_2}{\longrightarrow}& \Omega^2 I_3 \\ &\nearrow& \downarrow^{\mathrlap{p_1}} \\ && X_1 &\stackrel{\kappa_1}{\longrightarrow}& \Omega I_2 \\ &\nearrow& \downarrow^{\mathrlap{p_0}} \\ X &\underset{}{\longrightarrow}& X_0 = I_0 &\stackrel{\kappa_0}{\longrightarrow}& I_1 }$

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

$\array{ && \vdots \\ && \downarrow^{\mathrlap{[Y,p_2]}} \\ && [Y,X_2] &\stackrel{[Y,\kappa_2]}{\longrightarrow}& [Y,\Omega^2 I_3] \\ &\nearrow& \downarrow^{\mathrlap{[Y,p_1]}} \\ && [Y,X_1] &\stackrel{[Y,\kappa_1]}{\longrightarrow}& [Y,\Omega I_2] \\ &\nearrow& \downarrow^{\mathrlap{[Y,p_0]}} \\ [Y,X] &\underset{}{\longrightarrow}& [Y,X_0] = [Y,I_0] &\stackrel{[Y,\kappa_0]}{\longrightarrow}& [Y,I_1] } \,.$

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

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

with

$\mathcal{D} \coloneqq \oplus_{s,t} \mathcal{D}^{s,t} \coloneqq \oplus_{s,t} \pi_{t-s}([Y,X_s])$
$\mathcal{E} \coloneqq \oplus_{s,t} \mathcal{E}^{s+1,t} \coloneqq \oplus_{s,t} \pi_{t-s}([Y,\Omega^s I_{s+1}])$

and

$p \colon \pi_{t-s}([Y,X_{s+1}]) \stackrel{\pi_{t-s}([Y,p_s])}{\longrightarrow} X_{t-s}([Y,X_s])$
$\kappa \colon \pi_{t-s}([Y,X_s]) \stackrel{\pi_{t-s}([Y,\kappa_s])}{\longrightarrow} \pi_{t-s}([Y,\Omega^s I_{s+1}])$
$\partial \colon \pi_{t-s}([Y,\Omega^s I_{s+1}]) \stackrel{\pi_{t-s}([Y,\partial_s])}{\longrightarrow} \pi_{t-s}([Y,\Sigma X_{s+1}]) \,.$

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

Proposition

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

Proposition

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

1. the tower spectral sequence of its associated $E$-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 $E$-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 $E^{\wedge^\bullet} \wedge X$.

The first page

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

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

with the differentials being the image under $\pi_\bullet$ 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 $[Y,E^{\wedge (s+1)}\wedge X]$ may alternatively be computed as morphisms of $E$-homology equipped with suitable comodule structure over a Hopf algebroid structure on the dual $E$-Steenrod operations $E_\bullet(E)$. Then below we discuss that, as a result, the $d_1$-homology of the $\mathcal{E}_1$-page is seen to compute the Ext-groups from the $E$-homology of $Y$ to the $E$-homology of $X$, regarded as $E_\bullet(E)$-comodules. This re-formulation of the $\mathcal{E}_2$-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 $E$ flat if one, equivalently both, of the morphisms

$\eta_L \coloneqq \pi_\bullet(e \wedge id) \;\colon\; E_\bullet \longrightarrow E_\bullet(E)$
$\eta_r \;\coloneqq\; \pi_\bullet(id \wedge e) \colon E_\bullet \longrightarrow E_\bullet(E)$

is a flat morphism.

Example

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

Example

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

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

Proposition

If $E$ is flat, def. 8, then for all spectra $X$ there is a natural isomorphisms

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

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

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

The desired natural homomorphism

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

is given on $[\alpha] \in \pi_\bullet(E \wedge E)$ and $[\beta] \in \pi_\bullet(E \wedge X)$ by $([\alpha, \beta])\mapsto [(id \wedge \mu \wedge id) \circ (\alpha \wedge \beta)]$.

To see that this is an isomorphism, observe that by flatness of $E$, the assignment $X \mapsto E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(-)$ is a generalized homology functor, hence represented by some spectrum. The above morphism, natural in $X$, 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 $n$ by induction:

\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${}^{op}$ of commutative rings, regarded with its cartesian monoidal category structure.

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

Remark

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

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

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

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

$inv \;\colon\; Spec(\Gamma) \longrightarrow Spec(\Gamma)$

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 $CRing^{op}$, see at CRing – Properties – Cocartesian comonoidal structure:

$Spec(R_1) \underset{Spec(R_3)}{\times} Spec(R_2) \simeq Spec(R_1 \otimes_{R_3} R_2) \,.$

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

$\array{ \Gamma \underset{A}{\otimes} \Gamma \\ \uparrow^{\mathrlap{\Psi}} \\ \Gamma \\ {}^{\mathllap{\eta_L}}\uparrow \downarrow^{\mathrlap{\epsilon}} \; \uparrow^{\mathrlap{\eta_R}} \\ A }$

as well as

$c \; \colon \; \Gamma \longrightarrow \Gamma$

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

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

• $\epsilon$ is called the co-unit;

• $\Psi$ is called the comultiplication;

• $c$ is called the antipode or conjugation

Remark

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

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

Dually, a commutative Hopf algebroid $\Gamma \stackrel{\overset{\eta_L}{\longleftarrow}}{\underset{\eta_R}{\longleftarrow}} A$ for which $\eta_L$ and $\eta_R$ happen to coincide is equivalently a commutative Hopf algebra $\Gamma$ over $A$.

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
1. two commutative rings, $A$ and $\Gamma$;

2. ring homomorphisms

1. (left/right unit)

$\eta_L,\eta_R \colon A \longrightarrow \Gamma$;

2. (comultiplication)

$\Psi \colon \Gamma \longrightarrow \Gamma \underset{A}{\otimes} \Gamma$;

3. (counit)

$\epsilon \colon \Gamma \longrightarrow A$;

4. (conjugation)

$c \colon \Gamma \longrightarrow \Gamma$

such that

1. (co-unitality)

1. $\epsilon \circ \eta_L = \epsilon \circ \eta_R = id_A$;

2. $(id_\Gamma\otimes_A\epsilon) \circ \Delta = (\epsilon \otimes_A id_\Gamma) \circ \Delta = id_\Gamma$;

2. (co-associativity) $(id_\Gamma \otimes_A \Psi) \circ \Psi = (\Psi \otimes_A id_\Gamma) \circ \Psi$;

3. (inverses)

1. $c \circ c = id_\Gamma$;

2. $c\circ \eta_L = \eta_R$; $c \circ \eta_R = \eta_L$;

3. the universally induced $\nabla_c \colon \Gamma \otimes_A \Gamma \longrightarrow \Gamma$ satifies

$\nabla_c \circ \Psi = \epsilon \circ \eta_L = \epsilon \circ \eta_R$.

Comodules and cotensor product
Definition

Given a commutative Hopf algebroid $\Gamma$ over $A$ as in def. 10, hence an internal groupoid in $CRing^{op}$, then a comodule ring over it is an action in $CRing^{op}$ 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 $\Gamma$ over $A$, def. 10, then a left comodule over $\Gamma$ is

1. an $A$-module $N$;

2. an $A$-module homomorphism (co-action)

$\Psi_N \;\colon\; N \longrightarrow \Gamma \otimes_A N$;

such that

1. $(\epsilon \otimes_A id_N) \circ \Psi_N = id_N$;

2. $(\Psi \otimes_A id_N) \circ \Psi_N = (id_\Gamma \otimes_A \Psi_N)\circ \Psi_N$.

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

$\Gamma CoMod$

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

Example

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

$\Psi_A \;\colon\; A \overset{\eta_L}{\longrightarrow} \Gamma \simeq \Gamma \otimes_A A$

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

$\Psi_A \;\colon\; A \overset{\eta_R}{\longrightarrow} \Gamma \simeq \Gamma \otimes_A A \,.$
Proposition

Given a commutative Hopf algebroid $\Gamma$ over $A$, there is a free-forgetful adjunction

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

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

The co-free $\Gamma$-comodule on an $A$-module $N$ is $\Gamma \otimes_A N$ equipped with the coaction induced by the comultiplication $\Psi$ in $\Gamma$.

Proposition

Consider a commutative Hopf algebroid $\Gamma$ over $A$, def. 10. Any left comodule $N$ over $\Gamma$ (def. 12) becomes a right comodule via the coaction

$N \overset{\Psi}{\longrightarrow} \Gamma \otimes_A N \overset{\simeq}{\longrightarrow} N \otimes_A \Gamma \overset{id \otimes_A c}{\longrightarrow} N \otimes_A \Gamma \,,$

where the isomorphism in the middle the is braiding in $A Mod$ and where $c$ is the conjugation map of $\Gamma$.

Dually, a right comodule $N$ becoomes a left comodule with the coaction

$N \overset{\Psi}{\longrightarrow} N \otimes_A \Gamma \overset{\simeq}{\longrightarrow} \Gamma \otimes_A N \overset{c \otimes_A id}{\longrightarrow} \Gamma \otimes_A N \,.$
Definition

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

$N_1 \Box_\Gamma N_2 \;\coloneqq\; ker \left( N_1 \otimes_A N_2 \overset{\Psi_{N_1}\otimes_{A} id - id \otimes_A \Psi_{N_2} }{\longrightarrow} \right) \,.$

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

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

Example

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

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

is called the primitive elements of $N$:

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

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

$N_1 \Box N_2 \;\simeq\; N_2 \Box N_1 \,.$

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

Lemma

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

1. The morphism

$Hom_A(N_1, A) \overset{f \mapsto (id \otimes_A f) \circ \Psi_{N_1}}{\longrightarrow} Hom_A(N_1, \Gamma \otimes_A A) \simeq Hom_A(N_1, \Gamma) \simeq Hom_A(N_1, A) \otimes_A \Gamma$

gives $Hom_A(N_1,A)$ the structure of a right $\Gamma$-comodule;

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

$Hom_A(N_1, A) \Box_\Gamma N_2 \simeq Hom_\Gamma(N_1, N_2) \,.$

Hence in particular

$A \Box_\Gamma N_2 \;\simeq\; Hom_\Gamma(A,N_2)$

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

Remark

In computing the second page of $E$-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 $E$-Steenrod operations

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

Proposition

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

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

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

Proof

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

$\array{ \pi_\bullet(E\wedge E \wedge E) &\simeq& E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(E) \\ \uparrow^{\mathrlap{\pi_\bullet(id \wedge e \wedge id)}} && \uparrow^{\mathrlap{\Psi}} \\ \pi_\bullet(E \wedge E) &=& E_\bullet(E) \\ {}^{\mathllap{\pi_\bullet(e \wedge id)}}\uparrow \downarrow^{\mathrlap{\pi_\bullet(\mu)}} \;\;\;\;\;\; \uparrow^{\mathrlap{\pi_\bullet(id \wedge e)}} && {}^{\mathllap{\eta_L}}\uparrow \downarrow^{\mathrlap{\epsilon}} \uparrow^{\mathrlap{\eta_R}} \\ \pi_\bullet(E) &=& \pi_\bullet(E) } \,.$

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

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

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

$E =$

• H$\mathbb{F}_p$,

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 $E$:

Example

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

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

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

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

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

Proposition

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

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

(from the stable homotopy group of the mapping spectrum to the hom groups of $\pi_\bullet(E)$-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 $E$-homology:

Proposition

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

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

is an isomorphism.

Proof

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

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

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

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

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

The second page

Theorem

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

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

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

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

By prop. 3 it is sufficient to show this for the standard $E$-Adams resolution of prop. 1. For that case the $\mathcal{E}_1$ 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 $E_\bullet(E) CoMod$ is an abelian category;

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

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 $\Gamma$ over $A$, def. 9, 10 is such that $\eta_L, \eta_R \colon A \longrightarrow \Gamma$ is a flat morphism, then the category $\Gamma CoMod$ of comodules over $\Gamma$, def. 12, is an abelian category.

Proof

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

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

$\array{ N_1 &\stackrel{f}{\longrightarrow}& N_2 \\ \downarrow^{\mathrlap{\Psi_{N_1}}} && \downarrow^{\mathrlap{\Psi_{N_2}}} \\ \Gamma \otimes_A N_1 &\stackrel{id_\Gamma \otimes_A f}{\longrightarrow}& \Gamma \otimes_A N_2 } \,.$

Consider the kernel $ker(f)$ of $f$ in $A$Mod and its image under $\Gamma \otimes_A (-)$

$\array{ ker(f) &\longrightarrow& N_1 &\stackrel{f}{\longrightarrow}& N_2 \\ \downarrow && \downarrow^{\mathrlap{\Psi_{N_1}}} && \downarrow^{\mathrlap{\Psi_{N_2}}} \\ \Gamma \otimes_A ker(f) &\longrightarrow& \Gamma \otimes_A N_1 &\stackrel{id_\Gamma \otimes_A f}{\longrightarrow}& \Gamma \otimes_A N_2 } \,.$

By the assumption that $\Gamma$ is a flat module over $A$, also $\Gamma \otimes_A ker(f) \simeq ker(\Gamma \otimes_A f)$ 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 $A$-module $ker(f)$ uniquely inherits the structure of a $\Gamma$-comodule such as to make $ker(f) \to N_1$ a comodule homomorphism. By the same universal property it follows that $ker(f)$ with this comodule structure is in fact the kernel of $f$ in $\Gamma CoMod$.

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

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

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

Proposition

If a commutative Hopf algebroid $\Gamma$ over $A$, def. 9, 10 is such that $\eta_L, \eta_R \colon A \longrightarrow \Gamma$ is a flat morphism, then

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

2. $\Gamma CoMod$ 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 $A Mod$ has enough injectives (see this proposition). Now by prop. 6 we have the adjunction

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

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 $I \in A Mod$ an injective module, then the co-free comodule $\Gamma \otimes_A I$ is an injective object in $\Gamma CoMod$;

2. for $N \in \Gamma CoMod$ any object, and for $i \colon U(N) \hookrightarrow I$ a monomorphism of $A$-modules into an injective $A$-module, then the adjunct $\tilde i \colon N \hookrightarrow \Gamma\otimes_A I$ is a monomorphism in $\Gamma CoMod$ (and into an injective comodule).

Proposition

Let $\Gamma$ be a commutative Hopf algebroid over $A$, def. 9, 10, such that $\eta_L, \eta_R \colon A \longrightarrow \Gamma$ is a flat morphism, Let $N \in \Gamma CoMod$ be a Hopf comodule, def. 12, such that the underlying $A$-module is a projective module (a projective object in $A$Mod).

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

Proof

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

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

\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 $A$-module underlying $N$ 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 $X$ a spectrum and $E$ a ring spectrum, consider the inverse sequence

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

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

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

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

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

Remark

There is a canonical map

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

from the $E$-Bousfield localization of spectra of $X$ into the totalization.

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

Definition

For $R$ a ring, its core $c R$ is the equalizer in

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

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

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

• $\mathbb{Z}_n$ for $n \geq 2$.

Then the map in remark 11 is an equivalence

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

(Bousfield 79).

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

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

(where $M A$ denotes the Moore spectrum of the abelian group $A$).

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

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

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

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

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

Theorem

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

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

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

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

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

Examples

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

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

This is the classical Adams spectral sequence.

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

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

This is the Adams-Novikov spectral sequence.

As derived descent in higher algebra

We discuss the general definition of $E$-Adams-Novikov spectral sequences for suitable E-∞ rings $E$ 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 $E$ over $S$ and $S$-∞-module $X$ a spectral sequence converging to the homotopy groups of the $E$-localization of $X$, and this is

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

Spectral sequences computing homotopy groups of filtered objects

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

For instance

Definition

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

$\cdots X_{n+1} \to X_n \to X_{n-1} \to \cdots \,.$

Or rather, the object being filtered is the homotopy limit

$X \coloneqq \underset{\leftarrow}{\lim}_n X_n$

and the sequential diagram exhibits the filtering.

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

Definition

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

$F_n \coloneqq fib(X_n \to X_{n+1})$

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

$\array{ && \pi_\bullet(F_\bullet) \\ & \swarrow && \nwarrow \\ \pi_\bullet(X_\bullet) && \stackrel{}{\longrightarrow} && \pi_\bullet(X_\bullet) }$

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

$\cdots \to \pi_\bullet(X_{n+1}) \to \pi_\bullet(F_n) \to \pi_\bullet(X_n) \to \pi_\bullet(X_{n+1}) \to \pi_{\bullet+1}(F_n) \to \cdots$

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

Proposition

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

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

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

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

$Y \;\colon\; \Delta \longrightarrow \mathcal{C}$

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

$Tot Y \to \cdots \to Tot (cosk_2 Y) \to Tot (cosk_1 Y) \to Tot (cosk_0 Y) \to 0 \,.$
Proposition

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

$E_2^{p,q} = H^p(\pi_q(Tot (cosk_\bullet(Y)))) \Rightarrow \pi_{p-q} Tot(Y) \,.$

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

Canonical cosimplicial resolution of $E_\infty$-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 $S$ be an E-∞ ring and let $E$ be an E-∞ algebra over $S$, hence an E-∞ ring equipped with a homomorphism

$S \longrightarrow E \,.$

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 $S$:

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

More generally, for $X$ an $S$-∞-module, the canonical cosimplicial object is

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

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

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

This makes $(E_\bullet, E_\bullet(E))$ be the Hopf algebroid formed by the $E$-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 (…), $\pi_\bullet$ of the canonical cosimplicial object provides a resolution of comodule tensor product and hence computes the Ext-groups over the Hopf algebroid:

$H^p(\pi_q(Tot(cosk_\bullet(E^{\wedge^{\bullet+1}_S } \wedge_S X)))) \simeq Ext^p_{E_\bullet(E)}(\Sigma^q E_\bullet, E_\bullet(X)) \,.$

(…)

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

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

from the $E$-Bousfield localization of spectra of $X$ into the totalization.

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

Definition

For $R$ a ring, its core $c R$ is the equalizer in

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

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

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

• $\mathbb{Z}_n$ for $n \geq 2$.

Then the map in remark 11 is an equivalence

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

(Bousfield 79).

The $E$-Adams-Novikov spectral sequence

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

Corollary

Let $E$ 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

$E^{p,q}_\bullet \Rightarrow \pi_{q-p} L_{c \pi_0 E} X$

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

$E^{p,q}_\bullet = Ext^p_{E_\bullet(E)}(\Sigma^q E_\bullet, E_\bullet X) \,.$

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

• lecture 8, The Adams Spectral Sequence (pdf)

• lecture 9, The Adams Spectral Sequence for $MU$ (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 $Ext$-structure on $E^2$

• appendix A.1 of Ravenel 86

• Andrew Baker, Brave new Hopf algebroids and the Adams spectral sequence for $R$-modules (pdf)

• Andrew Baker, Andrey Lazarev, On the Adams Spectral Sequence for $R$-modules, Algebr. Geom. Topol. 1 (2001) 173-199 (arXiv:math/0105079)

• Andrew Baker and Alain Jeanneret, Brave new Hopf algebroids and extensions of $MU$-algebras, Homology Homotopy Appl. Volume 4, Number 1 (2002), 163-173. (Euclid)

• Mark Hovey, Homotopy theory of comodules over a Hopf algebroid (arXiv:math/0301229)

Further examples with more general coefficients

For tmf

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

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

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