nLab Adams spectral sequence

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.

For a detailed introduction see at Introduction to the Adams Spectral Sequence.

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 terminal morphism

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

in E-∞ geometry/spectral geometry.

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

1. Via the canonical resolution

2. Via injective resolutions

3. As derived descent in higher algebra

The $E$-Adams spectral sequence

We here discuss Adams spectral sequences for computation of $E$-localization of mapping spectra $[Y,X]$ for $E$ a commutative ring spectrum which is flat in a certain sense (def. below), via the “canonical” $E$-Adams resolution introduced in (Adams 74, theorem 15.1). There are other resolutions which lead to the same spectral sequence, this we discuss below in the section on E-Injective resolutions.

The classical Adams spectral sequence is the special case of this general concept of $E$-Adams spectral sequences given by setting $Y = X = \mathbb{S}$ the sphere spectrum and $E =$ H$\mathbb{F}_p$ the Eilenberg-MacLane spectrum of a prime field. This is discussed below.

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

The spectral sequence

Filtered spectra

We introduce the types of spectral sequences of which the $E$-Adams spectral sequences (def. below) are an example.

Definition

A filtered spectrum is a spectrum $Y \in Ho(Spectra)$ equipped with a sequence $Y_\bullet \colon (\mathbb{N}, \gt) \longrightarrow Ho(Spectra)$ in the stable homotopy category (def.) of the form

$$\cdots \longrightarrow Y_3 \overset{f_2}{\longrightarrow} Y_2 \overset{f_1}{\longrightarrow} Y_1 \overset{f_0}{\longrightarrow} Y_0 \coloneqq Y \,.$$

Remark

More generally a filtering on an object $X$ in (stable or not) homotopy theory is a $\mathbb{Z}$-graded sequence $X_\bullet$ such that $X$ is the homotopy colimit $X\simeq \underset{\longrightarrow}{\lim} X_\bullet$. But for the present purpose we stick with the simpler special case of def. .

Remark

There is no condition on the morphisms in def. . In particular, they are not required to be n-monomorphisms or n-epimorphisms for any $n$.

On the other hand, while they are also not explicitly required to have a presentation by cofibrations or fibrations, this follows automatically: by the existence of the model structure on topological sequential spectra (thm.) or equivalently (thm.) the model structure on orthogonal spectra (thm.), every filtering on a spectrum is equivalent to one in which all morphisms are represented by cofibrations or by fibrations.

This means that we may think of a filtration on a spectrum in the sense of def. as equivalently being a tower of fibrations over that spectrum.

The following definition unravels the structure encoded in a filtration on a spectrum, and motivates the concepts of exact couples and their spectral sequences from these.

Definition

(exact couple of a filtered spectrum)

Consider a spectrum $X \in Ho(Spectra)$ and a filtered spectrum $Y_\bullet$ as in def. .

Write $A_k$ for the homotopy cofiber of its $k$th stage, such as to obtain the diagram

$$\array{ \cdots &\stackrel{}{\longrightarrow}& Y_3 &\stackrel{f_2}{\longrightarrow}& Y_2 &\stackrel{f_1}{\longrightarrow} & Y_1 &\stackrel{f_0}{\longrightarrow}& Y \\ && \downarrow^{\mathrlap{g_3}} && \downarrow^{\mathrlap{g_2}} && \downarrow^{\mathrlap{g_1}} && \downarrow^{\mathrlap{g_0}} \\ && A_3 && A_2 && A_1 && A_0 }$$

where each stage

$$\array{ Y_{k+1} &\stackrel{f_k}{\longrightarrow}& Y_k \\ && \downarrow^{\mathrlap{g_k}} \\ && A_k }$$

is a homotopy cofiber sequence (def.), hence equivalently (prop.) a homotopy fiber sequence, hence where

$$Y_{k+1} \overset{f_k}{\longrightarrow} Y_k \overset{g_k}{\longrightarrow} A_k \overset{h_k}{\longrightarrow} \Sigma Y_{k+1}$$

is an exact triangle (prop.).

Apply the graded hom-group functor $[X,-]_\bullet$ (def.) to the above tower. This yields a diagram of $\mathbb{Z}$-graded abelian groups of the form

$$\array{ \cdots &\stackrel{}{\longrightarrow}& [X,Y_3]_\bullet &\stackrel{[X,f_2]_\bullet)}{\longrightarrow}& [X,Y_2]_\bullet &\stackrel{[X,f_1]_\bullet}{\longrightarrow} & [X,Y_1]_\bullet &\stackrel{[X,f_0]_\bullet}{\longrightarrow}& [X,Y_0]_\bullet \\ && \downarrow^{\mathrlap{[X,g_3]_\bullet}} && \downarrow^{\mathrlap{[X,g_2]_\bullet}} && \downarrow^{\mathrlap{[X,g_1]_\bullet}} && \downarrow^{\mathrlap{[X,g_0]_\bullet}} \\ && [X,A_3]_\bullet && [X,A_2]_\bullet && [X,A_1]_\bullet && [X,A_0]_\bullet } \,,$$

where each hook at stage $k$ extends to a long exact sequence of homotopy groups (prop.) via connecting homomorphisms $[X,h_k]_\bullet$

$$\cdots \to [X,A_k]_{\bullet+1} \stackrel{ [X,h_k]_{\bullet+1} }{\longrightarrow} [X,Y_{k+1}]_{\bullet} \stackrel{[X,f_k]_\bullet}{\longrightarrow} [X,Y_k]_\bullet \stackrel{[X,g_k]_\bullet}{\longrightarrow} [X,A_k]_\bullet \stackrel{ [X,h_k]_\bullet }{\longrightarrow} [X,Y_{k+1}]_{\bullet-1} \to \cdots \,.$$

If we regard the connecting homomorphism $[X,h_k]$ as a morphism of degree -1, then all this information fits into one diagram of the form

$$\array{ \cdots &\stackrel{}{\longrightarrow}& [X,Y_3]_\bullet &\stackrel{[X,f_2]_\bullet}{\longrightarrow}& [X,Y_2]_\bullet &\stackrel{[X,f_1]_\bullet}{\longrightarrow} & [X,Y_1]_\bullet &\stackrel{[X,f_0]_\bullet}{\longrightarrow}& [X,Y_0]_\bullet \\ && \downarrow &{}_{\mathllap{ }} \underset{[X,h_2]_\bullet}{\nwarrow} & \downarrow &{}_{\mathllap{ }} \underset{[X,h_1]_\bullet}{\nwarrow} & \downarrow &{}_{\mathllap{ }} \underset{[X,h_0]_\bullet}{\nwarrow} & \downarrow^{\mathrlap{[X,g_0]_\bullet}} \\ && [X,A_3]_\bullet && [X,A_2]_\bullet && [X,A_1]_\bullet && [X,A_0]_\bullet } \,,$$

where each triangle is a rolled-up incarnation of a long exact sequence of homotopy groups (and in particular is not a commuting diagram!).

If we furthermore consider the bigraded abelian groups $[X,Y_\bullet]_{\bullet}$ and $[X,A_\bullet]_{\bullet}$, then this information may further be rolled-up to a single diagram of the form

$$\array{ [X,Y_\bullet]_\bullet & \stackrel{[X,f_\bullet]_\bullet}{\longrightarrow} & [X,Y_\bullet]_{\bullet} \\ & {}_{\mathllap{ [X, h_\bullet]_\bullet }}\nwarrow & \downarrow^{\mathrlap{[X, g_\bullet]_\bullet }} \\ && [X,A_\bullet]_\bullet } \,.$$

Specifically, regard the terms here as bigraded in the following way

$$\begin{aligned} D^{s,t}(X,Y) & \coloneqq [X,Y_s]_{t-s} \\ E^{s,t}(X,Y) & \coloneqq [X,A_s]_{t-s} \end{aligned} \,.$$

Then the bidegree of the morphisms is

morphism	bidegree
$[X,f]$	$(-1,-1)$
$[X,g]$	$(0,0)$
$[X,h]$	$(1,0)$

This way $t$ counts the cycles of going around the triangles:

$$\cdots \to D^{s+1,t+1}(X,Y) \stackrel{[X,f]}{\longrightarrow} D^{s,t}(X,Y) \stackrel{[X,g]}{\longrightarrow} E^{s,t}(X,Y) \stackrel{[X,h]}{\longrightarrow} D^{s+1,t}(X,Y) \to \cdots$$

Data of this form is called an exact couple, def. below.

Definition

An unrolled exact couple (of Adams-type) is a diagram of abelian groups of the form

$$\array{ \cdots &\stackrel{}{\longrightarrow}& \mathcal{D}^{3,\bullet} &\stackrel{i_2}{\longrightarrow}& \mathcal{D}^{2,\bullet} &\stackrel{i_1}{\longrightarrow} & \mathcal{D}^{1,\bullet} &\stackrel{i_0}{\longrightarrow}& \mathcal{D}^{0,\bullet} \\ && \downarrow^{\mathrlap{}} &{}_{ } \underset{\mathllap{k_2}}{\nwarrow} & {}^{\mathllap{j_2}}\downarrow & \underset{k_1}{\nwarrow} & {}^{\mathllap{j_1}}\downarrow &{}_{} \underset{\mathllap{k_0}}{\nwarrow} & \downarrow_{\mathrlap{j_0}} \\ && \mathcal{E}^{3,\bullet} && \mathcal{E}^{2,\bullet} && \mathcal{E}^{1,\bullet} && \mathcal{E}^{0,\bullet} }$$

such that each triangle is a rolled-up long exact sequence of abelian groups of the form

$$\cdots \to \mathcal{D}^{s+1,t+1} \stackrel{i_s}{\longrightarrow} \mathcal{D}^{s,t} \stackrel{j_s}{\longrightarrow} \mathcal{E}^{s,t} \stackrel{k_s}{\longrightarrow} \mathcal{D}^{s+1,t} \to \cdots \,.$$

The collection of this “un-rolled” data into a single diagram of abelian groups is called the corresponding exact couple.

Definition

An exact couple is a diagram (non-commuting) of abelian groups of the form

$$\array{ \mathcal{D} &\stackrel{i}{\longrightarrow}& \mathcal{D} fto \\ & {}_{\mathllap{k}}\nwarrow & \downarrow^{\mathrlap{j}} \\ && \mathcal{E} } \,,$$

such that this is exact in each position, hence such that the kernel of every morphism is the image of the preceding one.

The concept of exact couple so far just collects the sequences of long exact sequences given by a filtration. Next we turn to extracting information from this sequence of sequences.

Remark

The sequence of long exact sequences in def. is inter-locking, in that every $[X,Y_s]_{t-s}$ appears twice:

$$\array{ && & \searrow && \nearrow \\ && && [X,Y_{s+1}]_{t-s-1} \\ && & {}^{\mathllap{ [X,h] }}\nearrow && \searrow^{\mathrlap{ [X,g] }} && && && \nearrow \\ && [X,A_s]_{t-s} && \underset{d_1 }{\longrightarrow} && [X, A_{s+1}]_{t-s-1} && \stackrel{d_1 }{\longrightarrow} && [X,A_{s+2}]_{t-s-2} \\ & \nearrow && && && {}_{\mathllap{ [X,h] }}\searrow && \nearrow_{\mathrlap{ [X,g] }} \\ && && && && [X,Y_{s+2}]_{t-s-2} \\ && && && & \nearrow && \searrow }$$

This gives rise to the horizontal (“splicing”) composites $d_1$, as shown, and by the fact that the diagonal sequences are long exact, these are differentials in that they square to zero: $(d_1)^2 = 0$. Hence there is a cochain complex:

$$\array{ \cdots & \overset{}{\longrightarrow} && [X,A_s]_{t-s} && \overset{d_1}{\longrightarrow} && [X,A_{S+1}]_{t-s-1} && \stackrel{d_1 }{\longrightarrow} && [X,A_{s+2}]_{t-s-2} && \longrightarrow & \cdots } \,.$$

We may read off from these interlocking long exact sequences what these differentials mean, as follows. An element $c \in [X,A_s]_{t-s}$ lifts to an element $\hat c \in[X,Y_{s+2}]_{t-s-1}$ precisely if $d_1 c = 0$:

$$\array{ &\hat c \in & [X, Y_{s+2}]_{t-s-1} \\ && & \searrow^{\mathrlap{ [X,f] }} \\ && && [X,Y_{s+1}]_{t-s-1} \\ && & {}^{\mathllap{ [X,h] }}\nearrow && \searrow^{\mathrlap{ [X,g] }} \\ & c \in & [X,A_s]_{t-s} && \underset{ d_1 }{\longrightarrow} && [X,A_{s+1}]_{t-s-1} }$$

In order to organize this observation, notice that in terms of the exact couple of remark , the differential

$$d_1 \;\coloneqq \; [X,g] \circ [X,h]$$

is the composite

$$d \coloneqq j \circ k \,.$$

Some terminology:

Definition

Given an exact couple, def. ,

$$\array{ \mathcal{D}^{\bullet,\bullet} &\stackrel{i}{\longrightarrow}& \mathcal{D}^{\bullet,\bullet} \\ & {}_{\mathllap{k}}\nwarrow & \downarrow^{\mathrlap{j}} \\ && \mathcal{E}^{\bullet,\bullet} }$$

observe that the composite

$$d \coloneqq j \circ k$$

is a differential in that it squares to 0, due to the exactness of the exact couple:

$$\begin{aligned} d \circ d & = j \circ \underset{= 0}{\underbrace{k \circ j}} \circ k \\ & = 0 \end{aligned} \,.$$

One says that the page of the exact couple is the graded chain complex

$$(\mathcal{E}^{\bullet,\bullet}, d \coloneqq j \circ k) \,.$$

Given a cochain complex like this, we are to pass to its cochain cohomology. Since the cochain complex here has the extra structure that it arises from an exact couple, its cohomology groups should still remember some of that extra structure. Indeed, the following says that the remaining extract structure on the cohomology of the page of an exact couple is itself again an exact couple, called the “derived exact couple”.

Definition

Given an exact couple, def. , then its derived exact couple is the diagram

$$\array{ \widetilde {\mathcal{D}} &\stackrel{\tilde i}{\longrightarrow}& \widetilde {\mathcal{D}} \\ & {}_{\mathllap{\tilde k}}\nwarrow & \downarrow^{\mathrlap{\tilde j}} \\ && \widetilde{\mathcal{E}} } \;\;\;\; \coloneqq \;\;\;\; \array{ im(i) &\stackrel{i}{\longrightarrow}& im(i) \\ & {}_{\mathllap{k}}\nwarrow & \downarrow {\mathrlap{j \circ i^{-1}}} \\ && H(\mathcal{E}, j \circ k) }$$

with

1. $\tilde{\mathcal{E}} \coloneqq ker(d)/im(d)$ (with $d \coloneqq j \circ k$ from def. );

2. $\tilde {\mathcal{D}} \coloneqq im(i)$;

3. $\tilde i \coloneqq i|_{im(i)}$;

4. $\tilde j \coloneqq j \circ i^{-1}$ (where $i^{-1}$ is the operation of choosing any preimage under $i$);

5. $\tilde k \coloneqq k|_{ker(d)}$.

Lemma

The derived exact couple in def. is well defined and is itself an exact couple, def. .

Proof

This is straightforward to check. For completeness we spell it out:

First consider that the morphisms are well defined in the first place.

It is clear that $\tilde i$ is well-defined.

That $\tilde j$ lands in $ker(d)$: it lands in the image of $j$ which is in the kernel of $k$, by exactness, hence in the kernel of $d$ by definition.

That $\tilde j$ is independent of the choice of preimage: For any $x \in \tilde {\mathcal{D}} = im(i)$, let $y, y' \in \mathcal{D}$ be two preimages under $i$, hence $i(y) = i(y') = x$. This means that $i(y'-y) = 0$, hence that $y'-y \in ker(i)$, hence that $y'-y \in im(k)$, hence there exists $z \in \mathcal{E}$ such that $y' = y + k(z)$, hence $j(y') = j(y) + j(k(z)) = j(y) + d(z)$, but $d(z) = 0$ in $\tilde{\mathcal{E}}$.

That $\tilde k$ vanishes on $im(d)$: because $im(d) \subset im(j)$ and hence by exactness.

That $\tilde k$ lands in $im(i)$: since it is defined on $ker(d) = ker(j \circ k)$ it lands in $ker(j)$. By exactness this is $im(i)$.

That the sequence of maps is again exact:

The kernel of $\tilde j$ is those $x \in \im(i)$ such that their preimage $i^{-1}(x)$ is still in $im(x)$ (by exactness of the original exact couple) hence such that $x \in im(i|_{im(i)})$, hence such that $x \in im(\tilde i)$.

The kernel of $\tilde k$ is the intersection of the kernel of $k$ with the kernel of $d = j \circ k$, wich is still the kernel of $k$, hence the image of $j$, by exactness. Indeed this is also still the image of $\tilde j$, since for every $x \in \mathcal{D}$ then $\tilde j(i(x)) = j(x)$.

The kernel of $\tilde i$ is $ker(i) \cap im(i) \simeq im(k) \cap im(i)$, by exactness. Let $x \in \mathcal{E}$ such that $k(x) \in im(i)$, then by exactness $k(x) \in ker(j)$ hence $j(k(x)) = d(x) = 0$, hence $x \in ker(d)$ and so $k(x) = \tilde k(x)$.

Definition

Given an exact couple, def. , then the induced spectral sequence of the exact couple is the sequence of pages, def. , of the induced sequence of derived exact couples, def. , lemma .

The $r$th page of the spectral sequence is the page (def. ) of the $r$th exact couple, denoted

$$\{\mathcal{E}_r, d_r\} \,.$$

Remark

So the spectral sequence of an exact couple (def. ) is a sequence of cochain complexes $(\mathcal{E}_r, d_r)$, where the cohomology of one is the terms of the next one:

$$\mathcal{E}_{r+1} \simeq H(\mathcal{E}_r,d_r) \,.$$

In practice this is used as a successive stagewise approximation to the computation of a limiting term $\mathcal{E}_\infty$. What that limiting term is, if it exists at all, is the subject of convergence of the spectral sequence, we come to this below.

Def. makes sense without a (bi-)grading on the terms of the exact couple, but much of the power of spectral sequences comes from the cases where such a bigrading is given, since together with the sequence of pages of the spectral sequence, this tends to organize computation of the successive cohomology groups in an efficient way. Therefore consider:

Definition

Given a filtered spectrum as in def. ,

$$\array{ \cdots &\stackrel{}{\longrightarrow}& X_3 &\stackrel{f_2}{\longrightarrow}& X_2 &\stackrel{f_1}{\longrightarrow} & X_1 &\stackrel{f_0}{\longrightarrow}& X \\ && \downarrow^{\mathrlap{g_3}} && \downarrow^{\mathrlap{g_2}} && \downarrow^{\mathrlap{g_1}} && \downarrow^{\mathrlap{g_0}} \\ && A_3 && A_2 && A_1 && A_0 }$$

and given another spectrum $X \in Ho(Spectra)$, the induced spectral sequence of a filtered spectrum is the spectral sequence that is induced, by def. from the exact couple (def. ) given by def. :

$$\array{ \mathcal{D} &\stackrel{i}{\longrightarrow}& \mathcal{D} \\ &{}_{\mathllap{k}}\nwarrow& \downarrow^{\mathrlap{j}} \\ && \mathcal{E} } \;\;\;\;\;\; \coloneqq \;\;\;\;\;\; \array{ \underset{s,t}{\oplus} D^{s,t}(X,Y) &\overset{[X,f]}{\longrightarrow}& \underset{s,t}{\oplus} D^{s,t}(X,Y) \\ & {}_{\mathllap{ [X, h] }}\nwarrow & \downarrow^{[X,g]} \\ && \underset{s,t}{\oplus} E^{s,t}(X,Y) } \;\;\;\; \coloneqq \;\;\;\; \array{ \underset{s,t}{\oplus} [X,Y_s]_{t-s} &\overset{[X,f]}{\longrightarrow}& \underset{s,t}{\oplus} [X,Y_s]_{t-s} \\ & {}_{\mathllap{ [X,h] }}\nwarrow & \downarrow^{\mathrlap{[X,g]}} \\ && [X,A_s]_{t-s} }$$

with the following bidegree of the differentials:

$$deg = \;\;\;\; \array{ \mathcal{D} &\stackrel{(-1,-1)}{\longrightarrow}& \mathcal{D} \\ &{}_{\mathllap{(1,0)}}\nwarrow& \downarrow^{\mathrlap{(0,0)}} \\ && \mathcal{E} } \,.$$

In particular the first page is

$$\mathcal{E}^{s,t}_1 = [X,A_s]_{t-s}$$

$$d_1 = [X, g \circ h] \,.$$

As we pass to derived exact couples, by def. , the bidegree of $i$ and $k$ is preserved, but that of $j$ increases by $(1,1)$ with each page, since (by def. )

$$\begin{aligned} deg(\tilde j) & = deg( j \circ i^{-1}) \\ &= deg(j) - deg(i) \\ & = deg(j) + (1,1) \end{aligned} \,.$$

Similarly the first differential has degree

$$\begin{aligned} deg(j \circ k ) & = deg(j) + deg(k) \\ &= (1,0) + (0,0) \\ & = (1,0) \end{aligned}$$

and so the differentials on the $r$th page are of the form

$$d_r \;\colon\; \mathcal{E}_r^{s,t} \longrightarrow \mathcal{E}_r^{s+r, t+r-1} \,.$$

It is conventional to depict this in tables where $s$ increases vertically and upwards and $t-s$ increases horizontally and to the right, so that $d_r$ goes up $r$ steps and always one step to the left. This is the “Adams type” grading convention for spectral sequences (different from the Serre-Atiyah-Hirzebruch spectral sequence convention (prop.)). One also says that

• $s$ is the filtration degree;

• $t-s$ is the total degree;

• $t$ is the internal degree.

A priori all this is $\mathbb{N}\times \mathbb{Z}$-graded, but we regard it as being $\mathbb{Z} \times \mathbb{Z}$-graded by setting

$$\mathcal{D}^{s \lt 0,\bullet} \coloneqq 0 \;\;\;\;\,, \;\;\;\; \mathcal{E}^{s \lt 0, \bullet} \coloneqq 0$$

and trivially extending the definition of the differentials to these zero-groups.

$E$-Adams filtrations

Given a homotopy commutative ring spectrum $(E,\mu,e)$, then an $E$-Adams spectral sequence is a spectral sequence as in def. , where each cofiber is induced from the unit morphism $e \;\colon\; \mathbb{S} \longrightarrow E$:

Definition

Let $X,Y \in Ho(Spectra)$ be two spectra (def.), and let $(E,\mu,e) \in CMon(Ho(Spectra),\wedge, \mathbb{S})$ be a homotopy commutative ring spectrum (def.) in the tensor triangulated stable homotopy category $(Ho(Spectra), \wedge, \mathbb{S})$ (prop.).

Then the $E$-Adams spectral sequence for the computation of the graded abelian group

$$[X,Y]_\bullet \coloneqq [X, \Sigma^{-\bullet} Y]$$

of morphisms in the stable homotopy category (def.) is the spectral sequence of a filtered spectrum (def. ) of the image under $[X,-]_\bullet$ of the tower

$$\array{ & \vdots \\ & {}^{\mathllap{f_0}}\downarrow \\ & Y_3 &\overset{g_3}{\longrightarrow}& E \wedge Y_3 = A_3 \\ & {}^{\mathllap{f_0}}\downarrow \\ & Y_2 &\overset{g_2}{\longrightarrow}& E \wedge Y_2 = A_2 \\ & {}^{\mathllap{f_0}}\downarrow \\ & Y_1 &\overset{g_1}{\longrightarrow}& E \wedge Y_1 = A_1 \\ & {}^{\mathllap{f_0}}\downarrow \\ Y = & Y_0 &\overset{g_0}{\longrightarrow}& E \wedge Y_0 = A_0 } \,,$$

where each hook is a homotopy fiber sequence (equivalently a homotopy cofiber sequence, prop.), hence where each

$$Y_{n+1} \overset{f_n}{\longrightarrow} Y_n \overset{g_n}{\longrightarrow} A_n \overset{h_n}{\longrightarrow} \Sigma Y_{n+1}$$

is an exact triangle (prop.), where inductively

$$A_n \coloneqq E \wedge Y_n$$

is the derived smash product of spectra (corollary) of $E$ with the stage $Y_n$ (cor.), and where

$$g_n \;\colon\; Y_n \underoverset{\simeq}{\ell^{-1}_{Y_n}}{\longrightarrow} \mathbb{S} \wedge Y_n \overset{e \wedge id}{\longrightarrow} E \wedge Y_{n}$$

is the composition of the inverse derived unitor on $Y_n$ (cor.) with the derived smash product of spectra of the unit $e$ of $E$ and the identity on $Y_n$.

Hence, by def , the first page is

$$E_1^{s,t}(X,Y) \;\coloneqq\; [X, A_s ]_{t-s}$$

$$d_1 = [X, g \circ h]$$

and the differentials are of the form

$$d_r \;\colon\; E_r^{s,t}(X,Y) \longrightarrow E_r^{s+r, t+r-1}(X,Y) \,.$$

A priori $E_r^{\bullet,\bullet}(X,Y)$ is $\mathbb{N}\times \mathbb{Z}$-graded, but we regard it as being $\mathbb{Z} \times \mathbb{Z}$-graded by setting

$$E_r^{s \lt 0,\bullet}(X,Y) \coloneqq 0$$

and trivially extending the definition of the differentials to these zero-groups.

(Adams 74, theorem 15.1 page 318)

Remark

The morphism

$$[X,g_k] \;\colon\; [X,Y_k]_\bullet \overset{[X,e \wedge id_{Y_k}] }{\longrightarrow} [X, E \wedge Y_k]_\bullet$$

in def. is sometimes called the Boardman homomorphism (Adams 74, p. 58).

For $X = \mathbb{S}$ the sphere spectrum it reduces to a canonical morphism from stable homotopy to generalized homology (rmk.)

$$\pi_\bullet(g_k) \;\colon\; \pi_\bullet(Y_k) \longrightarrow E_\bullet(Y_k) \,.$$

For $E =$ HA an Eilenberg-MacLane spectrum (def.) this in turn reduces to the Hurewicz homomorphism for spectra.

This way one may think of the $E$-Adams filtration on $Y$ in def. as the result of filtering any spectrum $Y$ by iteratively projecting out all its $E$-homology. This idea was historically the original motivation for the construction of the classical Adams spectral sequence by John Frank Adams, see the first pages of (Bruner 09) for a historical approach.

It is convenient to adopt the following notation for $E$-Adams spectral sequences (def. ):

Definition

For $(E,\mu,e) \in CMon(Ho(Spectra),\wedge, \mathbb{S})$ a homotopy commutative ring spectrum (def.), write $\overline{E}$ for the homotopy fiber of its unit $e \colon \mathbb{S}\to E$, i.e. such that there is a homotopy fiber sequence (equivalently a homotopy cofiber sequence, prop.) in the stable homotopy category $Ho(Spectra)$ of the form

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

equivalently an exact triangle (prop.) of the form

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

(Adams 74, theorem 15.1 page 319) Beware that for instance (Hopkins 99, proof of corollary 5.3) uses “$\overline{E}$” not for the homotopy fiber of $\mathbb{S} \overset{e}{\to} E$ but for its homotopy cofiber, hence for what is $\Sigma \overline{E}$ according to (Adams 74).

Lemma

In terms of def. , the spectra entering the definition of the $E$-Adams spectral sequence in def. are equivalently

$$Y_p \;\simeq\; \overline{E}^p \wedge Y$$

and

$$A_p \;\simeq\; E \wedge \overline{E}^p \wedge Y \,.$$

where we write

$$\overline{E}^p \coloneqq \underset{ p\; factors }{ \underbrace{ \overline{E} \wedge \cdots \wedge \overline{E} }} \wedge Y \,.$$

Hence the first page of the $E$-Adams spectral sequence reads equivalently

$$E^{s,t}_1(X,Y) \simeq [X, E \wedge {\overline{E}^s} \wedge Y]_{t-s} \,.$$

(Adams 74, theorem 15.1 page 319)

Proof

By definition the statement holds for $p = 0$. Assume then by induction that it holds for some $p \geq 0$. Since the smash product of spectra-functor $(-) \wedge \overline{E}^p \wedge Y$ preserves homotopy cofiber sequences (lemma, this is part of the tensor triangulated structure of the stable homotopy category), its application to the homotopy cofiber sequence

$$\overline{E} \longrightarrow \mathbb{S} \overset{e}{\longrightarrow} E$$

from def. yields another homotopy cofiber sequence, now of the form

$$\overline{E}^{p+1} \wedge Y \longrightarrow \overline{E}^p \wedge Y \overset{g_p}{\longrightarrow} E \wedge \overline{E}^p \wedge Y$$

where the morphism on the right is identified as $g_p$ by the induction assumption, hence $A_{p+1}\simeq E \wedge \overline{E}^p \wedge Y$. Since $Y_{p+1}$ is defined to be the homotopy fiber of $g_p$, it follows that $Y_{p+1} \simeq \overline{E}^{p+1} \wedge Y$.

Remark

Terminology differs across authors. The filtration in def. in the rewriting by lemma is due to (Adams 74, theorem 15.1), where it is not give any name. In (Ravenel 84, p. 356) it is called the (canonical) Adams tower while in (Ravenel 86, def. 2.21) it is called the canonical Adams resolution. Several authors follow the latter usage, for instance (Rognes 12, def. 4.1). But (Hopkins 99) uses “Adams resolution” for the “$E$-injective resolutions” that we discuss below and uses “Adams tower” for yet another concept, def. below. See also remark .

We proceed now to analyze the first two pages and then the convergence properties of $E$-Adams spectral sequences of def. .

The first page

By lemma the first page of an $E$-Adams spectral sequence (def. ) looks like

$$\begin{aligned} E^{s,t}_1(X,Y) & \simeq [X, E \wedge \overline{E}^s \wedge Y]_{s-t} \end{aligned} \,.$$

We discuss now how, under favorable conditions, these hom-groups 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 $E_\bullet(E)$ (The $E$-generalized homology of $E$ (rmk.)). Then below we discuss that, as a result, the $d_1$-cohomology of the first page computes the Ext-groups from the $E$-homology of $Y$ to the $E$-homology of $X$, regarded as $E_\bullet(E)$-comodules.

The condition needed for this to work is the following.

Flat homotopy ring spectra

Definition

Call a homotopy commutative ring spectrum $(E,\mu,e)$ (def.) flat if the canonical right $\pi_\bullet(E)$-module structure on $E_\bullet(E)$ (prop.) (equivalently the canonical left module struture, see prop. below) is a flat module.

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

Proposition

Let $(E,\mu,e)$ be a homotopy commutative ring spectrum (def.) and let $X \in Ho(Spectra)$ be any spectrum. Then there is a homomorphism of graded abelian groups of the form

$$E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(X) \longrightarrow [\mathbb{S}, E \wedge E \wedge X]_\bullet = \pi_\bullet(E \wedge E \wedge X)$$

(for $E_\bullet(-)$ the canonical $\pi_\bullet(E)$-modules from this prop.) given on elements

$$\Sigma^{n_1}\mathbb{S} \overset{\alpha_1}{\longrightarrow} E \wedge E \;\;\,, \;\; \Sigma^{n_2} \mathbb{S} \overset{\alpha_2}{\longrightarrow} E \wedge X$$

by

$$\alpha_1 \cdot \alpha_2 \;\colon\; \Sigma^{n_1 + n_2}\mathbb{S} \overset{\simeq}{\longrightarrow} \Sigma^{n_1} \mathbb{S} \wedge \Sigma^{n_2}\mathbb{S} \overset{\alpha_1 \wedge \alpha_2}{\longrightarrow} E \wedge E \wedge E \wedge X \overset{id_E \wedge \mu \wedge id_X}{\longrightarrow} E \wedge E \wedge X \,.$$

If $E_\bullet(E)$ is a flat module over $\pi_\bullet(E)$ then this is an isomorphism.

(Adams 69, lecture 3, lemma 1 (p. 68), Adams 74, part III, lemma 12.5)

Proof

First of all, that the given pairing is a well defined homomorphism (descends from $E_\bullet(E) \times E_\bullet(X)$ to $E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(X)$) follows from the associativity of $\mu$.

We discuss that it is an isomorphism when $E_\bullet(E)$ is flat over $\pi_\bullet(E)$:

First consider the case that $X \simeq \Sigma^{n} \mathbb{S}$ is a suspension of the sphere spectrum. Then (by this example, using the tensor triangulated stucture on the stable homotopy category (prop.))

$$E_\bullet(X) = E_\bullet(\Sigma^n X) \simeq \pi_{\bullet-n}(E)$$

and

$$\pi_\bullet(E \wedge E \wedge X) = \pi_\bullet(E \wedge E \wedge \Sigma^n \mathbb{S}) \simeq E_{\bullet-n}(E)$$

and

$$E_\bullet(E) \otimes_{\pi_\bullet(E)} \pi_{\bullet-n}(E) \simeq E_{\bullet-n}(E)$$

Therefore in this case we have an isomorphism for all $E$.

For general $X$, we may without restriction assume that $X$ is represented by a sequential CW-spectrum (prop.). Then the homotopy cofibers of its cell attachment maps are suspensions of the sphere spectrum (rmk.).

First consider the case that $X$ is a CW-spectrum with finitely many cells. Consider the homotopy cofiber sequence of the $(k+1)$st cell attachment (by that remark):

$$\array{ \Sigma^{n_k-1}\mathbb{S} &\longrightarrow& X_k &\longrightarrow& X_{k+1} &\longrightarrow& \Sigma^{n_k}\mathbb{S} &\longrightarrow& \Sigma X_k }$$

and its image under the natural morphism $E_\bullet(E) \otimes_{\pi_\bullet(E)}E_\bullet(-) \to \pi_\bullet([\mathbb{S}, E \wedge E \wedge (-)])$, which is a commuting diagram of the form

$$\array{ E_\bullet(E) \otimes_{\pi_\bullet(E)}E_\bullet(\Sigma^{n_k-1}\mathbb{S}) &\longrightarrow& E_\bullet(E) \otimes_{\pi_\bullet(E)}E_\bullet(X_k) &\longrightarrow& E_\bullet(E) \otimes_{\pi_\bullet(E)}E_\bullet(X_{k+1}) &\longrightarrow& E_\bullet(E) \otimes_{\pi_\bullet(E)}E_\bullet(\Sigma^{n_k}\mathbb{S}) &\longrightarrow& E_\bullet(E) \otimes_{\pi_\bullet(E)}E_\bullet(\Sigma X_k) \\ \downarrow && \downarrow && \downarrow && \downarrow && \downarrow \\ [\mathbb{S}, E \wedge E \wedge \Sigma^{n_k-1}\mathbb{S}]_\bullet &\longrightarrow& [\mathbb{S}, E \wedge E \wedge X_k]_{\bullet} &\longrightarrow& [\mathbb{S}, E \wedge E \wedge X_{k+1}]_{\bullet} &\longrightarrow& [\mathbb{S}, E \wedge E \wedge \Sigma^{n_k}\mathbb{S}]_{\bullet} &\longrightarrow& [\mathbb{S}, E \wedge E \wedge \Sigma X_k]_{\bullet} } \,.$$

Here the bottom row is a long exact sequence since $E \wedge E \wedge (-)$ preserves homotopy cofiber sequences (by this lemma, part of the tensor triangulated structure on $Ho(Spectra)$ prop.), and since $[\mathbb{S},-]_\bullet \simeq \pi_\bullet(-)$ sends homotopy cofiber sequences to long exact sequences (prop.). By the same reasoning, $E_\bullet(-)$ of the homotopy cofiber sequence is long exact; and by the assumption that $E_\bullet(E)$ is flat, the functor $E_\bullet(E)\otimes_{\pi_\bullet(E)}(-)$ preserves this exactness, so that also the top row is a long exact sequence.

Now by induction over the cells of $X$, the outer four vertical morphisms are isomorphisms. Hence the 5-lemma implies that also the middle morphism is an isomorphism.

This shows the claim inductively for all finite CW-spectra. For the general statement, now use that

1. every CW-spectrum is the filtered colimit over its finite CW-subspectra;

2. the symmetric monoidal smash product of spectra $\wedge$ (def.) preserves colimits in its arguments separately (since it has a right adjoint (prop.));

3. $[\mathbb{S},-]_\bullet \simeq \pi_\bullet(-)$ commutes over filtered colimits of CW-spectrum inclusions (by this lemma, since spheres are compact);

4. $E_\bullet(E) \otimes_{\pi_\bullet(E)}(-)$ distributes over colimits (it being a left adjoint).

Using prop. , we find below (theorem ) that the first page of the $E$-Adams spectral sequence may be equivalently rewritten as hom-groups of comodules over $E_\bullet(E)$ regarded as a graded commutative Hopf algebroid. We now first discuss what this means.

The $E$-Steenrod algebra

We discuss here all the extra structure that exists on the $E$-self homology $E_\bullet(E)$ of a flat homotopy commutative ring spectrum. For $E = H \mathbb{F}_p$ the Eilenberg-MacLane spectrum on a prime field this reduces to the classical structure in algebraic topology called the dual Steenrod algebra $\mathcal{A}^\ast_p$. Therefore one may generally speak of $E_\bullet(E)$ as being the dual $E$-Steenrod algebra.

Without the qualifier “dual” then “$E$-Steenrod algebra” refers to the $E$-self-cohomology $E^\bullet(E)$. For $E = H \mathbb{F}_p$ this Steenrod algebra $\mathcal{A}_p$ (without “dual”) is traditionally considered first, and the classical Adams spectral sequence was originally formulated in terms of $\mathcal{A}_p$ instead of $\mathcal{A}_p^\ast$. But one observes (Adams 74, p. 280) that the “dual” Steenrod algebra $E_\bullet(E)$ is much better behaved, at least as long as $E$ is flat in the sense of def. .

Moreover, the dual $E$-Steenrod algebra $E_\bullet(E)$ is more fundamental in that it reflects a stacky geometry secretly underlying the $E$-Adams spectral sequence (Hopkins 99). This is the content of the concept of “commutative Hopf algebroid” (def. below) which is equivalently the formal dual of a groupoid internal to affine schemes, def. .

A simple illustrative archetype of the following construction of commutative Hopf algebroids from homotopy commutative ring spectra is the following situation:

For $X$ a finite set consider

$$\array{ X \times X \times X \\ \downarrow^{\mathrlap{\circ = (pr_1, pr_3)}} \\ X \times X \\ {}^{\mathllap{s = pr_1}}\downarrow \uparrow \downarrow^{\mathrlap{t = pr_2}} \\ X }$$

as the (“codiscrete”) groupoid with $X$ as objects and precisely one morphism from every object to every other. Hence the composition operation $\circ$, and the source and target maps are simply projections as shown. The identity morphism (going upwards in the above diagram) is the diagonal.

Then consider the image of this structure under forming the free abelian groups $\mathbb{Z}[X]$, regarded as commutative rings under pointwise multiplication.

Since

$$\mathbb{Z}[X \times X] \simeq \mathbb{Z}[X] \otimes \mathbb{Z}[X]$$

this yields a diagram of homomorphisms of commutative rings of the form

$$\array{ (\mathbb{Z}[X] \otimes \mathbb{Z}[X] ) \otimes_{\mathbb{Z}[X]} (\mathbb{Z}[X] \otimes \mathbb{Z}[X]) \\ \uparrow^{\mathrlap{} } \\ \mathbb{Z}[X] \otimes \mathbb{Z}[X] \\ \uparrow \downarrow \uparrow \\ \mathbb{Z}[X] }$$

satisfying some obvious conditions. Observe that here

1. the two morphisms $\mathbb{Z}[X] \rightrightarrows \mathbb{Z}[X] \otimes \mathbb{Z}[X]$ are $f \mapsto f \otimes e$ and $f \mapsto e \otimes f$, respectively, where $e$ denotes the unit element in $\mathbb{Z}[X]$;

2. the morphism $\mathbb{Z}[X] \otimes \mathbb{Z}[X] \to \mathbb{Z}[X]$ is the multiplication in the ring $\mathbb{Z}[X]$;

3. the morphism

   $$\mathbb{Z}[X] \otimes \mathbb{Z}[X] \longrightarrow \mathbb{Z}[X] \otimes \mathbb{Z}[C] \otimes \mathbb{Z}[C] \overset{\simeq}{\longrightarrow} (\mathbb{Z}[X] \otimes \mathbb{Z}[X] ) \otimes_{\mathbb{Z}[X]} (\mathbb{Z}[X] \otimes \mathbb{Z}[X])$$

   is given by $f \otimes g \mapsto f \otimes e \otimes g$.

All of the following rich structure is directly modeled on this simplistic example. We simply

1. replace the commutative ring $\mathbb{Z}[X]$ with any flat homotopy commutative ring spectrum $E$,

2. replace tensor product of abelian groups $\otimes$ with derived smash product of spectra;

3. and form the stable homotopy groups $\pi_\bullet(-)$ of all resulting expressions.

Definition

Let $(E, \mu, e)$ be a homotopy commutative ring spectrum (def.) which is flat according to def. .

Then the dual $E$-Steenrod algebra is the pair of graded abelian groups

$$(E_\bullet(E), \pi_\bullet(E))$$

(rmk.) equipped with the following structure:

1. the graded commutative ring structure

   $$\pi_\bullet(E) \otimes \pi_\bullet(E) \longrightarrow \pi_\bullet(E)$$

   induced from $E$ being a homotopy commutative ring spectrum (prop.);

2. the graded commutative ring structure

   $$E_\bullet(E) \otimes E_\bullet(E) \longrightarrow E_\bullet(E)$$

   induced from the fact that with $E$ also $E \wedge E$ is canonically a homotopy commutative ring spectrum (exmpl.), so that also $E_\bullet(E) = \pi_\bullet(E \wedge E)$ is a graded commutative ring (prop.);

3. the homomorphism of graded commutative rings

   $$\Psi \;\colon\; E_\bullet(E) \longrightarrow E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(E)$$

   induced under $\pi_\bullet(-)$ from

   $$E \wedge E \overset{id \wedge e \wedge id}{\longrightarrow} E \wedge E \wedge E$$

   via prop. ;

4. the homomorphisms of graded commutative rings

   $$\eta_L \;\colon\; \pi_\bullet(E) \longrightarrow E_\bullet(E)$$

   and

   $$\eta_R \;\colon\; \pi_\bullet(E) \longrightarrow E_\bullet(E)$$

   induced under $\pi_\bullet(-)$ from the homomorphisms of commutative ring spectra

   $$E \underoverset{\simeq}{r_{E}^{-1}}{\to} E \wedge \mathbb{S} \overset{id \wedge e}{\longrightarrow} E \wedge E$$

   and

   $$E \underoverset{\simeq}{\ell_E^{-1}}{\to} \mathbb{S} \wedge E \overset{id \wedge e}{\longrightarrow} E \wedge E \,,$$

   respectively (exmpl.);

5. the homomorphism of graded commutative rings

   $$\epsilon \;\colon\; E_\bullet(E) \longrightarrow \pi_\bullet(E)$$

   induced under $\pi_\bullet(-)$ from

   $$\mu \;\colon\; E \wedge E \longrightarrow E$$

   regarded as a homomorphism of homotopy commutative ring spectra (exmpl.);

6. the homomorphisms graded commutative rings

   $$c \;\colon\; E_\bullet(E) \longrightarrow E_\bullet(E)$$

   induced under $\pi_\bullet(-)$ from the braiding

   $$\tau_{E,E} \;\colon\; E \wedge E \longrightarrow E \wedge E$$

   regarded as a homomorphism of homotopy commutative ring spectra (exmpl.).

(Adams 69, lecture 3, pages 66-68)

Notice that (as verified by direct unwinding of the definitions):

Lemma

For $(E, \mu, e)$ a homotopy commutative ring spectrum (def.), consider $E_\bullet(E)$ with its canonical left and right $\pi_\bullet(E)$-module structure as in this prop.. These module structures coincide with those induced by the ring homomorphisms $\eta_L$ and $\eta_R$ from def. .

These two actions need not strictly coincide, but they are isomorphic:

Proposition

For $(E, \mu, e)$ a homotopy commutative ring spectrum (def.), consider $E_\bullet(E)$ with its canonical left and right $\pi_\bullet(E)$-module structure (prop.). Since $E$ is a commutative monoid, this right module structure may equivalently be regarded as a left-module, too. Then the braiding

$$E_\bullet(E) \simeq \pi_\bullet(E \wedge E) \overset{\pi_\bullet(\tau_{E,E})}{\longrightarrow} \pi_\bullet(E \wedge E) \simeq E_\bullet(E)$$

constitutes a module isomorphism (def.) between these two left module structures.

Proof

On representatives as in the proof of (this propo.), the original left action is given by (we are notationally suppressing associators throughout)

$$E \wedge E \wedge E \overset{\mu \wedge id}{\longrightarrow} E \wedge E \,,$$

while the other left action, induced from the canonical right action, is given by

$$E \wedge E \wedge E \underoverset{\simeq}{\tau_{E, E \wedge E}}{\longrightarrow} E \wedge E \wedge E \overset{id \wedge \mu}{\longrightarrow} E \wedge \,.$$

So in order that $\tau_{E,E}$ represents a module homomorphism under $\pi_\bullet(-)$, it is sufficient that the following diagram commutes (we write $E_i \coloneqq E$ for $i \in \{1,2,3\}$ to make the action of the braiding more manifest)

$$\array{ E_1 \wedge E_2 \wedge E_3 &\overset{id \wedge \tau_{E_2,E_3}}{\longrightarrow}& E_1 \wedge E_3 \wedge E_2 \\ {}^{\mathllap{id}}\downarrow && \downarrow^{\mathrlap{\tau_{E_1, E_3 \wedge E_2}}} \\ && E_3 \wedge E_2 \wedge E_1 \\ {}^{\mathllap{\mu \wedge id}}\downarrow && \downarrow^{\mathrlap{id \wedge \mu}} \\ E \wedge E_3 &\underset{\tau_{E,E_3}}{\longrightarrow}& E_3 \wedge E } \,.$$

But since $(E,\mu,e)$ is a commutative monoid (def.), it satisfies $\mu = \mu \circ \tau$ so that we may factor this diagram as follows:

$$\array{ E_1 \wedge E_2 \wedge E_3 &\overset{id \wedge \tau_{E_2,E_3}}{\longrightarrow}& E_1 \wedge E_3 \wedge E_2 \\ {}^{\mathllap{\tau_{E_1, E_2} \wedge id}}\downarrow && \downarrow^{\mathrlap{\tau_{E_1, E_3 \wedge E_2}}} \\ E_2 \wedge E_1 \wedge E_3 &\overset{\tau_{E_2 \wedge E_1, E_3}}{\longrightarrow}& E_3 \wedge E_2 \wedge E_1 \\ {}^{\mathllap{\mu \wedge id}}\downarrow && \downarrow^{\mathrlap{id \wedge \mu}} \\ E \wedge E_3 &\underset{\tau_{E,E_3}}{\longrightarrow}& E_3 \wedge E } \,.$$

Here the top square commutes by coherence of the braiding (rmk) since both composite morphisms correspond to the same permutation, while the bottom square commutesm due to the naturality of the braiding. Hence the total rectangle commutes.

The dual $E$-Steenrod algebras of def. evidently carry a lot of structure. The concept organizing this is that of_commutative Hopf algebroids.

Definition

A graded commutative Hopf algebroid is an internal groupoid in the opposite category $gCRing^{op}$ of $\mathbb{Z}$-graded commutative rings, regarded with its cartesian monoidal category structure.

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

Remark

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

An internal category in $gCRing^{op}$ is a diagram in $gCRing^{op}$ 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 in order to express this equivalently in terms of algebra 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 $gCRing$ 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. below. Here

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

An internal groupoid $\mathcal{G}_1 \stackrel{\overset{s}{\longrightarrow}}{\underset{t}{\longrightarrow}} \mathcal{G}_0$ 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 yields the following equivalent but maybe more explicit definition of commutative Hopf algebroids, def.

Definition

A commutative Hopf algebroid is

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. (identity morphisms respect source and target)

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

   2. (identity morphisms are units for composition)

      $(id_\Gamma \otimes_A \epsilon) \circ \Psi = (\epsilon \otimes_A id_\Gamma) \circ \Psi = id_\Gamma$;

   3. (composition respects source and target)

      1. $\Psi \circ \eta_R = (id \otimes_A \eta_R) \circ \eta_R$;

      2. $\Psi \circ \eta_L = (\eta_L \otimes_A id) \circ \eta_L$

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

3. (inverses)

   1. (inverting twice is the identity)

      $c \circ c = id_\Gamma$;

   2. (inversion swaps source and target)

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

   3. (inverse morphisms are indeed left and right inverses for composition)

      the morphisms $\alpha$ and $\beta$ induced via the coequalizer property of the tensor product from $(-) \cdot c(-)$ and $c(-)\cdot (-)$, respectively

      $$\array{ \Gamma \otimes A \otimes \Gamma & \underoverset {\longrightarrow} {\longrightarrow} {} & \Gamma \otimes \Gamma & \overset{coeq}{\longrightarrow} & \Gamma \otimes_A \Gamma \\ && {}_{\mathllap{(-)\cdot c(-)}}\downarrow & \swarrow_{\mathrlap{\alpha}} \\ && \Gamma }$$

      and

      $$\array{ \Gamma \otimes A \otimes \Gamma & \underoverset {\longrightarrow} {\longrightarrow} {} & \Gamma \otimes \Gamma & \overset{coeq}{\longrightarrow} & \Gamma \otimes_A \Gamma \\ && {}_{\mathllap{c(-)\cdot (-)}}\downarrow & \swarrow_{\mathrlap{\beta}} \\ && \Gamma }$$

      satisfy

      $\alpha \circ \Psi = \eta_L \circ \epsilon$

      and

      $\beta \circ \Psi = \eta_R \circ \epsilon$.

(Adams 69, lecture 3, pages 62-66, Ravenel 86, def. A1.1.1)

Remark

In (Adams 69, lecture 3, page 60) the terminology used is “Hopf algebra in a fully satisfactory sense” with emphasis that the left and right module structure may differ. According to (Ravenel 86, first page of appendix A1) the terminology “Hopf algebroid” for this situation is due to Haynes Miller.

Example

For $R$ a commutative ring, then $R \otimes R$ becomes a commutative Hopf algebroid over $R$, formally dual (via def. ) to the pair groupoid on $Spec(R) \in CRing^{op}$.

For $X$ a finite set and $R = \mathbb{Z}[X]$, then this reduces to the motivating example from above.

It is now straightforward, if somewhat tedious, to check that:

Proposition

Let $(E, \mu, e)$ be a homotopy commutative ring spectrum (def.) which is flat according to def. , then the dual $E$-Steenrod algebra $(E_\bullet(E), \pi_\bullet(E))$ with the structure maps $(\eta_L, \eta_R, \epsilon, c, \Psi)$ from prop. is a graded commutative Hopf algebroid according to def. :

$$(E_\bullet(E), \pi_\bullet(E)) \;\in\; CommHopfAlgd$$

(Adams 69, lecture 3, pages 67-71, Ravenel 86, chapter II, prop. 2.2.8)

Proof

One observes that $E \wedge E$ satisfies the axioms of a commutative Hopf algebroid object in homotopy commutative ring spectra, over $E$, by direct analogy to example (one just has to verify that the symmetric braidings go along coherently, which works by use of the coherence theorem for symmetric monoidal categories (rmk.)). Applying the functor $\pi_\bullet(-)$ that forms stable homotopy groups to all structure morphisms of $E \wedge E$ yields the claimed structure morphisms of $E_\bullet(E)$.

We close this subsection on commutative Hopf algebroids by discussion of their isomorphism classes, when regarded dually as affine groupoids:

Definition

Given an internal groupoid in $gCRing^{op}$ (def. , remark )

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

then its affife scheme $Spec(A)_{/\sim}$ of isomorphism classes of objects is the coequlizer? of the source and target morphisms

$$Spec(\Gamma) \underoverset {\underset{t}{\longrightarrow}} {\overset{s}{\longleftarrow}} {\phantom{AA}} Spec(A) \overset{equ}{\longrightarrow} Spec(A)_{/\sim} \,.$$

Hence this is the formal dual of the equalizer of the left and right unit (def. )

$$A \underoverset {\underset{\eta_R}{\longrightarrow}} {\overset{\eta_L}{\longrightarrow}} {\phantom{AA}} \Gamma \,.$$

By example every commutative ring gives rise to a commutative Hopf algebroid $R \otimes R$ over $R$. The core of $R$ is the formal dual of the corresponding affine scheme of isomorphism classes according to def. :

Definition

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

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

A ring which is isomorphic to its core is called a solid ring.

(Bousfield-Kan 72, §1, def. 2.1, Bousfield 79, 6.4)

Proposition

The core of any ring $R$ is solid (def. ):

$$c c R \simeq c R \,.$$

(Bousfield-Kan 72, prop. 2.2)

Proposition

The following is the complete list of solid rings (def. ) up to isomorphism:

1. The localization of the ring of integers at a set $J$ of prime numbers (def. )

   $$\mathbb{Z}[J^{-1}] \,;$$

2. the integers modulo n

   $$\mathbb{Z}/n\mathbb{Z}$$

   for $n \geq 2$;

3. the product rings

   $$\mathbb{Z}[J^{-1}] \times \mathbb{Z}/n\mathbb{Z} \,,$$

   for $n \geq 2$ such that each prime factor of $n$ is contained in the set of primes $J$;

4. the ring cores of product rings

   $$c(\mathbb{Z}[J^{-1}] \times \underset{p \in K}{\prod} \mathbb{Z}/p^{e(p)}) \,,$$

   where $K \subset J$ are infinite sets of primes and $e(p)$ are positive natural numbers.

(Bousfield-Kan 72, prop. 3.5, Bousfield 79, p. 276)

Comodules over the $E$-Steenrod algebra

Definition

Let $(E, \mu, e)$ be a homotopy commutative ring spectrum (def.) which is flat according to def. .

For $X \in Ho(Spectra)$ any spectrum, say that the comodule structure on $E_\bullet(X)$ (rmk.)) over the dual $E$-Steenrod algebra (def. ) is

1. the canonical structure of a $\pi_\bullet(E)$-module (according to this prop.);

2. the homomorphism of $\pi_\bullet(E)$-modules

   $$\Psi_{E_\bullet(X)} \;\colon\; E_\bullet(X) \longrightarrow E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(X)$$

   induced under $\pi_\bullet(-)$ and via prop. from the morphism of spectra

   $$E \wedge X \simeq E \wedge \mathbb{S} \wedge X \overset{id \wedge e \wedge id}{\longrightarrow} E \wedge E \wedge X \,.$$

Definition

Given a graded commutative Hopf algebroid $\Gamma$ over $A$ as in def. , hence an internal groupoid in $gCRing^{op}$, 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 $\mathbb{Z}$-graded commutative Hopf algebroid $\Gamma$ over $A$ (def. ) then a left comodule over $\Gamma$ is

1. a $\mathbb{Z}$-graded $A$-module $N$;

2. (co-action) a homomorphism of graded $A$-modules

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

such that

1. (co-unitality)

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

2. (co-action property)

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

A homomorphism between graded comodules $f \colon N_1 \to N_2$ is a homomorphism of underlying graded $A$-modules such that the following diagram commutes

$$\array{ N_1 &\overset{f}{\longrightarrow}& N_1 \\ {}^{\mathllap{\Psi_{N_1}}}\downarrow && \downarrow^{\mathrlap{\Psi_{N_2}}} \\ \Gamma \otimes_A N_1 &\underset{id \otimes_A f}{\longrightarrow}& \Gamma \otimes_A N_2 } \,.$$

We write

$$\Gamma CoMod$$

for the resulting category of left comodules over $\Gamma$. Analogously for right comodules. The notation for the hom-sets in this category is abbreviated to

$$Hom_\Gamma(-,-) \coloneqq Hom_{\Gamma CoMod}(-,-) \,.$$

A priori this is an Ab-enriched category, but it is naturally further enriched in graded abelian groups:

we may drop in the above definition of comodule homomorphisms $f\colon N_1 \to N_2$ the condition that the underlying morphism be grading-preserving. Say that $f$ has degree $n$ if it increases degree by $n$. This gives a $\mathbb{Z}$-graded hom-group

$$Hom^\bullet_\Gamma(-,-) \,.$$

Example

For $(\Gamma,A)$ a commutative Hopf algebroid, then $A$ becomes a left $\Gamma$-comodule (def. ) with coaction given by the right unit

$$A \overset{\eta_R}{\longrightarrow} \Gamma \simeq \Gamma \otimes_A A \,.$$

Proof

The required co-unitality property is the dual condition in def.

$$\epsilon \circ \eta_R = id_A$$

of the fact in def. that identity morphisms respect sources:

$$id \;\colon\; A \overset{\eta_R}{\longrightarrow} \Gamma \simeq \Gamma \otimes_A A \overset{\epsilon \otimes_A id}{\longrightarrow} A \otimes_A A \simeq A$$

The required co-action property is the dual condition

$$\Psi \circ \eta_R = (id \otimes_A \eta_R) \circ \eta_R$$

of the fact in def. that composition of morphisms in a groupoid respects sources

$$\array{ A &\overset{\eta_R}{\longrightarrow}& \Gamma \\ {}^{\mathllap{\eta_R}}\downarrow && \downarrow^{\mathrlap{\Psi}} \\ \Gamma \simeq \Gamma \otimes_A A &\underset{id \otimes_A \eta_R}{\longrightarrow}& \Gamma \otimes_A \Gamma } \,.$$

Proposition

Let $(E, \mu, e)$ be a homotopy commutative ring spectrum (def.) which is flat according to def. , and for $X \in Ho(Spectra)$ any spectrum, then the morphism $\Psi_{E_\bullet(X)}$ from def. makes $E_\bullet(X)$ into a comodule (def. ) over the dual $E$-Steenrod algebra (def. )

$$E_\bullet(X) \;\in\; E_\bullet(E) CoMod \,.$$

(Adams 69, lecture 3, pages 67-71, Ravenel 86, chapter II, prop. 2.2.8)

Example

Given a commutative Hopf algebroid $\Gamma$ over $A$, def. , then $A$ itself becomes a left $\Gamma$-comodule (def. ) 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

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

More generally:

Proposition

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

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

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

Moreover:

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

2. The adjunct $\tilde f$ of a comodule homomorphism

   $$N \overset{f}{\longrightarrow} \Gamma \otimes_A C$$

   is its composite with the counit $\epsilon$ of $\Gamma$

   $$\tilde f \;\colon\; N \overset{f}{\longrightarrow} \Gamma \otimes_A C \overset{\epsilon \otimes_A id}{\longrightarrow} A \otimes_A C \simeq C \,.$$

The proof is formally dual to the proof that shows that constructing free modules is left adjoint to the forgetful functor from a category of modules to the underlying monoidal category (prop.). But since the details of the adjunction isomorphism are important for the following discussion, we spell it out:

Proof

A homomorphism into a co-free $\Gamma$-comodule is a morphism of $A$-modules of the form

$$f \;\colon\; N \longrightarrow \Gamma \otimes_A C$$

making the following diagram commute

$$\array{ N &\overset{f}{\longrightarrow}& \Gamma \otimes_A C \\ {}^{\mathllap{\Psi_N}}\downarrow && \downarrow^{\mathrlap{\Psi \otimes_A id}} \\ \Gamma \otimes_A N &\underset{id \otimes_A f}{\longrightarrow}& \Gamma \otimes_A \Gamma \otimes_A C } \,.$$

Consider the composite

$$\tilde f \;\colon\; N \overset{f}{\longrightarrow} \Gamma \otimes_A C \overset{\epsilon \otimes_A id}{\longrightarrow} A \otimes_A C \simeq C \,,$$

i.e. the “corestriction” of $f$ along the counit of $\Gamma$. By definition this makes the following square commute

$$\array{ \Gamma \otimes_A N &\overset{id \otimes_A f}{\longrightarrow}& \Gamma \otimes_A \Gamma \otimes_A C \\ {}^{\mathllap{=}}\downarrow && \downarrow^{\mathrlap{id \otimes_A \epsilon \otimes_A id}} \\ \Gamma \otimes_A N &\underset{id \otimes_A \tilde f}{\longrightarrow}& \Gamma \otimes_A C } \,.$$

Pasting this square onto the bottom of the previous one yields

$$\array{ N &\overset{f}{\longrightarrow}& \Gamma \otimes_A C \\ {}^{\mathllap{\Psi_N}}\downarrow && \downarrow^{\mathrlap{\Psi \otimes_A id}} \\ \Gamma \otimes_A N &\underset{id \otimes_A f}{\longrightarrow}& \Gamma \otimes_A \Gamma \otimes_A C \\ {}^{\mathllap{=}}\downarrow && \downarrow^{\mathrlap{id \otimes_A \epsilon \otimes_A id}} \\ \Gamma \otimes_A N &\underset{id \otimes_A \tilde f}{\longrightarrow}& \Gamma \otimes_A C } \,.$$

Now due to co-unitality, the right right vertical composite is the identity on $\Gamma \otimes_A C$. But this means by the commutativity of the outer rectangle that $f$ is uniquely fixed in terms of $\tilde f$ by the relation

$$f = (id \otimes_A f) \circ \Psi \,.$$

This establishes a natural bijection

$$\frac{ N \overset{f}{\longrightarrow} \Gamma \otimes_A C }{ N \overset{\tilde f}{\longrightarrow} C }$$

and hence the adjunction in question.

Proposition

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

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

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. .

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 . 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. , and given $N_1, N_2$ two left $\Gamma$-comodules (def. ), then their cotensor product (def. ) 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. , and given $N_1, N_2$ two left $\Gamma$-comodules (def. ), 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. ) 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 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.

Universal coefficient theorem

The key use of the Hopf coalgebroid structure of prop. for the present purpose is that it is extra structure inherited by morphisms in $E$-homology from morphisms of spectra. Namely forming $E$-homology $f_\ast \colon E_\bullet(X) \to E_\bullet(Y)$ of a morphism of a spectra $f \colon X \to Y$ does not just produce a morphism of $E$-homology groups

$$[X,Y]_\bullet \longrightarrow Hom_{Ab^{\mathbb{Z}}}(E_\bullet(X), E_\bullet(Y))$$

but in fact produces homomorphisms of comodules over $E_\bullet(E)$

$$\alpha \;\colon\; [X,Y]_\bullet \longrightarrow Hom_{E_\bullet(E)}(E_\bullet(X), E_\bullet(Y)) \,.$$

This is the statement of lemma below. The point is that $E_\bullet(E)$-comodule homomorphism are much more rigid than general abelian group homomorphisms and hence closer to reflecting the underlying morphism of spectra $f \colon X \to Y$.

In good cases such an approximation of homotopy by homology is in fact accurate, in that $\alpha$ is an isomorphism. In such a case (Adams 74, part III, section 13) speaks of a “universal coefficient theorem” (the coefficients here being $E$.)

One such case is exhibited by prop. below. This allows to equivalently re-write the first page of the $E$-Adams spectral sequence in terms of $E$-homology homomorphisms in theorem below.

Lemma

For $X,Y \in Ho(Spectra)$ any two spectra, the morphism (of $\mathbb{Z}$-graded abelian) generalized homology groups given by smash product with $E$ (rmk.)

$$\array{ \pi_\bullet(E \wedge -) &\colon& [X,Y]_\bullet &\longrightarrow& Hom^\bullet_{Ab^{\mathbb{Z}}}(E_\bullet(X), E_\bullet(Y)) \\ && (X \overset{f}{\longrightarrow} Y) &\mapsto& \left( E_\bullet(X) \overset{f_\ast}{\longrightarrow} E_\bullet(Y) \right) }$$

factors through the forgetful functor from $E_\bullet(E)$-comodule homomorphisms (def. ) over the dual $E$-Steenrod algebra (def. ):

$$\array{ && Hom^\bullet_{E_\bullet(E)}(E_\bullet(X), E_\bullet(Y)) \\ & {}^{\mathllap{\exists}}\nearrow & \downarrow^{\mathrlap{forget}} \\ [X,N]_\bullet &\underset{\pi_\bullet(E \wedge -)}{\longrightarrow}& Hom^\bullet_{Ab^{\mathbb{Z}}}(E_\bullet(X), E_\bullet(Y)) } \,,$$

where $E_\bullet(X)$ and $E_\bullet(Y)$ are regarded as $E$-Steenrod comodules according to def. , prop. .

Proof

By def. we need to show that for $X \overset{f}{\longrightarrow} Y$ a morphism in $Ho(Spectra)$ then the following diagram commutes

$$\array{ E_\bullet(X) &\overset{f_\ast}{\longrightarrow}& E_\bullet(Y) \\ {}^{\mathllap{\Psi_{E_\bullet(X)}}}\downarrow && \downarrow^{\mathrlap{\Psi_{E_\bullet(Y)}}} \\ E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(X) &\underset{id \otimes_{\pi_\bullet(E)} f_\ast }{\longrightarrow}& E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(Y) } \,.$$

By def. and prop. this is the image under foming stable homotopy groups $\pi_\bullet(-)$ of the following diagram in $Ho(Spectra)$:

$$\array{ E \wedge X &\overset{id \wedge f}{\longrightarrow}& E \wedge Y \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\simeq}} \\ E \wedge \mathbb{S} \wedge X && E \wedge \mathbb{S} \wedge Y \\ {}^{\mathllap{id \wedge e \wedge id}}\downarrow && \downarrow^{\mathrlap{id \wedge e \wedge id}} \\ E \wedge E \wedge X &\underset{id \wedge id \wedge f}{\longrightarrow}& E \wedge E \wedge Y } \,.$$

But that this diagram commutes is simply the functoriality of the derived smash product of spectra as a functor on the product category $Ho(Spectra) \times Ho(Spectra)$.

Proposition

Let $(E, \mu, e)$ be a homotopy commutative ring spectrum (def.), and let $X, Y \in Ho(Spectra)$ be two spectra such that $E_\bullet(X)$ is a projective module over $\pi_\bullet(E)$ (via this prop.).

Then the homomorphism of graded abelian groups

$$\phi_{UC} \;\colon\; [X, E \wedge Y]_\bullet \stackrel{}{\longrightarrow} Hom_{\pi_\bullet(E)}^\bullet(E_\bullet(X), E_\bullet(Y))_\bullet$$

given by

$$(X \overset{f}{\longrightarrow} E \wedge Y) \;\mapsto\; \pi_\bullet ( E \wedge X \overset{id \wedge f}{\longrightarrow} E \wedge E \wedge Y \overset{\mu \wedge id}{\longrightarrow} E \wedge Y )$$

is an isomorphism.

(Schwede 12, chapter II, prop. 6.20)

Proof

First of all we claim that the morphism in question factors as

$$\beta \;\colon\; [X, E \wedge Y]_\bullet \overset{\simeq}{\longrightarrow} Hom^\bullet_{E Mod}( E \wedge X , E \wedge Y) \overset{\pi_\bullet}{\longrightarrow} Hom^\bullet_{\pi_\bullet(E)}( E_\bullet(X), E_\bullet(Y) ) \,,$$

where

1. $E Mod = E Mod(Ho(Spectra), \wedge, \mathbb{S})$ denotes the category of homotopy module spectra over $E$ (def.)

2. the first morphisms is the free-forgetful adjunction isomorphism for forming free (prop.) $E$-homotopy module spectra

3. the second morphism is the respective component of the composite of the forgetful functor from $E$-homotopy module spectra back to $Ho(Spectra)$ with the functor $\pi_\bullet$ that forms stable homotopy groups.

This is because (by this prop.) the first map is given by first smashing with $E$ and then postcomposing with the $E$-action on the free module $E \wedge X$, which is the pairing $E \wedge E \overset{\mu}{\to} E$ (prop.).

Hence it is sufficient to show that the morphism on the right is an isomorphism.

We show more generally that for $N_1, N_2$ any two $E$-homotopy module spectra (def.) such that $\pi_\bullet(N_1)$ is a projective module over $\pi_\bullet(E)$, then

$$Hom^\bullet_{E Mod}( N_1 , N_2) \overset{\pi_\bullet}{\longrightarrow} Hom^\bullet_{\pi_\bullet(E)}( \pi_\bullet(N_1), \pi_\bullet(N_2) )$$

is an isomorphism.

To see this, first consider the case that $\pi_\bullet(N_1)$ is in fact a $\pi_\bullet(E)$-free module.

This implies that there is a basis $\mathcal{B} = \{x_i\}_{i \in I}$ and a homomorphism

$$\underset{i \in I}{\vee} \Sigma^{\vert x_i\vert} E \longrightarrow N_1$$

of $E$-homotopy module spectra, such that this is a stable weak homotopy equivalence.

Observe that this sits in a commuting diagram of the form

$$\array{ Hom^\bullet_{E Mod}( \underset{i \in I}{\vee} \Sigma^{\vert x_i\vert} E, N_2 ) &\overset{\pi_\bullet}{\longrightarrow}& Hom^\bullet_{\pi_\bullet(E)}(\pi_\bullet(\underset{i \in I}{\vee}\Sigma^{\vert x_i\vert} E) , \pi_\bullet(N_2) ) \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\simeq}} \\ \underset{i \in I}{\prod} [ \Sigma^{\vert x_i\vert}\mathbb{S}, N_2 ]_\bullet &\underset{\simeq}{\longrightarrow}& \underset{i \in I}{\prod} \pi_{\bullet + \vert x_i \vert}(N_2) }$$

where

1. the left vertical isomorphism exhibits wedge sum of spectra as the coproduct in the stable homotopy category (lemma);

2. the bottom isomorphism is from this prop.;

3. the right vertical isomorphism is that of the free-forgetful adjunction for modules over $\pi_\bullet(E)$.

Hence the top horizontal morphism is an isomorphism, which was to be shown.

Now consider the general case that $\pi_\bullet(N_1)$ is a projective module over $\pi_\bullet(E)$. Since (graded) projective modules are precisely the retracts of (graded) free modules (prop.), there exists a diagram of $\pi_\bullet(E)$-modules of the form

$$id \;\colon\; \pi_\bullet(N_1) \longrightarrow \pi_\bullet( \underset{i \in I}{\vee} \Sigma^{\vert x_i \vert} E ) \longrightarrow \pi_\bullet(N_1)$$

which induces the corresponding split idempotent of $\pi_\bullet(E)$-modules

$$\pi_\bullet( \underset{i \in I}{\vee} \Sigma^{\vert x_i \vert} E ) \longrightarrow \pi_\bullet(N_1) \longrightarrow \pi_\bullet( \underset{i \in I}{\vee} \Sigma^{\vert x_i \vert} E ) \,.$$

As before, by freeness this is actually the image under $\pi_\bullet$ of an idempotent of homotopy ring spectra

$$e_\bullet \;\colon\; \underset{i \in I}{\vee} \Sigma^{\vert x_i \vert} E \longrightarrow \underset{i \in I}{\vee} \Sigma^{\vert x_i \vert} E$$

and so in particular of spectra.

Now in the stable homotopy category $Ho(Spectra)$ all idempotents split (prop.), hence there exists a diagram of spectra of the form

$$e \;\colon\; \underset{i \in I}{\vee} \Sigma^{\vert x_i \vert} E \longrightarrow X \longrightarrow \underset{i \in I}{\vee} \Sigma^{\vert x_i \vert} E$$

with $\pi_\bullet(e) = e_\bullet$.

Consider the composite

$$X \longrightarrow \underset{i \in I}{\vee} \Sigma^{\vert x_i \vert} E \longrightarrow N_1 \,.$$

Since $\pi_\bullet(e) = e_\bullet$ it follows that under $\pi_\bullet$ this is an isomorphism, then that $X \simeq N_1$ in the stable homotopy category.

In conclusion this exhibits $N_1$ as a retract of an free $E$-homotopy module spectrum

$$id \;\colon\; N_1 \longrightarrow \underset{i \in I}{\vee} \Sigma^{\vert x_i \vert} E \longrightarrow N_1 \,,$$

hence of a spectrum for which the morphism in question is an isomorphism. Since the morphism in question is natural, its value on $N_1$ is a retract in the arrow category of an isomorphism, hence itself an isomorphism (lemma).

Remark

A stronger version of the statement of prop. , with the free homotopy $E$-module spectrum $E \wedge Y$ replaced by any homotopy $E$-module spectrum $F$, is considered in (Adams 74, chapter III, prop. 13.5) (“universal coefficient theorem”). Strong conditions are considered that ensure that

$$F^\bullet(X) = [X,F]_\bullet \longrightarrow Hom^\bullet_{\pi_\bullet(E)}(E_\bullet(X), \pi_\bullet(F))$$

is an isomormphism (expressing the $F$-cohomology of $X$ as the $\pi_\bullet(E)$-linear dual of the $E$-homology of $X$).

For the following we need only the weaker but much more general statement of prop. , and in fact this is all that (Adams 74, p. 323) ends up using, too.

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

Proposition

Let $(E, \mu, e)$ be a homotopy commutative ring spectrum (def.), and let $X, Y \in Ho(Spectra)$ be two spectra such that

1. $E$ is flat according to def. ;

2. $E_\bullet(X)$ is a projective module over $\pi_\bullet(E)$ (via this prop.).

Then the morphism from lemma

$$[X, E \wedge Y]_\bullet \stackrel{\pi_\bullet(E \wedge -)}{\longrightarrow} Hom_{E_\bullet(E)}^\bullet(E_\bullet(X), E_\bullet( E \wedge Y))) \simeq Hom_{E_\bullet(E)}^\bullet(E_\bullet(X), E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(Y) ))$$

is an isomorphism (where the isomophism on the right is that of prop. ).

(Adams 74, part III, page 323)

Proof

Observe that the following diagram commutes:

$$\array{ [X, E \wedge Y]_\bullet && \overset{\pi_\bullet(E \wedge -)}{\longrightarrow} && Hom_{E_\bullet(E)}^\bullet(E_\bullet(X), E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(Y) )) \\ & {}_{\mathllap{\phi_{UC}}}\searrow && \swarrow_{\mathrlap{\epsilon \otimes id \circ (-) }} \\ && Hom^\bullet_{\pi_\bullet(E)}(E_\bullet(X), E_\bullet(Y)) } \,,$$

where

1. the top morphism is the one from lemma ;

2. the right vertical morphism is the adjunction isomorphism from prop. ;

3. the left diagonal morphism is the one from prop. .

To see that this indeed commutes, notice that

1. the top morphism sends $(X \overset{f}{\to} E \wedge Y)$ to $E_\bullet(X) \overset{E_\bullet(f)}{\to} E_\bullet(E \wedge Y) \simeq \pi_\bullet(E \wedge E \wedge Y)$ by definition;

2. the right vertical morphism sends this further to $E_\bullet(X) \overset{E_\bullet(f)}{\to} \pi_\bullet(E \wedge E \wedge Y) \overset{\pi_\bullet(\mu \wedge id)}{\to} \pi_\bullet(E \wedge Y)$, by the proof of prop. (which says that the map is given by postcomposition with the counit of $E_\bullet(E)$) and def. (which says that this counit is represented by $\mu$);

3. by prop. this is the same as the action of the left diagonal morphism.

But now

1. the right vertical morphism is an isomorphism by prop. ;

2. the left diagonal morphism is an isomorphism by prop.

and so it follows that the top horizontal morphism is an isomorphism, too.

In conclusion:

Theorem

Let $(E, \mu, e)$ be a homotopy commutative ring spectrum (def.), and let $X, Y \in Ho(Spectra)$ be two spectra such that

1. $E$ is flat according to def. ;

2. $E_\bullet(X)$ is a projective module over $\pi_\bullet(E)$ (via this prop.).

Then the first page of the $E$-Adams spectral sequence, def. , for $[Y,X]_\bullet$ is isomorphic to the following chain complex of graded homs of comodules (def. ) over the dual $E$-Steenrod algebra $(E_\bullet(E), \pi_\bullet(E))$ (prop. ):

$$E_1^{s,t}(X,Y) \;\simeq\; Hom^t_{E_\bullet(E)}(E_\bullet(X), E_{\bullet-s}(A_s)) \;\;\,, \;\;\;\;\; d_1 \;=\; Hom_{E_\bullet(E)}(E_\bullet(X), E_\bullet( g \circ h ))$$

$\,$

$$0 \to Hom_{E_\bullet(E)}^t(E_\bullet(X),E_\bullet(A_0)) \stackrel{d_1}{\longrightarrow} Hom_{E_\bullet(E)}^t( E_\bullet(X), E_{\bullet-1}(A_1) ) \stackrel{d_1}{\longrightarrow} Hom_{E_\bullet(E)}^t( E_\bullet(X), E_{\bullet-2}(A_2) ) \stackrel{d_1}{\longrightarrow} \cdots \,.$$

(Adams 74, theorem 15.1 page 323)

Proof

This is prop. applied to def. :

$$\begin{aligned} E_1^{s,t}(X,Y) & = [X, \underset{A_s}{\underbrace{E \wedge Y_s}}]_{t-s} \\ & \simeq Hom^{t-s}_{E_\bullet(E)}( E_\bullet(X), E_\bullet(\underset{A_s}{\underbrace{E \wedge Y_s}}) ) \\ &\simeq Hom^{t}_{E_\bullet(E)}( E_\bullet(X), E_{\bullet-s}(A_s) ) \end{aligned}$$

The second page

Theorem

Let $(E, \mu, e)$ be a homotopy commutative ring spectrum (def.), and let $X, Y \in Ho(Spectra)$ be two spectra such that

1. $E$ is flat according to def. ;

2. $E_\bullet(X)$ is a projective module over $\pi_\bullet(E)$ (via this prop.).

Then the entries of the second page of the $E$-Adams spectral sequence for $[X,Y]_\bullet$ (def. ) are the Ext-groups of commutative Hopf algebroid-comodules (def. ) over the commutative Hopf algebroid structure on the dual $E$-Steenrod algebra $E_\bullet(E)$ from prop. :

$$E_2^{s,t}(X,Y) \simeq Ext^{s,t}_{E_\bullet(E)}(E_\bullet(X), E_\bullet(Y)) \,.$$

(On the right $s$ is the degree that goes with any Ext-functor, and the “internal degree” $t$ is the additional degree of morphisms between graded modules from def. .)

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

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

(Adams 74, theorem 15.1, page 323)

Proof

By theorem , under the given assumptions the first page reads

$$E_1^{s,t}(X,Y) \;\simeq\; Hom^t_{E_\bullet(E)}(E_\bullet(X), E_{\bullet-s}(A_s)) \;\;\,, \;\;\;\;\; d_1 \;=\; Hom_{E_\bullet(E)}(E_\bullet(X), E_\bullet( g \circ h ))$$

$\,$

$$0 \to Hom_{E_\bullet(E)}^t(E_\bullet(X),E_\bullet(A_0)) \stackrel{d_1}{\longrightarrow} Hom_{E_\bullet(E)}^t( E_\bullet(X), E_{\bullet-1}(A_1) ) \stackrel{d_1}{\longrightarrow} Hom_{E_\bullet(E)}^t( E_\bullet(X), E_{\bullet-2}(A_2) ) \stackrel{d_1}{\longrightarrow} \cdots \,.$$

By remark the second page is the cochain cohomology of this complex. Hence 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 $E_\bullet(E) CoMod$ (def. , prop. ) is an abelian category with enough injectives (so that all right derived functors on $E_\bullet(E) CoMod$ exist);

2. the first page graded chain complex $(E^{\bullet,t}_1(X,Y), d_1)$ is the image under the hom-functor $F \coloneqq Hom_{E_\bullet(E)}(E_\bullet(Y),-)$ of an $F$-acyclic resolution of $E_\bullet(X)$ (so that its cohomology indeed computes the Ext-derived functor (theorem)).

That $E_\bullet(E) CoMod$ is an abelian category is lemma below, and that it has enough injectives is lemma .

Lemma below shows that $E_\bullet(A_\bullet)$ is a resolution of $E_\bullet(Y)$ in $E_\bullet(E) CoMod$. By prop. it is a resolution by cofree comodules (def. ). That these are $F$-acyclic is lemma below.

$E$-Adams resolutions

We discuss that the first page of the $E$-Adams spectral sequence indeed exhibits a resolution as required by the proof of theorem .

Lemma

Given an $E$-Adams spectral sequence $(E^{s,t}_r(X,Y),d_r)$ as in def. , then the sequences of morphisms

$$0 \to E_\bullet(Y_p) \overset{E_\bullet(g_p)}{\longrightarrow} E_\bullet(A_p) \overset{E_\bullet(h_p)}{\longrightarrow} E_{\bullet-1}(Y_{p+1}) \to 0$$

are short exact, hence their splicing of short exact sequences

$$\array{ 0 &\to& E_\bullet(Y) && \overset{E_\bullet(g_0)}{\longrightarrow} && E_\bullet(A_0) && \overset{\partial}{\longrightarrow} && E_{\bullet-1}(A_1) && \overset{\partial}{\longrightarrow} && E_{\bullet-2}(A_2) && \longrightarrow && \cdots \\ && && && & {}_{\mathllap{E_\bullet(h_0)}}\searrow && \nearrow_{\mathrlap{E_\bullet(g_1)}} && {}_{\mathllap{E_\bullet(h_1)}}\searrow && \nearrow_{\mathrlap{E_\bullet(g_2)}} \\ && && && && E_{\bullet-1}(Y_1) && && E_{\bullet-2}(Y_2) }$$

is a long exact sequence, exhibiting the graded chain complex $(E_\bullet(A_\bullet), \partial)$ as a resolution of $E_\bullet(Y)$.

(Adams 74, theorem 15.1, page 322)

Proof

Consider the image of the defining homotopy cofiber sequence

$$Y_p \overset{g_p}{\longrightarrow} E \wedge Y_p \overset{h_p}{\longrightarrow} \Sigma Y_{p+1}$$

under the functor $E \wedge (-)$. This is itself a homotopy cofiber sequence of the form

$$E \wedge Y_p \overset{E \wedge g_p}{\longrightarrow} E \wedge E \wedge Y_p \overset{E \wedge h_p}{\longrightarrow} \Sigma E \wedge Y_{p + 1}$$

(due to the tensor triangulated structure of the stable homotopy category, prop.).

Applying the stable homotopy groups functor $\pi_\bullet(-) \simeq [\mathbb{S},-]_\bullet$ (lemma) to this yields a long exact sequence (prop.)

$$\cdots \longrightarrow E_{\bullet}(Y_{p+1}) \overset{ E_\bullet(f_p) }{\longrightarrow} E_\bullet(Y_p) \overset{ E_\bullet(g_p) }{\longrightarrow} E_\bullet(A_p) \overset{ E_\bullet(h_p) }{\longrightarrow} E_{\bullet-1}(Y_{p+1}) \overset{ E_{\bullet-1}(f_p) }{\longrightarrow} E_{\bullet-1}(Y_{p}) \overset{ E_{\bullet-1}(g_p) }{\longrightarrow} E_{\bullet-1}(A_p) \longrightarrow \cdots \,.$$

But in fact this splits: by unitality of $(E,\mu,e)$, the product operation $\mu$ on the homotopy commutative ring spectrum $E$ is a left inverse to $g_p$ in that

$$id \;\colon\; E \wedge Y_p \overset{E \wedge g_p}{\longrightarrow} E \wedge E \wedge Y_p \overset{\mu \wedge id}{\longrightarrow} E \wedge Y_p \,.$$

Therefore $E_\bullet(g_p)$ is a monomorphism, hence its kernel is trivial, and so by exactness $E_\bullet(f_p) = 0$. This means that the above long exact sequence collapses to short exact sequences.

Homological co-algebra

We discuss basic aspects of homological algebra in categories of comodules (def. ) over commutative Hopf algebroids (def. ), needed in the proof of theorem .

Lemma

Let $(\Gamma, A)$ be a commutative Hopf algebroid $\Gamma$ over $A$ (def. , ), such that the right $A$-module structure on $\Gamma$ induced by $\eta_R$ is a flat module.

Then the category $\Gamma CoMod$ of comodules over $\Gamma$ (def. ) is an abelian category.

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

Proof

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

Next we need to show if $\Gamma$ is flat over $A$, that then this is also a pre-abelian category, in that kernels and cokernels exist.

To that end, 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 $\Gamma \otimes_A (-)$

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

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. Hence by the universal property of kernels and the commutativity of the square o the right, there exists a unique vertical morphism as shown on the left, making the left square 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. Hence $\Gamma CoMod$ is a pre-abelian category.

But it also follows from this construction 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 by the fact that the category of modules $A Mod$ is an abelian category, hence it is an isomorphism also in $\Gamma CoMod$. So the latter is in fact an abelian category itself.

Lemma

Let $(\Gamma, A)$ be a commutative Hopf algebroid $\Gamma$ over $A$ (def. , ), such that the right $A$-module structure on $\Gamma$ induced by $\eta_R$ is a flat module.

Then

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

2. $\Gamma CoMod$ has enough injectives (def.) 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 (by this proposition).

Now by prop. we have the adjunction

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

Observe that the left adjoint is a faithful functor (being a forgetful functor) and that, by the proof of lemma , it is an exact functor. This implies that

1. for $I \in A Mod$ an injective module, then the co-free comodule $\Gamma \otimes_A I$ is an injective object in $\Gamma CoMod$ (by this lemma);

2. for $N \in \Gamma CoMod$ any object, and for $i \colon forget(N) \hookrightarrow I$ 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 (by this lemma)) hence a monomorpism into an injective comodule, by the previous item.

Hence $\Gamma CoMod$ has enough injective objects (def.).

Lemma

Let $(\Gamma, A)$ be a commutative Hopf algebroid $\Gamma$ over $A$ (def. , ), such that the right $A$-module structure on $\Gamma$ induced by $\eta_R$ is a flat module. Let $N \in \Gamma CoMod$ be a $\Gamma$-comodule (def. ) such that the underlying $A$-module is a projective module (a projective object in $A$Mod).

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

Proof

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

To that end, let $I^\bullet$ be an injective resolution of $K$ in $A Mod$. By lemma 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

$$\begin{aligned} \mathbb{R}^{\bullet \geq 1} Hom_\Gamma(N, \Gamma \otimes_A K) & \simeq H_{\bullet \geq 1}( Hom_\Gamma( N, \Gamma \otimes_A I^\bullet ) ) \\ & \simeq H_{\bullet \geq 1}( Hom_A(N, I^\bullet) ) \\ & \simeq 0 \end{aligned} \,.$$

Here the second equivalence is the cofree/forgetful adjunction isomorphism of prop. , 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 lemma the proof of theorem is completed.

Convergence

We discuss the convergence of $E$-Adams spectral sequences (def. ), i.e. the identification of the “limit”, in an appropriate sense, of the terms $E_r^{s,t}(X,Y)$ on the $r$th page of the spectral sequence as $r$ goes to infinity.

If an $E$-Adams spectral sequence converges, then it converges not necessarily to the full stable homotopy groups $[X,Y]_\bullet$, but to some localization of them. This typically means, roughly, that only certain $p$-torsion subgroups in $[X,Y]_\bullet$ for some prime numbers $p$ are retained. We give a precise discussion below in Localization and adic completion of abelian groups.

If one knows that $[X,Y]_q$ is a finitely generated abelian group (as is the case notably for $\pi_q^s = [\mathbb{S},\mathbb{S}]_q$ by the Serre finiteness theorem) then this allows to recover the full information from its pieces: by the fundamental theorem of finitely generated abelian groups (prop. below) these groups are direct sums of powers $\mathbb{Z}^n$ of the infinite cyclic group $\mathbb{Z}$ with finite cyclic groups of the form $\mathbb{Z}/p^k \mathbb{Z}$, and so all one needs to compute is the powers $k$ “one prime $p$ at a time”. This we review below in Primary decomposition of abelian groups.

The deeper reason that $E$-Adams spectral sequences tend to converge to localizations of the abelian groups $[X,Y]_\bullet$ of morphisms of spectra, is that they really converge to (the actual homotopy groups of) localizations of spectra. This is more than just a reformulation, because the localization at the level of spectra determies the filtration which controls the nature of the convergence. We discuss this localization of spectra below in Localization and nilpotent completion of spectra.

Then we state convergence properties of $E$-Adams spectral sequences below in Convergence statements.

Primary decomposition of abelian groups

An $E$-Adams spectral sequence typically converges (discussed below) not to the full stable homotopy groups $[X,Y]_\bullet$, but just to some piece which on the finite direct summands consists only of p-primary groups for some prime numbers $p$ that depend on the nature of the homotopy ring spectrum $E$ . Here we review basic facts about $p$-primary decomposition of finite abelian groups and introduce their graphical calculus (remark below) which will allow to read off these $p$-primary pieces from the stable page of the $E$-Adams spectral sequnce.

Theorem

(fundamental theorem of finitely generated abelian groups)

Every finitely generated abelian group $A$ is isomorphic to a direct sum of p-primary cyclic groups $\mathbb{Z}/p^k \mathbb{Z}$ (for $p$ a prime number and $k$ a natural number ) and copies of the infinite cyclic group $\mathbb{Z}$:

$$A \;\simeq\; \mathbb{Z}^n \oplus \underset{i}{\bigoplus} \mathbb{Z}/p_i^{k_i} \mathbb{Z} \,.$$

The summands of the form $\mathbb{Z}/p^k \mathbb{Z}$ are also called the p-primary components of $A$. Notice that the $p_i$ need not all be distinct.

fundamental theorem of finite abelian groups:

In particular every finite abelian group is of this form for $n = 0$, hence is a direct sum of cyclic groups.

fundamental theorem of cyclic groups:

In particular every cyclic group $\mathbb{Z}/n\mathbb{Z}$ is a direct sum of cyclic groups of the form

$$\mathbb{Z}/n\mathbb{Z} \simeq \underset{i}{\bigoplus} \mathbb{Z}/ p_i^{k_i} \mathbb{Z}$$

where all the $p_i$ are distinct and $k_i$ is the maximal power of the prime factor $p_i$ in the prime decomposition of $n$.

Specifically, for each natural number $d$ dividing $n$ it contains $\mathbb{Z}/d\mathbb{Z}$ as the subgroup generated by $n/d \in \mathbb{Z}\to \mathbb{Z}/n\mathbb{Z}$. In fact the lattice of subgroups of $\mathbb{Z}/n\mathbb{Z}$ is the formal dual of the lattice of natural numbers $\leq n$ ordered by inclusion.

(e.g. Roman 12, theorem 13.4, Navarro 03) for cyclic groups e.g. (Aluffi 09, pages 83-84)

This is a special case of the structure theorem for finitely generated modules over a principal ideal domain.

Example

For $p$ a prime number, there are, up to isomorphism, two abelian groups of order $p^2$, namely

$$(\mathbb{Z}/p\mathbb{Z}) \oplus (\mathbb{Z}/p\mathbb{Z})$$

and

$$\mathbb{Z}/p^2 \mathbb{Z} \,.$$

Example

For $p_1$ and $p_2$ two distinct prime numbers, $p_1 \neq p_2$, then there is, up to isomorphism, precisely one abelian group of order $p_1 p_2$, namely

$$\mathbb{Z}/p_1 \mathbb{Z} \oplus \mathbb{Z}/p_2 \mathbb{Z} \,.$$

This is equivalently the cyclic group

$$\mathbb{Z}/p_1 p_2 \mathbb{Z} \simeq \mathbb{Z}/p_1 \mathbb{Z} \oplus \mathbb{Z}/p_2 \mathbb{Z} \,.$$

The isomorphism is given by sending $1$ to $(p_2,p_1)$.

Example

Moving up, for two distinct prime numbers $p_1$ and $p_2$, there are exactly two abelian groups of order $p_1^2 p_2$, namely $(\mathbb{Z}/p_1 \mathbb{Z})\oplus (\mathbb{Z}/p_1 \mathbb{Z}) \oplus (\mathbb{Z}/p_2 \mathbb{Z})$ and $(\mathbb{Z}/p_1^2 \mathbb{Z})\oplus (\mathbb{Z}/p_2 \mathbb{Z})$. The latter is the cyclic group of order $p_1^2 p_2$. For instance, $\mathbb{Z}/12\mathbb{Z} \cong (\mathbb{Z}/4 \mathbb{Z})\oplus (\mathbb{Z}/3 \mathbb{Z})$.

Example

Similarly, there are four abelian groups of order $p_1^2 p_2^2$, three abelian groups of order $p_1^3 p_2$, and so on.

More generally, theorem may be used to compute exactly how many abelian groups there are of any finite order $n$ (up to isomorphism): write down its prime factorization, and then for each prime power $p^k$ appearing therein, consider how many ways it can be written as a product of positive powers of $p$. That is, each partition of $k$ yields an abelian group of order $p^k$. Since the choices can be made independently for each $p$, the numbers of such partitions for each $p$ are then multiplied.

Of all these abelian groups of order $n$, of course, one of them is the cyclic group of order $n$. The fundamental theorem of cyclic groups says it is the one that involves the one-element partitions $k= [k]$, i.e. the cyclic groups of order $p^k$ for each $p$.

Remark

(graphical representation of $p$-primary decomposition)

Theorem says that for any prime number $p$, the p-primary part of any finitely generated abelian group is determined uniquely up to isomorphism by

• a total number $k \in \mathbb{N}$ of powers of $p$;

• a partition $k = k_1 + k_2 + \cdots + k_q$.

The corresponding p-primary group is

$$\underoverset{i = 1}{q}{\bigoplus} \mathbb{Z}/p^{k_i} \mathbb{Z} \,.$$

In the context of Adams spectral sequences it is conventional to depict this information graphically by

• $k$ dots;

• of which sequences of length $k_i$ are connected by vertical lines, for $i \in \{1, \cdots, q\}$.

For example the graphical representation of the $p$-primary group

$$\mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Z}/p^3 \mathbb{Z} \oplus \mathbb{Z}/p^4\mathbb{Z}$$

is

$$\array{ &&& \bullet \\ && & \vert \\ && \bullet & \bullet \\ && \vert & \vert \\ && \bullet & \bullet \\ && \vert & \vert \\ \bullet & \bullet & \bullet & \bullet } \,.$$

This notation comes from the convention of drawing stable pages of multiplicative Adams spectral sequences and reading them as encoding the extension problem for computing the homotopy groups that the spectral sequence converges to:

• a dot at the top of a vertical sequence of dots denotes the group $\mathbb{Z}/p\mathbb{Z}$;

• inductively, a dot vetically below a sequence of dots denotes a group extension of $\mathbb{Z}/p\mathbb{Z}$ by the group represented by the sequence of dots above;

• a vertical line between two dots means that the generator of the group corresponding to the upper dot is, regarded after inclusion into the group extension, the product by $p$ of the generator of the group corresponding to the lower dot, regarded as represented by the generator of the group extension.

So for instance

$$\array{ \bullet \\ \vert \\ \bullet }$$

stands for an abelian group $A$ which forms a group extension of the form

$$\array{ \mathbb{Z}/p\mathbb{Z} \\ \downarrow \\ A \\ \downarrow \\ \mathbb{Z}/p\mathbb{Z} }$$

such that multiplication by $p$ takes the generator of the bottom copy of $\mathbb{Z}/p\mathbb{Z}$, regarded as represented by the generator of $A$, to the generator of the image of the top copy of $\mathbb{Z}/p\mathbb{Z}$.

This means that of the two possible choices of extensions (by example ) $A$ corresponds to the non-trivial extension $A = \mathbb{Z}/p^2\mathbb{Z}$. Because then in

$$\array{ \mathbb{Z}/p\mathbb{Z} & \\ \downarrow \\ \mathbb{Z}/p^2 \mathbb{Z} & \\ \downarrow \\ \mathbb{Z}/p\mathbb{Z} & }$$

the image of the generator 1 of the first group in the middle group is $p = p \cdot 1$.

Conversely, the notation

$$\array{ \bullet \\ \\ \bullet }$$

stands for an abelian group $A$ which forms a group extension of the form

$$\array{ \mathbb{Z}/p\mathbb{Z} \\ \downarrow \\ A \\ \downarrow \\ \mathbb{Z}/p\mathbb{Z} }$$

such that multiplication by $p$ of the generator of the top group in the middle group does not yield the generator of the bottom group.

This means that of the two possible choices (by example ) $A$ corresponds to the trivial extension $A = \mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z}$. Because then in

$$\array{ \mathbb{Z}/p\mathbb{Z} \\ \downarrow \\ \mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z} \\ \downarrow \\ \mathbb{Z}/p\mathbb{Z} }$$

the generator 1 of the top group maps to the element $(1,0)$ in the middle group, and multiplication of that by $p$ is $(0,0)$ instead of $(0,1)$, where the latter is the generator of the bottom group.

Similarly

$$\array{ \bullet \\ \vert \\ \bullet \\ \vert \\ \bullet }$$

is to be read as the result of appending to the previous case a dot below, so that this now indicates a group extension of the form

$$\array{ \mathbb{Z}/p^2 \mathbb{Z} \\ \downarrow \\ A \\ \downarrow \\ \mathbb{Z}/p \mathbb{Z} }$$

such that $p$-times the generator of the bottom group, regarded as represented by the generator of the middle group, is the image of the generator of the top group. This is again the case for the unique non-trivial extension, and hence in this case the diagram stands for $A = \mathbb{Z}/p^3 \mathbb{Z}$.

And so on.

For example the stable page of the $\mathbb{F}_2$-classical Adams spectral sequence for computation of the 2-primary part of the stable homotopy groups of spheres $\pi_{t-s}(\mathbb{S})$ has in (“internal”) degree $t-s \leq 13$ the following non-trivial entries:

(graphics taken from (Schwede 12))

Ignoring here the diagonal lines (which denote multiplication by the element $h_1$ that encodes the additional ring structure on $\pi_\bullet(\mathbb{S})$ which here we are not concerned with) and applying the above prescription, we read off for instance that $\pi_3(\mathbb{S}) \simeq \mathbb{Z}/8\mathbb{Z}$ (because all three dots are connected) while $\pi_8(\mathbb{S}) \simeq \mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}$ (because here the two dots are not connected). In total

$k =$	0	1	2	3	4	5	6	7	8	9	10	11	12	13
$\pi_k(\mathbb{S})_{(2)} =$	$\mathbb{Z}_{(2)}$	$\mathbb{Z}/2$	$\mathbb{Z}/2$	$\mathbb{Z}/8$	$0$	$0$	$\mathbb{Z}/2$	$\mathbb{Z}/16$	$(\mathbb{Z}/2)^2$	$(\mathbb{Z}/2)^3$	$\mathbb{Z}/2$	$\mathbb{Z}/8$	$0$	$0$

Here the only entry that needs further explanation is the one for $k = 0$. We discuss the relevant concepts for this below in the section Localization and adic completion of abelian groups, but for completeness, here is the quick idea:

The symbol $\mathbb{Z}_{(2)}$ refers to the 2-adic integers (def. ), i.e. for the limit of abelian groups

$$\mathbb{Z}_{(2)} = \underset{\longleftarrow}{\lim}_{n \geq 1} \mathbb{Z}/2^n \mathbb{Z}$$

This is not 2-primary, but it does arise when applying 2-adic completion of abelian groups (def. ) to finitely generated abelian groups as in theorem . The 2-adic integers is the abelian group associated to the diagram

$$\array{ \vdots \\ \vert \\ \bullet \\ \vert \\ \bullet \\ \vert \\ \bullet \\ \vert \\ \bullet }$$

as in the above figure. Namely by the above prescrption, this infinite sequence should encode an abelian group $A$ such that it is an extension of $\mathbb{Z}/p\mathbb{Z}$ by itself of the form

$$0 \to A \overset{p \cdot(-)}{\longrightarrow} A \longrightarrow \mathbb{Z}/p\mathbb{Z}$$

(Because it is supposed to encode an extension of $\mathbb{Z}/p\mathbb{Z}$ by the group corresponding to the result of chopping off the lowest dot, which however in this case does not change the figure.)

Indeed, by lemma below we have a short exact sequence

$$0 \to \mathbb{Z}_{(p)} \overset{p \cdot (-)}{\longrightarrow} \mathbb{Z}_{(p)} \longrightarrow \mathbb{Z}/p\mathbb{Z} \to 0 \,.$$

Localization and adic completion of abelian groups

Remark

Recall that Ext-groups $\Ext^\bullet(A,B)$ between abelian groups $A, B \in$ Ab are concentrated in degrees 0 and 1 (prop.). Since

$$Ext^0(A,B) \simeq Hom(A,B)$$

is the plain hom-functor, this means that there is only one possibly non-vanishing Ext-group $Ext^1$, therefore often abbreviated to just “$Ext$”:

$$Ext(A,B) \coloneqq Ext^1(A,B) \,.$$

Definition

Let $K$ be an abelian group.

Then an abelian group $A$ is called $K$-local if all the Ext-groups from $K$ to $A$ vanish:

$$Ext^\bullet(K,A) \simeq 0$$

hence equivalently (remark ) if

$$Hom(K,A) \simeq 0 \;\;\;\;\; and \;\;\;\;\; Ext(K,A) \simeq 0 \,.$$

A homomorphism of abelian groups $f \colon B \longrightarrow C$ is called $K$-local if for all $K$-local groups $A$ the function

$$Hom(f,A) \;\colon\; Hom(B,A) \longrightarrow Hom(A,A)$$

is a bijection.

(Beware that here it would seem more natural to use $Ext^\bullet$ instead of $Hom$, but we do use $Hom$. See (Neisendorfer 08, remark 3.2).

A homomorphism of abelian groups

$$\eta \;\colon\; A \longrightarrow L_K A$$

is called a $K$-localization if

1. $L_K A$ is $K$-local;

2. $\eta$ is a $K$-local morphism.

We now discuss two classes of examples of localization of abelian groups

1. Classical localization at/away from primes;

2. Formal completion at primes.

$\,$

Classical localization at/away from primes

For $n \in \mathbb{N}$, write $\mathbb{Z}/n\mathbb{Z}$ for the cyclic group of order $n$.

Lemma

For $n \in \mathbb{N}$ and $A \in Ab$ any abelian group, then

1. the hom-group out of $\mathbb{Z}/n\mathbb{Z}$ into $A$ is the $n$-torsion subgroup of $A$

   $$Hom(\mathbb{Z}/n\mathbb{Z}, A) \simeq \{ a \in A \;\vert\; p \cdot a = 0 \}$$

2. the first Ext-group out of $\mathbb{Z}/n\mathbb{Z}$ into $A$ is

   $$Ext^1(\mathbb{Z}/n\mathbb{Z},A) \simeq A/n A \,.$$

Proof

Regarding the first item: Since $\mathbb{Z}/p\mathbb{Z}$ is generated by its element 1 a morphism $\mathbb{Z}/p\mathbb{Z} \to A$ is fixed by the image $a$ of this element, and the only relation on 1 in $\mathbb{Z}/p\mathbb{Z}$ is that $p \cdot 1 = 0$.

Regarding the second item:

Consider the canonical free resolution

$$0 \to \mathbb{Z} \overset{p \cdot (-)}{\longrightarrow} \mathbb{Z} \longrightarrow \mathbb{Z}/p\mathbb{Z} \to 0 \,.$$

By the general discusson of derived functors in homological algebra this exhibits the Ext-group in degree 1 as part of the following short exact sequence

$$0 \to Hom(\mathbb{Z},A) \overset{Hom(n\cdot(-),A)}{\longrightarrow} Hom(\mathbb{Z}, A) \longrightarrow Ext^1(\mathbb{Z}/n\mathbb{Z},A) \to 0 \,,$$

where the morphism on the left is equivalently $A \overset{n \cdot (-)}{\to} A$.

Example

An abelian group $A$ is $\mathbb{Z}/p\mathbb{Z}$-local precisely if the operation

$$p \cdot (-) \;\colon\; A \longrightarrow A$$

of multiplication by $p$ is an isomorphism, hence if “$p$ is invertible in $A$”.

Proof

By the first item of lemma we have

$$Hom(\mathbb{Z}/p\mathbb{Z}, A) \simeq \{ a \in A \;\vert\; p \cdot a = 0 \}$$

By the second item of lemma we have

$$Ext^1(\mathbb{Z}/p\mathbb{Z},A) \simeq A/p A \,.$$

Hence by def. $A$ is $\mathbb{Z}/p\mathbb{Z}$-local precisely if

$$\{ a \in A \;\vert\; p \cdot a = 0 \} \simeq 0$$

and if

$$A / p A \simeq 0 \,.$$

Both these conditions are equivalent to multiplication by $p$ being invertible.

Definition

For $J \subset \mathbb{N}$ a set of prime numbers, consider the direct sum $\underset{p \in J}{\oplus} \mathbb{Z}/p\mathbb{Z}$ of cyclic groups of order $p$.

The operation of $\underset{p \in J}{\otimes} \mathbb{Z}/p\mathbb{Z}$-localization of abelian groups according to def. is called inverting the primes in $J$.

Specifically

1. for $J = \{p\}$ a single prime then $\mathbb{Z}/p\mathbb{Z}$-localization is called localization away from $p$;

2. for $J$ the set of all primes except $p$ then $\underset{p \in J}{\otimes} \mathbb{Z}/p\mathbb{Z}$-localization is called localization at $p$;

3. for $J$ the set of all primes, then $\underset{p \in J}{\otimes} \mathbb{Z}/p\mathbb{Z}$-localizaton is called rationalization..

Definition

For $J \subset Primes \subset \mathbb{N}$ a set of prime numbers, then

$$\mathbb{Z}[J^{-1}] \hookrightarrow \mathbb{Q}$$

denotes the subgroup of the rational numbers on those elements which have an expression as a fraction of natural numbers with denominator a product of elements in $J$.

This is the abelian group underlying the localization of a commutative ring of the ring of integers at the set $J$ of primes.

If $J = Primes - \{p\}$ is the set of all primes except $p$ one also writes

$$\mathbb{Z}_{(p)} \coloneqq \mathbb{Z}[Primes - \{p\}] \,.$$

Notice the parenthesis, to distinguish from the notation $\mathbb{Z}_{p}$ for the p-adic integers (def. below).

Remark

The terminology in def. is motivated by the following perspective of arithmetic geometry:

Generally for $R$ a commutative ring, then an $R$-module is equivalently a quasicoherent sheaf on the spectrum of the ring $Spec(R)$.

In the present case $R = \mathbb{Z}$ is the integers and abelian groups are identified with $\mathbb{Z}$-modules. Hence we may think of an abelian group $A$ equivalently as a quasicoherent sheaf on Spec(Z).

The “ring of functions” on Spec(Z) is the integers, and a point in $Spec(\mathbb{Z})$ is labeled by a prime number $p$, thought of as generating the ideal of functions on Spec(Z) which vanish at that point. The residue field at that point is $\mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}$.

Inverting a prime means forcing $p$ to become invertible, which, by this characterization, it is precisely away from that point which it labels. The localization of an abelian group at $\mathbb{Z}/p\mathbb{Z}$ hence corresponds to the restriction of the corresponding quasicoherent sheaf over $Spec(\mathbb{Z})$ to the complement of the point labeled by $p$.

Similarly localization at $p$ is localization away from all points except $p$.

See also at function field analogy for more on this.

Proposition

For $J \subset \mathbb{N}$ a set of prime numbers, a homomorphism of abelian groups $f \;\colon\; A \lookrightarrow B$ is local (def. ) with respect to $\underset{p \in J}{\oplus} \mathbb{Z}/p\mathbb{Z}$ (def. ) if under tensor product of abelian groups with $\mathbb{Z}[J^{-1}]$ (def. ) it becomes an isomorphism

$$f \otimes \mathbb{Z}[J^{-1}] \;\colon\; X \otimes \mathbb{Z}[J^{-1}] \overset{\simeq}{\longrightarrow} Y \otimes \mathbb{Z}[J^{-1}] \,.$$

Moreover, for $A$ any abelian group then its $\underset{p \in J}{\oplus} \mathbb{Z}/p\mathbb{Z}$-localization exists and is given by the canonical projection morphism

$$A \longrightarrow A \otimes \mathbb{Z}[J^{-1}] \,.$$

(e.g. Neisendorfer 08, theorem 4.2)

$\,$

Formal completion at primes

We have seen above in remark that classical localization of abelian groups at a prime number is geometrically interpreted as restricting a quasicoherent sheaf over Spec(Z) to a single point, the point that is labeled by that prime number.

Alternatively one may restrict to the “infinitesimal neighbourhood” of such a point. Technically this is called the formal neighbourhood, because its ring of functions is, by definition, the ring of formal power series (regarded as an adic ring or pro-ring). The corresponding operation on abelian groups is accordingly called formal completion or adic completion or just completion, for short, of abelian groups.

It turns out that if the abelian group is finitely generated then the operation of p-completion coincides with an operation of localization in the sense of def. , namely localization at the p-primary component $\mathbb{Z}(p^\infty)$ of the group $\mathbb{Q}/\mathbb{Z}$ (def. below). On the one hand this equivalence is useful for deducing some key properties of p-completion, this we discuss below. On the other hand this situation is a shadow of the relation between localization of spectra and nilpotent completion of spectra, which is important for characterizing the convergence properties of Adams spectral sequences.

Definition

For $p$ a prime number, then the p-adic completion of an abelian group $A$ is the abelian group given by the limit

$$A^\wedge_p \coloneqq \underset{\longleftarrow}{\lim} \left( \cdots \longrightarrow A / p^3 A \longrightarrow A / p^2 A \longrightarrow A/p A \right) \,,$$

where the morphisms are the evident quotient morphisms. With these understood we often write

$$A^\wedge_p \coloneqq \underset{\longleftarrow}{\lim}_n A/p^n A$$

for short. Notice that here the indexing starts at $n = 1$.

Example

The p-adic completion (def. ) of the integers $\mathbb{Z}$ is called the p-adic integers, often written

$$\mathbb{Z}_p \coloneqq \mathbb{Z}^\wedge_p \coloneqq \underset{\longleftarrow}{\lim}_n \mathbb{Z}/p^n \mathbb{Z} \,,$$

which is the original example that gives the general concept its name.

With respect to the canonical ring-structure on the integers, of course $p \mathbb{Z}$ is a prime ideal.

Compare this to the ring $\mathcal{O}_{\mathbb{C}}$ of holomorphic functions on the complex plane. For $x \in \mathbb{C}$ any point, it contains the prime ideal generated by $(z-x)$ (for $z$ the canonical coordinate function on $\mathbb{z}$). The formal power series ring $\mathbb{C}[ [(z.x)] ]$ is the adic completion of $\mathcal{O}_{\mathbb{C}}$ at this ideal. It has the interpretation of functions defined on a formal neighbourhood of $X$ in $\mathbb{C}$.

Analogously, the p-adic integers $\mathbb{Z}_p$ may be thought of as the functions defined on a formal neighbourhood of the point labeled by $p$ in Spec(Z).

Lemma

There is a short exact sequence

$$0 \to \mathbb{Z}_p \overset{p \cdot (-)}{\longrightarrow} \mathbb{Z}_p \longrightarrow \mathbb{Z}/p\mathbb{Z} \to 0 \,.$$

Proof

Consider the following commuting diagram

$$\array{ \vdots && \vdots && \vdots \\ \downarrow && \downarrow && \downarrow \\ \mathbb{Z}/p^3\mathbb{Z} &\overset{p\cdot (-)}{\longrightarrow}& \mathbb{Z}/p^4 \mathbb{Z} &\longrightarrow& \mathbb{Z}/p\mathbb{Z} \\ \downarrow && \downarrow && \downarrow \\ \mathbb{Z}/p^2\mathbb{Z} &\overset{p\cdot (-)}{\longrightarrow}& \mathbb{Z}/p^3 \mathbb{Z} &\longrightarrow& \mathbb{Z}/p\mathbb{Z} \\ \downarrow && \downarrow && \downarrow \\ \mathbb{Z}/p\mathbb{Z} &\overset{p\cdot (-)}{\longrightarrow}& \mathbb{Z}/p^2 \mathbb{Z} &\longrightarrow& \mathbb{Z}/p\mathbb{Z} \\ \downarrow && \downarrow && \downarrow \\ 0 &\longrightarrow& \mathbb{Z}/p\mathbb{Z} &\longrightarrow& \mathbb{Z}/p\mathbb{Z} } \,.$$

Each horizontal sequence is exact. Taking the limit over the vertical sequences yields the sequence in question. Since limits commute over limits, the result follows.

We now consider a concept of $p$-completion that is in general different from def. , but turns out to coincide with it in finitely generated abelian groups.

Definition

For $p$ a prime number, write

$$\mathbb{Z}[1/p] \coloneqq \underset{\longrightarrow}{\lim} \left( \mathbb{Z} \overset{p \cdot (-)}{\longrightarrow} \mathbb{Z} \overset{p \cdot (-)}{\longrightarrow} \mathbb{Z} \overset{}{\longrightarrow} \cdots \right)$$

for the colimit (in Ab) over iterative applications of multiplication by $p$ on the integers.

This is the abelian group generated by formal expressions $\frac{1}{p^k}$ for $k \in \mathbb{N}$, subject to the relations

$$(p \cdot n) \frac{1}{p^{k+1}} = n \frac{1}{p^k} \,.$$

Equivalently it is the abelian group underlying the ring localization $\mathbb{Z}[1/p]$.

Definition

For $p$ a prime number, then localization of abelian groups (def. ) at $\mathbb{Z}[1/p]$ (def. ) is called $p$-completion of abelian groups.

Lemma

An abelian group $A$ is $p$-complete according to def. precisely if both the limit as well as the lim^1 of the sequence

$$\cdots \overset{}{\longrightarrow} A \overset{p}{\longrightarrow} A \overset{p}{\longrightarrow} A \overset{p}{\longrightarrow} A$$

vanishes:

$$\underset{\longleftarrow}{\lim} \left( \cdots \overset{}{\longrightarrow} A \overset{p}{\longrightarrow} A \overset{p}{\longrightarrow} A \overset{p}{\longrightarrow} A \right) \simeq 0$$

and

$$\underset{\longleftarrow}{\lim}^1 \left( \cdots \overset{}{\longrightarrow} A \overset{p}{\longrightarrow} A \overset{p}{\longrightarrow} A \overset{p}{\longrightarrow} A \right) \simeq 0 \,.$$

Proof

By def. the group $A$ is $\mathbb{Z}[1/p]$-local precisely if

$$Hom(\mathbb{Z}[1/p], A) \simeq 0 \;\;\;\;\;\;\; and \;\;\;\;\;\;\; Ext^1(\mathbb{Z}[1/p], A) \simeq 0 \,.$$

Now use the colimit definition $\mathbb{Z}[1/p] = \underset{\longrightarrow}{\lim}_n \mathbb{Z}$ (def. ) and the fact that the hom-functor sends colimits in the first argument to limits to deduce that

$$\begin{aligned} Hom(\mathbb{Z}[1/p], A) & = Hom( \underset{\longrightarrow}{\lim}_n \mathbb{Z}, A ) \\ & \simeq \underset{\longleftarrow}{\lim}_n Hom(\mathbb{Z},A) \\ & \simeq \underset{\longleftarrow}{\lim}_n A \end{aligned} \,.$$

This yields the first statement. For the second, use that for every cotower over abelian groups $B_\bullet$ there is a short exact sequence of the form

$$0 \to \underset{\longleftarrow}{\lim}^1_n Hom(B_n, A) \longrightarrow Ext^1( \underset{\longrightarrow}{\lim}_n B_n, A ) \longrightarrow \underset{\longleftarrow}{\lim}_n Ext^1( B_n, A) \to 0$$

(by this lemma).

In the present case all $B_n \simeq \mathbb{Z}$, which is a free abelian group, hence a projective object, so that all the Ext-groups out of it vannish. Hence by exactness there is an isomorphism

$$Ext^1( \underset{\longrightarrow}{\lim}_n \mathbb{Z}, A ) \simeq \underset{\longleftarrow}{\lim}^1_n Hom(\mathbb{Z}, A) \simeq \underset{\longleftarrow}{\lim}^1_n A \,.$$

This gives the second statement.

Example

For $p$ a prime number, the p-primary cyclic groups of the form $\mathbb{Z}/p^n \mathbb{Z}$ are $p$-complete (def. ).

Proof

By lemma we need to check that

$$\underset{\longleftarrow}{\lim} \left( \cdots \overset{p}{\longrightarrow} \mathbb{Z}/p^n \mathbb{Z} \overset{p}{\longrightarrow} \mathbb{Z}/p^n \mathbb{Z} \overset{p}{\longrightarrow} \mathbb{Z}/p^n \mathbb{Z} \right) \simeq 0$$

and

$$\underset{\longleftarrow}{\lim}^1 \left( \cdots \overset{p}{\longrightarrow} \mathbb{Z}/p^n \mathbb{Z} \overset{p}{\longrightarrow} \mathbb{Z}/p^n \mathbb{Z} \overset{p}{\longrightarrow} \mathbb{Z}/p^n \mathbb{Z} \right) \simeq 0 \,.$$

For the first statement observe that $n$ consecutive stages of the tower compose to the zero morphism. First of all this directly implies that the limit vanishes, secondly it means that the tower satisfies the Mittag-Leffler condition (def.) and this implies that the $\lim^1$ also vanishes (prop.).

Definition

For $p$ a prime number, write

$$\mathbb{Z}(p^\infty) \coloneqq \mathbb{Z}[1/p]/\mathbb{Z}$$

(the p-primary part of $\mathbb{Q}/\mathbb{Z}$), where $\mathbb{Z}[1/p] = \underset{\longrightarrow}{\lim}(\mathbb{Z}\overset{p}{\to} \mathbb{Z} \overset{p}{\to} \mathbb{Z} \to \cdots )$ from def. .

Since colimits commute over each other, this is equivalently

$$\mathbb{Z}(p^\infty) \simeq \underset{\longrightarrow}{\lim} ( 0 \hookrightarrow \mathbb{Z}/p\mathbb{Z} \hookrightarrow \mathbb{Z}/p^2 \mathbb{Z} \hookrightarrow \cdots ) \,.$$

Theorem

For $p$ a prime number, the $\mathbb{Z}[1/p]$-localization

$$A \longrightarrow L_{\mathbb{Z}[1/p]} A$$

of an abelian group $A$ (def. , def. ), hence the $p$-completion of $A$ according to def. , is given by the morphism

$$A \longrightarrow Ext^1( \mathbb{Z}(p^\infty), A )$$

into the first Ext-group into $A$ out of $\mathbb{Z}(p^\infty)$ (def. ), which appears as the first connecting homomorphism $\delta$ in the long exact sequence of Ext-groups

$$0 \to Hom(\mathbb{Z}(p^\infty),A) \longrightarrow Hom(\mathbb{Z}[1/p],A) \longrightarrow Hom(\mathbb{Z},A) \overset{\delta)}{\longrightarrow} Ext^1(\mathbb{Z}(p^\infty), A) \to \cdots \,.$$

that is induced (via this prop.) from the defining short exact sequence

$$0 \to \mathbb{Z} \longrightarrow \mathbb{Z}[1/p] \longrightarrow \mathbb{Z}(p^\infty) \to 0$$

(def. ).

e.g. (Neisendorfer 08, p. 16)

Proposition

If $A$ is a finitely generated abelian group, then its $p$-completion (def. , for any prime number $p$) is equivalently its p-adic completion (def. )

$$\mathbb{Z}[1/p] A \simeq A^\wedge_p \,.$$

Proof

By theorem the $p$-completion is $Ext^1(\mathbb{Z}(p^\infty),A)$. By def. there is a colimit

$$\mathbb{Z}(p^\infty) = \underset{\longrightarrow}{\lim} \left( \mathbb{Z}/p\mathbb{Z} \to \mathbb{Z}/p^2 \mathbb{Z} \to \mathbb{Z}/p^3 \mathbb{Z} \to \cdots \right) \,.$$

Together this implies, by this lemma, that there is a short exact sequence of the form

$$0 \to \underset{\longleftarrow}{\lim}^1 Hom(\mathbb{Z}/p^n \mathbb{Z},A) \longrightarrow X^\wedge_p \longrightarrow \underset{\longleftarrow}{\lim}_n Ext^1(\mathbb{Z}/p^n \mathbb{Z}, A) \to 0 \,.$$

By lemma the lim^1 on the left is over the $p^n$-torsion subgroups of $A$, as $n$ ranges. By the assumption that $A$ is finitely generated, there is a maximum $n$ such that all torsion elements in $A$ are annihilated by $p^n$. This means that the Mittag-Leffler condition (def.) is satisfied by the tower of $p$-torsion subgroups, and hence the lim^1-term vanishes (prop.).

Therefore by exactness of the above sequence there is an isomorphism

$$\begin{aligned} L_{\mathbb{Z}[1/p]}X & \simeq \underset{\longleftarrow}{\lim}_n Ext^1(\mathbb{Z}/p^n \mathbb{Z}, A) \\ & \simeq \underset{\longleftarrow}{\lim}_n A/p^n A \end{aligned} \,,$$

where the second isomorphism is by lemma .

Proposition

If $A$ is a $p$-divisible group in that $A \overset{p \cdot (-)}{\longrightarrow} A$ is an isomorphism, then its $p$-completion (def. ) vanishes.

Proof

By theorem the localization morphism $\delta$ sits in a long exact sequence of the form

$$0 \to Hom(\mathbb{Z}(p^\infty),A) \longrightarrow Hom(\mathbb{Z}[1/p],A) \overset{\phi}{\longrightarrow} Hom(\mathbb{Z},A) \overset{\delta}{\longrightarrow} Ext^1(\mathbb{Z}(p^\infty), A) \to \cdots \,.$$

Here by def. and since the hom-functor takes colimits in the first argument to limits, the second term is equivalently the limit

$$Hom(\mathbb{Z}[1/p],A) \simeq \underset{\longleftarrow}{\lim} \left( \cdots \to A \overset{p \cdot (-)}{\longrightarrow} A \overset{p \cdot (-)}{\longrightarrow} A \right) \,.$$

But by assumption all these morphisms $p \cdot (-)$ that the limit is over are isomorphisms, so that the limit collapses to its first term, which means that the morphism $\phi$ in the above sequence is an isomorphism. But by exactness of the sequence this means that $\delta = 0$.

Corollary

Let $p$ be a prime number. If $A$ is a finite abelian group, then its $p$-completion (def. ) is equivalently its p-primary part.

Proof

By the fundamental theorem of finite abelian groups, $A$ is a finite direct sum

$$A \simeq \underset{i}{\oplus} \mathbb{Z}/p_i^{k_i}\mathbb{Z}$$

of cyclic groups of ordr $p_i^{k_1}$ for $p_i$ prime numbers and $k_i \in \mathbb{N}$ (thm.).

Since finite direct sums are equivalently products (biproducts: Ab is an additive category) this means that

$$Ext^1( \mathbb{Z}(p^\infty), A ) \simeq \underset{i}{\prod} Ext^1( \mathbb{Z}(p^\infty), \mathbb{Z}/p_i^{k_1}\mathbb{Z} ) \,.$$

By theorem the $i$th factor here is the $p$-completion of $\mathbb{Z}/p_i^{k_i}\mathbb{Z}$, and since $p \cdot(-)$ is an isomorphism on $\mathbb{Z}/p_i^{k_i}\mathbb{Z}$ if $p_i \neq p$ (because its kernel evidently vanishes), prop. says that in this case the factor vanishes, so that only the factors with $p_i = p$ remain. On these however $Ext^1(\mathbb{Z}(p^\infty),-)$ is the identity by example .

Localization and nilpotent completion of spectra

We discuss

1. Bousfield localization of spectra

2. Nilpotent completion of spectra

which are the analogs in stable homotopy theory of the construction of localization of abelian groups discussed above.

Literature: (Bousfield 79)

$\,,$

Localization of spectra

Definition

Let $E \in Ho(Spectra)$ be be a spectrum. Say that

1. a spectrum $X$ is $E$-acyclic if the smash product with $E$ is zero, $E \wedge X \simeq 0$;

2. a morphism $f \colon X \to Y$ of spectra is an $E$-equivalence if $E \wedge f \;\colon\; E \wedge X \to E \wedge Y$ is an isomorphism in $Ho(Spectra)$, hence if $E_\bullet(f)$ is an isomorphism in $E$-generalized homology;

3. a spectrum $X$ is $E$-local if the following equivalent conditions hold

   1. for every $E$-equivalence $f$ then $[f,X]_\bullet$ is an isomorphism;

   2. every morphism $Y \longrightarrow X$ out of an $E$-acyclic spectrum $Y$ is zero in $Ho(Spectra)$;

(Bousfield 79, §1) see also for instance (Lurie, Lecture 20, example 4)

Lemma

The two conditions in the last item of def. are indeed equivalent.

Proof

Notice that $A \in Ho(Spectra)$ being $E$-acyclic means equivalently that the unique morphism $0 \longrightarrow A$ is an $E$-equivalence.

Hence one direction of the claim is trivial. For the other direction we need to show that for $[-,X]_\bullet$ to give an isomorphism on all $E$-equivalences $f$, it is sufficient that it gives an isomorphism on all $E$-equivalences of the form $0 \to A$.

Given a morphism $f \colon A \to B$, write $B \longrightarrow B/A$ for its homotopy cofiber. Then since $Ho(Spectra)$ is a triangulated category (prop.) the defining axioms of triangulated categories (def., lemma) give that there is a commuting diagram of the form

$$\array{ 0 &\longrightarrow& A &\overset{id}{\longrightarrow}& A &\overset{}{\longrightarrow}& 0 &\overset{}{\longrightarrow}& \Sigma A \\ \downarrow && \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{f}} && \downarrow && \downarrow^{\mathrlap{id}} \\ \Sigma^{-1} B/A &\longrightarrow& A &\underset{f}{\longrightarrow}& B &\longrightarrow& B/A &\longrightarrow& \Sigma A } \,,$$

where both the top as well as the bottom are homotopy cofiber sequences. Hence applying $[-,X]_\bullet$ to this diagram in $Ho(Spectra)$ yields a diagram of graded abelian groups of the form

$$\array{ 0 &\longleftarrow& [A,X]_\bullet &\longleftarrow& [A,X]_\bullet &\longleftarrow& 0 &\longleftarrow& [A,X]_{\bullet+1} \\ \uparrow && \uparrow^{\mathrlap{id}} && \uparrow^{\mathrlap{[f,X]_\bullet}} && \uparrow && \uparrow^{\mathrlap{id}} \\ [B/A,X]_{\bullet+1} &\longleftarrow& [A,X]_\bullet &\longleftarrow& [B,X]_\bullet &\longleftarrow& [B/A,X]_\bullet &\longleftarrow& [A,X]_{\bullet+1} } \,,$$

where now both horizontal sequences are long exact sequences (prop.).

Hence if $[B/A,X]_\bullet \longrightarrow 0$ is an isomorphism, then all four outer vertical morphisms in this diagram are isomorphisms, and then the five-lemma implies that also $[f,X]_\bullet$ is an isomorphism.

Hence it is now sufficient to observe that with $f \colon A \to B$ an $E$-equivalence, then its homotopy cofiber $B/A$ is $E$-acyclic.

To see this, notice that by the tensor triangulated structure on $Ho(Spectra)$ (prop.) the smash product with $E$ preserves homotopy cofiber sequences, so that there is a homotopy cofiber sequence

$$E \wedge A \overset{E \wedge f}{\longrightarrow} E \wedge B \longrightarrow E \wedge (B/A) \longrightarrow E \wedge \Sigma A \,.$$

But if the first morphism here is an isomorphism, then the axioms of a triangulated category (def.) imply that $E \wedge B / A \simeq 0$. In detail: by the axioms we may form the morphism of homotopy cofiber sequences

$$\array{ E \wedge A &\overset{E \wedge f}{\longrightarrow}& E \wedge B &\longrightarrow& E \wedge B/A &\longrightarrow& E \wedge \Sigma A \\ \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{(E\wedge f)^{-1}}} && \downarrow && \downarrow^{\mathrlap{id}} \\ E \wedge A &\underset{id}{\longrightarrow}& E \wedge A &\longrightarrow& 0 &\longrightarrow& E \wedge \Sigma A } \,.$$

Then since two of the three vertical morphisms on the left are isomorphisms, so is the third (lemma).

Definition

Given $E,X \in Ho(Spectra)$, then an $E$-Bousfield localization of spectra of $X$ is

1. an $E$-local spectrum $L_E X$

2. an $E$-equivalence $X \longrightarrow L_E X$.

according to def. .

We discuss now that $E$-Localizations always exist. The key to this is the following lemma , which asserts that a spectrum being $E$-local is equivalent to it being $A$-null, for some “small” spectrum $A$:

Lemma

For every spectrum $E$ there exists a spectrum $A$ such that any spectrum $X$ is $E$-local (def. ) precisely if it is $A$-null, i.e.

$$X \;is\; E\text{-local} \;\;\;\; \Leftrightarrow \;\;\;\; [A,X]_\ast = 0$$

and such that

1. $A$ is $E$-acyclic (def. );

2. there exists an infinite cardinal number $\kappa$ such that $A$ is a $\kappa$-CW spectrum (hence a CW spectrum (def.) with at most $\kappa$ many cells);

3. the class of $E$-acyclic spectra (def. ) is the class generated by $A$ under

   1. wedge sum

   2. the relation that if in a homotopy cofiber sequence $X_1 \to X_2 \to X_3$ two of the spectra are in the class, then so is the third.

(Bousfield 79, lemma 1.13 with lemma 1.14) review includes (Bauer 11, p.2,3, VanKoughnett 13, p. 8)

Proposition

For $E \in Ho(Spectra)$ any spectrum, every spectrum $X$ sits in a homotopy cofiber sequence of the form

$$G_E(X) \longrightarrow X \overset{\eta_X}{\longrightarrow} L_E(X) \,,$$

and natural in $X$, such that

1. $G_E(X)$ is $E$-acyclic,

2. $L_E(X)$ is $E$-local,

according to def. .

(Bousfield 79, theorem 1.1) see also for instance (Lurie, Lecture 20, example 4)

Proof

Consider the $\kappa$-CW-spectrum spectrum $A$ whose existence is asserted by lemma . Let

$$I_A \coloneqq \{A \to Cone(A)\}$$

denote the set containing as its single element the canonical morphism (of sequential spectra) from $A$ into the standard cone of $A$, i.e. the cofiber

$$Cone(A) \coloneqq cofib( A \to A \wedge I_+ ) \simeq A \wedge I$$

of the inclusion of $A$ into its standard cylinder spectrum (def.).

Since the standard cylinder spectrum on a CW-spectrum is a good cylinder object (prop.) this means (lemma) that for $X$ any fibrant sequential spectrum, and for $A \longrightarrow X$ any morphism, then an extension along the cone inclusion

$$\array{ A &\longrightarrow& X \\ \downarrow & \nearrow \\ Cone(A) }$$

equivalently exhibits a null-homotopy of the top morphism. Hence the $(A \to Cone(A))$-injective objects in $Ho(Spectra)$ are precisely those spectra $X$ for which $[A,X]_\bullet \simeq 0$.

Moreover, due to the degreewise nature of the smash tensoring $Cone(A) = A \wedge I$ (def), the inclusion morphism $A \to Cone(A)$ is degreewise the inclusion of a CW-complex into its standard cone, which is a relative cell complex inclusion (prop.).

By this lemma the $\kappa$-cell spectrum $A$ is $\kappa$-small object (def.) with respect to morphisms of spectra which are degreewise relative cell complex inclusion small object argument .

Hence the small object argument applies (prop.) and gives for every $X$ a factorization of the terminal morphism $X \to \ast$ as an $I_A$-relative cell complex (def.) followed by an $I_A$-injective morphism (def.)

$$X \overset{I_A Cell}{\longrightarrow} L_E X \overset{I_A Inj}{\longrightarrow} \ast \,.$$

By the above, this means that $[A, L_E X] = 0$, hence by lemma that $L_E X$ is $E$-local.

It remains to see that the homotopy fiber of $X \to L_E X$ is $E$-acyclic: By the tensor triangulated structure on $Ho(Spectra)$ (prop.) it is sufficient to show that the homotopy cofiber is $E$-acyclic (since it differs from the homotopy fiber only by suspension). By the pasting law, the homotopy cofiber of a transfinite composition is the transfinite composition of a sequence of homotopy pushouts. By lemma and applying the pasting law again, all these homotopy pushouts produce $E$-acyclic objects. Hence we conclude by observing that the transfinite composition of the morphisms between these $E$-acyclic objects is $E$-acyclic. Since by construction all these morphisms are relative cell complex inclusions, this follows again with the compactness of the $n$-spheres (lemma).

Lemma

The morphism $X \to L_E (X)$ in prop. exhibits an $E$-localization of $X$ according to def.

Proof

It only remains to show that $X \to L_E X$ is an $E$-equivalence. By the tensor triangulated structure on $Ho(Spectra)$ (prop.) the smash product with $E$ preserves homotopy cofiber sequences, so that

$$E \wedge G_E X \longrightarrow E \wedge X \overset{E \wedge \eta_X}{\longrightarrow} E \wedge L_E X \longrightarrow E \wedge \Sigma G_E X$$

is also a homotopy cofiber sequence. But now $E \wedge G_E X \simeq 0$ by prop. , and so the axioms (def.) of the triangulated structure on $Ho(Spectra)$ (prop.) imply that $E \wedge \eta$ is an isomorphism.

$\,$

Nilpotent completion of spectra

Definition

Let $(E, \mu, e)$ be a homotopy commutative ring spectrum (def.) and $Y \in Ho(Spectra)$ any spectrum. Write $\overline{E}$ for the homotopy fiber of the unit $\mathbb{S}\overset{e}{\to} E$ as in def. such that the $E$-Adams filtration of $Y$ (def. ) reads (according to lemma )

$$\array{ \vdots \\ \downarrow \\ \overline{E}^3 \wedge Y \\ \downarrow \\ \overline{E}^2 \wedge Y \\ \downarrow \\ \overline{E} \wedge Y \\ \downarrow \\ Y } \,.$$

For $s \in \mathbb{N}$, write

$$\overline{E}_{s-1} \coloneqq hocof( \overline{E}^s \overset{i^s}{\longrightarrow} \mathbb{S})$$

for the homotopy cofiber. Here $\overline{E}_{-1} \simeq 0$. By the tensor triangulated structure of $Ho(Spectra)$ (prop.), this homotopy cofiber is preserved by forming smash product with $Y$, and so also

$$\overline{E}_n \wedge Y \simeq hocof( \overline{E}^n \wedge Y \overset{}{\longrightarrow} Y) \,.$$

Now let

$$\overline{E}_s \overset{p_{s-1}}{\longrightarrow} \overline{E}_{s-1}$$

be the morphism implied by the octahedral axiom of the triangulated category $Ho(Spectra)$ (def., prop.):

$$\array{ \overline{E}^{s+1} &\overset{i}{\longrightarrow}& \overline{E}^s &\longrightarrow& E \wedge \overline{E}^s &\longrightarrow& \Sigma \overline{E}^{s+1} \\ {}^{\mathllap{=}}\downarrow && \downarrow^{\mathrlap{i^s}} && \downarrow^{} && \downarrow \\ \overline{E}^{s+1} &\longrightarrow& \mathbb{S} &\longrightarrow& \overline{E}_s &\longrightarrow& \Sigma \overline{E}^{s+1} \\ && \downarrow && \downarrow^{\mathrlap{p_{s-1}}} \\ && \overline{E}_{s-1} &\overset{=}{\longrightarrow}& \overline{E}_{s-1} \\ && \downarrow && \downarrow \\ && \Sigma \overline{E}^s &\longrightarrow& \Sigma E \wedge \overline{E}^s } \,.$$

By the commuting square in the middle and using again the tensor triangulated structure, this yields an inverse sequence under $Y$:

$$Y \simeq \mathbb{S} \wedge Y \longrightarrow \cdots \overset{p_3 \wedge id}{\longrightarrow} \overline{E}_3 \wedge Y \overset{p_2 \wedge id}{\longrightarrow} \overline{E}_2 \wedge Y \overset{p_1 \wedge id}{\longrightarrow} \overline{E}_1 \wedge Y$$

The E-nilpotent completion $Y^\wedge_E$ of $Y$ is the homotopy limit over the resulting inverse sequence

$$Y^\wedge_E \coloneqq \mathbb{R}\underset{\longleftarrow}{\lim}_n \overline{E}_n \wedge Y$$

or rather the canonical morphism into it

$$Y \longrightarrow Y^\wedge_E \,.$$

Concretely, if

$$Y \simeq \mathbb{S} \wedge Y \longrightarrow \cdots \overset{p_3 \wedge id}{\longrightarrow} \overline{E}_3 \wedge Y \overset{p_2 \wedge id}{\longrightarrow} \overline{E}_2 \wedge Y \overset{p_1 \wedge id}{\longrightarrow} \overline{E}_1 \wedge Y$$

is presented by a tower of fibrations between fibrant spectra in the model structure on topological sequential spectra, then $Y^\wedge_E$ is represented by the ordinary sequential limit over this tower.

(Bousfield 79, top, middle and bottom of page 272)

Remark

In (Bousfield 79) the $E$-nilpotent completion of $X$ (def. ) is denoted “$E^\wedge X$”. The notation “$X^\wedge_E$” which we use here is more common among modern authors. It emphasizes the conceptual relation to p-adic completion $A^\wedge_p$ of abelian groups (def. ) and is less likely to lead to confusion with the smash product of $E$ with $X$.

Remark

The nilpotent completion $X^\wedge_E$ is $E$-local. This induces a universal morphism

$$L_E X \overset{}{\longrightarrow} X^\wedge_E$$

from the $E$-Bousfield localization of spectra of $X$ into the $E$-nilmpotent completion

(Bousfield 79, top of page 273)

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

Proposition

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

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

• a ring of integers modulo n $c \pi_0(E) \simeq \mathbb{Z}/n\mathbb{Z}$, for $n \geq 2$.

Then the map in remark is an equivalence

$$L_E X \stackrel{\simeq}{\longrightarrow} X^\wedge_E \,.$$

(Bousfield 79, theorem 6.5, theorem 6.6).

Convergence theorems

We state the two main versions of Bousfield‘s convergence theorems for the $E$-Adams spectral sequence, below as theorem and theorem .

First we need to define the concepts that enter the convergence statement:

1. the infinity-page $E_\infty^{s,t}(X,Y)$ (def. ),

2. a filtration on $[X,Y^\wedge_E]_\bullet$ (def. )

3. what it means for the former to converge to the latter (def. ).

Broadly the statement will be that typically

1. the $E$-Adams spectral sequence $E_r^{s,t}(X,Y)$ computes the stable homotopy groups $[X,Y^\wedge_E]$ of maps from $X$ into the E-nilpotent completion of $Y$;

2. these groups are localizations of the full groups $[X,Y]_\bullet$ depending on the core of $\pi_0(E)$.

Literature: (Bousfield 79)

$\,$

Definition

Let $(E, \mu, e)$ be a homotopy commutative ring spectrum (def.) and $X,Y \in Ho(Spectra)$ two spectra with associated $E$-Adams spectral sequence $\{E_r^{s,t}, d_r\}$ (def. ).

Observe that

$$if \; r \gt s \; then \; \; E^{s,\bullet}_{r+1}(X,Y) \simeq ker(d_r|_{E_r^{s,\bullet}(X,Y)}) \subset E_r^{s,\bullet}(X,Y)$$

since the differential $d_r$ on the $r$th page has bidegree $(r,r-1)$, and since $E_r^{s \lt 0,\bullet(X,Y)} \simeq 0$, so that for $r \gt s$ the image of $d_r$ in $E_r^{s,t}(X,Y)$ vanishes.

Thus define the bigraded abelian group

$$E_\infty^{s,t}(X,Y) \coloneqq \underset{r \gt s}{\lim} E_r^{s,t}(X,Y) = \underset{r \gt s}{\cap} E_r^{s,t}(X,Y)$$

called the “infinity page” of the $E$-Adams spectral sequence.

Definition

Let $(E, \mu, e)$ be a homotopy commutative ring spectrum (def.) and $X,Y \in Ho(Spectra)$ two spectra with associated $E$-Adams spectral sequence $\{E_r^{s,t}, d_r\}$ (def. ) and E-nilpotent completion $Y^\wedge_E$ (def. ).

Define a filtration

$$\cdots \hookrightarrow F^3 [X, Y^\wedge_E]_\bullet \hookrightarrow F^2 [X, Y^\wedge_E]_\bullet \hookrightarrow F^1 [X,Y^\wedge_E]_\bullet = [X, Y^\wedge_E ]_\bullet$$

on the graded abelian group $[X,Y^\wedge_E]_\bullet$ by

$$F^s [X, Y^\wedge_E]_\bullet \;\coloneqq\; ker( \; [X, Y^\wedge_E]_\bullet \overset{[X, Y^\wedge_E \to \overline{E}_{s-1} \wedge Y ]}{\longrightarrow} [X, \overline{E}_{s-1} \wedge Y]_\bullet \; ) \,,$$

where the morphisms $Y^\wedge_E \to \overline{E}_{s-1} \wedge Y$ is the canonical one from def. .

Definition

Let $(E, \mu, e)$ be a homotopy commutative ring spectrum (def.) and $X,Y \in Ho(Spectra)$ two spectra with associated $E$-Adams spectral sequence $\{E_r^{s,t}, d_r\}$ (def. ) and E-nilpotent completion $Y^\wedge_E$ (def. ).

Say that the $E$-Adams spectral sequence $\{E_r^{s,t}, d_r\}$ converges completely to the E-nilpotent completion $[X,Y^\wedge_E]_\bullet$ if the following two canonical morphisms are isomorphisms

1. $[X, Y^\wedge_E]_\bullet \longrightarrow \underset{\longleftarrow}{\lim}_s [X, Y^\wedge_E]_\bullet / F^s [X, Y^\wedge_E]_\bullet$

   (where on the right we have the limit over the tower of quotients by the stages of the filtration from def. )

2. $F^s [X, Y^\wedge_E]_{t-s} / F^{s+1}[X, Y^\wedge]_{t-s} \longrightarrow E^{s,t}_\infty(X,Y) \;\;\;\;\;\;\;\; \forall s,t$

   (where $F^s [X, Y^\wedge_E]_\bullet$ is the filtration stage from def. and $E^{s,t}_\infty(X,Y)$ is the infinity-page from def. ).

Notice that the first morphism is always surjective, while the second is necessarily injective, hence the condition is equivalently that the first is also injective, and the second also surjective.

(Bousfield 79, §6)

$\,$

Now we state sufficient conditions for complete convergence of the $E$-Adams spectral sequence. It turns out that convergence is controled by the core (def. ) of the ring $\pi_0(E)$. By prop. these cores are either localizations of the integers $\mathbb{Z}[J^{-1}]$ at a set $J$ of primes (def. ) or are rings of integers modulo n, or cores of products of these. We discuss the first two cases.

Theorem

Let $(E,\mu,e)$ be a homotopy commutative ring spectrum (def.) and let $X,Y \in Ho(Spectra)$ be two spectra such that

1. the core (def. ) of the 0-th stable homotopy group ring of $E$ (prop.) is the localization of the integers at a set $J$ of primes (def. )

   $$c \pi_0(E) \simeq \mathbb{Z}[J^{-1}] \subset \mathbb{Q}$$

2. $X$ is a CW-spectrum (def.) with a finite number of cells (rmk.);

then the $E$-Adams spectral sequence for $[X,Y]_\bullet$ (def. ) converges completely (def. ) to the localization

$$[X, Y^\wedge_E]_\bullet = \mathbb{Z}[J^{-1}] \otimes [X,Y]_\bullet$$

of $[X,Y]_\bullet$.

(Bousfield 79, theorem 6.5)

Theorem

Let $(E,\mu,e)$ be a homotopy commutative ring spectrum (def.) and let $X,Y \in Ho(Spectra)$ be two spectra such that

1. the core (def. ) of the 0-th stable homotopy group ring of $E$ (prop.) is a prime field

   $c \pi_0(E) \simeq \mathbb{F}_p$

   for some prime number $p$;

2. $Y$ is a connective spectrum in that its stable homotopy groups $\pi_\bullet(Y)$ vanish in negative degree;

3. $X$ is a CW-spectrum (def.) with a finite number of cells (rmk.);

4. $[X,Y]_\bullet$ is degreewise a finitely generated group

then the $E$-Adams spectral sequence for $[X,Y]_\bullet$ (def. ) converges completely (def. ) to the $p$-adic completion (def. )

$$[X, Y^\wedge_E]_\bullet \simeq \underset{\longleftarrow}{\lim}_n [X,Y]_\bullet/p^n[X,Y]_\bullet$$

of $[X,Y]_\bullet$.

(Bousfield 79, theorem 6.6)

Examples

Example

Examples of commutative ring spectra $E$ for which the dual $E$-Steenrod algebra $E_\bullet(E)$ over $\pi_\bullet(E)$ of def. where the left and right action of $\pi_\bullet(E)$ are not just isomorphic (via prop. ) but actually equal according to remark , includes the case $E =$ H$\mathbb{F}_p$.

Example

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

• $\mathbb{S}$ – the sphere spectrum;

• $H \mathbb{F}_p$ – Eilenberg-MacLane spectra for prime fields;

• MO, MU, MSp – Thom spectra;

• KO, KU – topological K-theory spectra.

(Adams 69, lecture 1, lemma 28 (p. 45))

Proof of the first two items

For $E = \mathbb{S}$ we have $\mathbb{S}_\bullet(\mathbb{S}) \coloneqq \pi_\bullet(\mathbb{S} \wedge \mathbb{S}) \simeq \pi_\bullet(\mathbb{S})$, since the sphere spectrum $\mathbb{S}$ is the tensor unit for the derived smash product of spectra (cor.). Hence the statement follows since every ring is, clearly, flat over itself.

For $E = H \mathbb{F}_p$ we have that $\pi_\bullet(H \mathbb{F}_p) \simeq \mathbb{F}_p$ (prop.), hence a field (a prime field). Every module over a field is a projective module (prop.) and every projective module is flat (prop.).

Example

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

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

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

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

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

  $$X^\wedge_{MU} \simeq X \,.$$

• $E =$ BP at prime $p$. For every spectrum $X$ its $BP$-nilpotent completion is its p-localization

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

  (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).

Classical Adams spectral sequence ($E = H \mathbb{F}_2$, $X = \mathbb{S}$)

We consider now the example of the $E$-Adams spectral sequence $\{E_r^{s,t}(X,Y), d_r\}$ (def. ) for the case that

1. $E = H \mathbb{F}_p$ is the Eilenberg-MacLane spectrum (def.) with coefficients in a prime field, regarded in $Ho(Spectra)$ with its canonical struture of a homotopy commutative ring spectrum induced (via this corollary) from its canonical structure of an orthogonal ring spectrum (from this def.);

2. $X = Y = \mathbb{S}$ are both the sphere spectrum.

This example is called the classical Adams spectral sequence.

The $H\mathbb{F}_p$-dual Steenrod algebra according to the general definition turns out to be the classical dual Steenrod algebra $\mathcal{A}_p^\ast$ recalled below .

Notice that $H \mathbb{F}_2$ satisfies the two assumptions needed to identify the second page of the $H\mathbb{F}_p$-Adams spectral sequence according to theorem :

Lemma

The Eilenberg-MacLane spectrum $H\mathbb{F}_p$ is flat according to , and $H \mathbb{F}_p(\mathbb{S})$ is a projective module over $\pi_\bullet(H \mathbb{F}_p)$.

Proof

The stable homotopy groups of $H \mathbb{F}_p$ is the prime field $\mathbb{F}_p$ itself, regarded as a graded commutative ring concentrated in degree 0 (prop.)

$$\pi_\bullet(H\mathbb{F}_p) \simeq \mathbb{F}_p \,.$$

Since this is a field, all modules over it are projective modules (prop.), hence in particular flat modules (prop.).

Corollary

The classical Adams spectral sequence, i.e. the $E$-Adams spectral sequence (def. ) for $E = H \mathbb{F}_p$ (def.) and $X = Y = \mathbb{S}$, has on its second page the Ext-groups of classical dual Steenrod algebra comodules from $\mathbb{F}_p \simeq H\mathbb{F}_p(\mathbb{S})$ to itself, and converges completely (def. ) to the p-adic completion (def. ) of the stable homotopy groups of spheres, hence in degree 0 to the p-adic integers and in all other degrees to the $p$-primary part (theorem )

$$E_2^{s,t}(\mathbb{S},\mathbb{S}) \;=\; Ext^{s,t}_{\mathcal{A}_p^\ast}(\mathbb{F}_p, \mathbb{F}_p) \;\Rightarrow\; (\pi_{\bullet}(\mathbb{S}))_p \,.$$

Proof

By lemma the conditions of theorem are satisfied, which implies the form of the second page.

For the convergence statement, we check the assumptions in theorem :

1. By prop. and prop. the ring $\mathbb{F}_p = \pi_0(H \mathbb{F}_p)$ coincides with its core: $c \mathbb{F}_p \simeq \mathbb{F}_p$;

2. $\mathbb{S}$ is clearly a connective spectrum;

3. $\mathbb{S}$ is clearly a finite CW-spectrum;

4. the groups $\pi_\bullet(\mathbb{S}) \simeq [\mathbb{S},\mathbb{S}]_\bullet$ are degreewise finitely generated, by Serre's finiteness theorem?.

Hence theorem applies and gives the convergence as stated.

Finally, by prop. the dual $E$-Steenrod algebra in the present case is the classical dual Steenrod algebra.

We now use the classical Adams spectral sequence from corollary to compute the first dozen stable homotopy groups of spheres.

The dual Steenrod algebra

Definition

Let $p$ be a prime number. Write $\mathbb{F}_p$ for the corresponding prime field.

The mod $p$-Steenrod algebra $\mathcal{A}_{p}$ is the graded co-commutative Hopf algebra over $\mathbb{F}_p$ which is

• for $p = 2$ generated by elements denoted $Sq^n$ for $n \in \mathbb{N}$, $n \geq 1$;

• for $p \gt 2$ generated by elements denoted $\beta$ and $P^n$ for $\in \mathbb{N}$, $n \geq 1$

(called the Serre-Cartan basis elements)

whose product is subject to the following relations (called the Ádem relations):

for $p = 2$:

for $0 \lt h \lt 2k$ the

$$Sq^h Sq^k \;=\; \underoverset{i = 0}{[h/2]}{\sum} \left( \array{ k -i - 1 \\ h - 2i } \right) Sq^{h + k -i} Sq^i \,,$$

for $p \gt 2$:

for $0 \lt h \lt p k$ then

$$P^h P^k \;=\; \underoverset{i = 0}{[h/p]}{\sum} (-1)^{h+i} \left( \array{ (p-1)(k-i) - 1 \\ h - p i } \right) P^{h +k - i}P^i$$

and if $0 \lt h \lt p k$ then

$$\begin{aligned} P^h \beta P^k & =\; \underoverset{[h/p]}{i = 0}{\sum} (-1)^{h+i} \left( \array{ (p-1)(k-i) \\ h - p i } \right) \beta P^{h+k-i}P^i \\ & + \underoverset{[(h-1)/p]}{i = 0}{\sum} (-1)^{h+i-1} \left( \array{ (p-1)(k-i) - 1 \\ h - p i - 1 } \right) P^{h+k-i} \beta P^i \end{aligned}$$

and whose coproduct $\Psi$ is subject to the following relations:

for $p = 2$:

$$\Psi(Sq^n) \;=\; \underoverset{k = 0}{n}{\sum} Sq^k \otimes Sq^{n-k}$$

for $p \gt 2$:

$$\Psi(P^n) \;=\; \underoverset{n}{k = 0}{\sum} P^k \otimes P^{n-k}$$

and

$$\Psi(\beta) = \beta \otimes 1 + 1 \otimes \beta \,.$$

e.g. (Kochmann 96, p. 52)

Definition

The $\mathbb{F}_p$-linear dual of the mod $p$-Steenrod algebra (def. ) is itself naturally a graded commutative Hopf algebra (with coproduct the linear dual of the original product, and vice versa), called the dual Steenrod algebra $\mathbb{A}_{\mathbb{F}_p}^\ast$.

Proposition

There is an isomorphism

$$\mathcal{A}^\ast_{p} \simeq H_\bullet( H \mathbb{F}_p, \mathbb{F}_p ) = \pi_\bullet( H \mathbb{F}_p \wedge H \mathbb{F}_p ) \,.$$

(e.g. Ravenel 86, p. 49, Rognes 12, remark 7.24)

We now give the generators-and-relations description of the dual Steenrod algebra $\mathcal{A}^\ast_{p}$ from def. , in terms of linear duals of the generators for $\mathcal{A}_{p}$ itself, according to def. .

Theorem

(Milnor’s theorem)

The dual mod $2$-Steenrod algebra $\mathcal{A}^\ast_{2}$ (def. ) is, as an associative algebra, the free graded commutative algebra

$$\mathcal{A}^\ast_{p} \simeq Sym_{\mathbb{F}_p}(\xi_1, \xi_2, \cdots, )$$

on generators:

• $\xi_n$, $n \geq 1$ being the linear dual to $Sq^{p^{n-1}} Sq^{p^{n-2}} \cdots Sq^p Sq^1$,

  of degree $2^n -1$.

The dual mod $p$-Steenrod algebra $\mathcal{A}^\ast_{p}$ (def. ) is, as an associative algebra, the free graded commutative algebra

$$\mathcal{A}^\ast_{p} \simeq Sym_{\mathbb{F}_p}(\xi_1, \xi_2, \cdots, \;\tau_0, \tau_1, \cdots)$$

on generators:

• $\xi_n$, $n \geq 1$ being the linear dual to $P^{p^{n-1}} P^{p^{n-2}} \cdots P^p P^1$,

  of degree $2(p^n-1)$.

• $\tau_n$ being linear dual to $P^{p^{n-1}} P^{p^{n-2}} \cdots P^p P^1\beta$.

Moreover, the coproduct on $\mathcal{A}^\ast_{p}$ is given on generators by

$$\Psi(\xi_n) = \underoverset{k = 0}{n}{\sum} \xi_{n-k}^{p^k} \otimes \xi_k$$

and

$$\Psi(\tau_n) = \tau_n \otimes 1 + \underoverset{k=0}{n}{\sum} \xi_{n-k}^{p^k} \xi_{n-k}^{p^k}\otimes \tau_k \,,$$

where we set $\xi_0 \coloneqq 1$.

(This defines the coproduct on the full algbra by it being an algebra homomorphism.)

This is due to (Milnor 58). See for instance (Kochmann 96, theorem 2.5.1, Ravenel 86, chapter III, theorem 3.1.1)

The cobar complex

In order to compute the second page of the classical $H \mathbb{F}_p$-Adams spectral sequence (cor. ) we consider a suitable cochain complex whose cochain cohomology gives the relevant Ext-groups.

Definition

Let $(\Gamma,A)$ be a graded commutative Hopf algebra, hence a commutative Hopf algebroid for which the left and right units coincide $\eta \;\colon\; A \longrightarrow \Gamma$ (remark ).

Then the unit coideal of $\Gamma$ is the cokernel

$$\overline{\Gamma} \coloneqq coker( A \overset{\eta}{\longrightarrow} \Gamma) \,.$$

Remark

By co-unitality of graded commutative Hopf algebras (def. ) $\epsilon \circ \eta = id_A$ the defining projection of the unit coideal (def. )

$$A \overset{\eta}{\longrightarrow} \Gamma \overset{}{\longrightarrow} \overline{\Gamma}$$

forms a split exact sequence which exhibits a direct sum decomposition

$$\Gamma \simeq A \oplus \overline{\Gamma} \,.$$

Lemma

Let $(\Gamma,A)$ be a commutative Hopf algebra, hence a commutative Hopf algebroid for which the left and right units coincide $\eta \;\colon\; A \longrightarrow \Gamma$.

Then the unit coideal $\overline{\Gamma}$ (def. ) carries the structure of an $A$-bimodule such that the projection morphism

$$\Gamma \longrightarrow \overline{\Gamma}$$

is an $A$-bimodule homomorphism. Moreover, the coproduct $\Psi \;\colon\; \Gamma \longrightarrow \Gamma \otimes_A \Gamma$ descends to a morphism $\overline{\Gamma} \;\colon\; \overline{\Gamma} \longrightarrow \overline{\Gamma} \otimes_A \overline{\Gamma}$ such that the projection intertwines the two coproducts.

Proof

For the first statement, consider the commuting diagram

$$\array{ A \otimes A &\overset{A \otimes \eta}{\longrightarrow}& A \otimes \Gamma &\longrightarrow& A \otimes \overline{\Gamma} \\ \downarrow && \downarrow && \downarrow^{\mathrlap{\exists}} \\ A &\underset{\eta}{\longrightarrow}& \Gamma &\longrightarrow& \overline{\Gamma} } \,,$$

where the left commuting square exhibits the fact that $\eta$ is a homomorphism of left $A$-modules.

Since the tensor product of abelian groups $\otimes$ is a right exact functor it preserves cokernels, hence $A \otimes \overline{\Gamma}$ is the cokernel of $A \otimes A \to A\otimes \Gamma$ and hence the right vertical morphisms exists by the universal property of cokernels. This is the compatible left module structure on $\overline{\Gamma}$. Similarly the right $A$-module structure is obtained.

For the second statement, consider the commuting diagram

$$\array{ A &\overset{\eta}{\longrightarrow}& \Gamma &\longrightarrow& \overline{\Gamma} \\ {}^{\mathllap{\eta}}\downarrow && \downarrow^{\mathrlap{\Psi}} && \downarrow^{\mathrlap{\exists}} \\ \Gamma \simeq \Gamma \otimes_A A &\underset{id \otimes_A \eta}{\longrightarrow}& \Gamma \otimes_A \Gamma &\longrightarrow& \overline{\Gamma} \otimes_A \overline{\Gamma} } \,.$$

Here the left square commutes by one of the co-unitality conditions on $(\Gamma,A)$, equivalently this is the co-action property of $A$ regarded canonically as a $\Gamma$-comodule.

Since also the bottom morphism factors through zero, the universal property of the cokernel $\overline{\Gamma}$ implies the existence of the right vertical morphism as shown.

Definition

(cobar complex)

Let $(\Gamma,A)$ be a commutative Hopf algebra, hence a commutative Hopf algebroid for which the left and right units coincide $A \overset{\eta}{\longrightarrow} \Gamma$. Let $N$ be a left $\Gamma$-comodule.

The cobar complex $C^\bullet_\Gamma(N)$ is the cochain complex of abelian groups with terms

$$C^s_\Gamma(N) \coloneqq \underset{s\; factors}{ \underbrace{ \overline{\Gamma} \otimes_A \cdots \otimes_A \overline{\Gamma} } } \otimes_A N$$

(for $\overline{\Gamma}$ the unit coideal of def. , with its $A$-bimodule structure via lemma )

and with differentials $d_s \colon C^s_\Gamma(N) \longrightarrow C^{s+1}_\Gamma(N)$ given by the alternating sum of the coproducts via lemma .

(Ravenel 86, def. A1.2.11)

Proposition

Let $(\Gamma,A)$ be a commutative Hopf algebra, hence a commutative Hopf algebroid for which the left and right units coincide $A \overset{\eta}{\longrightarrow} \Gamma$. Let $N$ be a left $\Gamma$-comodule.

Then the cochain cohomology of the cobar complex $C^\bullet_\Gamma(N)$ (def. ) is the Ext-groups of comodules from $A$ (regarded as a left comodule via def. ) into $N$

$$H^\bullet(C^\bullet_\Gamma(N)) \;\simeq\; Ext^\bullet_\Gamma(A,N) \,.$$

(Ravenel 86, cor. A1.2.12, Kochman 96, prop. 5.2.1)

Proof idea

One first shows that there is a resolution of $N$ by co-free comodules given by the complex

$$D^\bullet_\Gamma(N) \coloneqq \Gamma \otimes_A \overline{\Gamma}^{\otimes_A^{\bullet}} \otimes_A N$$

with differentials given by the alternating sum of the coproducts. This is called the cobar resolution of $N$.

To see that this is indeed a resolution, one observes that a contracting homotopy is given by

$$s (\gamma \gamma_1\vert \cdots \vert \gamma_s n) \coloneqq \epsilon(\gamma) \gamma_1\vert \cdots \vert \gamma_s n$$

for $s \gt 0$ and

$$s(\gamma n) \coloneqq 0 \,.$$

Now from lemma , in view of remark , and since $A$ is trivially projective over itself, it follows that this is an $F$-acyclic resolution for $F \coloneqq Hom_\Gamma(A,-)$.

This means that the resolution serves to compute the Ext-functor in question and we get

$$\begin{aligned} Ext^\bullet_\Gamma(A,N) & \simeq H^\bullet(Hom_\Gamma(A, D^\bullet_\Gamma(N))) \\ & = H^\bullet( Hom_\Gamma(A, \Gamma \otimes_A \overline{\Gamma}^{\otimes_A^{\bullet}} \otimes_A N ) ) \\ &\simeq H^\bullet( Hom_A(A, \overline{\Gamma}^{\otimes_A^{\bullet}} \otimes_A N ) ) \\ & \simeq H^\bullet( \overline{\Gamma}^{\otimes_A^{\bullet}} \otimes_A N ) \,, \end{aligned} \,,$$

where the second-but-last equivalence is the isomorphism of the co-free/forgetful adjunction

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

from prop. , while the last equivalence is the isomorphism of the free/forgetful adjunction

$$A Mod \underoverset {\underset{forget}{\longrightarrow}} {\overset{free}{\longleftarrow}} {\bot} Ab$$

The May spectral sequence

The cobar complex (def. ) realizes the second page of the classical Adams spectral sequence (cor. ) as the cochain cohomology of a cochain complex. This is still hard to compute directly, but we now discuss that this cochain complex admits a filtration so that the induced spectral sequence of a filtered complex is computable and has trivial extension problem (rmk.). This is called the May spectral sequence.

We obtain this spectral sequence in prop. below. First we need to consider some prerequisites.

Lemma

Let $(\Gamma,A)$ be a graded commutative Hopf algebra, i.e. a graded commutative Hopf algebroid with left and right unit coinciding for which the underlying $A$-algebra of $\Gamma$ is a free graded commutative $A$-algebra on a set of generators $\{x_i\}_{i \in I}$

such that

1. all generators $x_i$ are primitive elements;

2. $A$ is in degree 0;

3. $(i \lt j) \Rightarrow (deg(x_i) \leq deg(x_j))$;

4. there are only finitely many $x_i$ in a given degree,

then the Ext of $\Gamma$-comodules from $A$ to itself is the free graded commutative algebra on these generators

$$Ext_\Gamma(A,A) \simeq A[\{x_i\}_{i \in I}] \,.$$

(Ravenel 86, lemma 3.1.9, Kochman 96, prop. 3.7.5)

Proof

Consider the co-free left $\Gamma$-comodule (prop.)

$$\Gamma \otimes_A A[\{y_i\}_{i \in I}]$$

and regard it as a chain complex of left comodules by defining a differential via

$$d \colon x_i \mapsto y_i$$

$$d \colon y_i \mapsto 0$$

and extending as a graded derivation.

We claim that $d$ is a homomorphism of left comodules: Due to the assumption that all the $x_i$ are primitive we have on generators that

$$\begin{aligned} (id,d) ( \Psi(x_i) ) & = (id,d) ( x_i \otimes 1 + 1 \otimes x_i ) \\ & = \underset{= 0}{x_i \otimes \underbrace{(d 1)} } + \underset{= y_i}{ 1 \otimes \underbrace{(d x_i)} } \\ & = \Psi( d x_i ) \end{aligned}$$

and

$$\begin{aligned} (id,d)( \Psi(y_i) ) & = (id,d) ( 1, y_i ) \\ & = (1, d y_i) \\ & = 0 \\ & = \Psi( 0 ) \\ & = \Psi(d y_i) \end{aligned} \,.$$

Since $d$ is a graded derivation on a free graded commutative algbra, and $\Psi$ is an algebra homomorphism, this implies the statement for all other elements.

Now observe that the canonical chain map

$$(\Gamma \otimes_A A[\{y_i\}_{i \in I}] ,\; d) \overset{\simeq_{qi}}{\longrightarrow} A$$

(which projects out the generators $x_i$ and $y_i$ and is the identity on $A$), is a quasi-isomorphism, by construction. Therefore it constitutes a co-free resolution of $A$ in left $\Gamma$-comodules.

Since the counit $\eta$ is assumed to be flat, and since $A[\{y_i\}_{i \in I}]$ is degreewise a free module over $A$, hence in particular a projective module, prop. says that the above is an acyclic resolution with respect to the functor $Hom_{\Gamma}(A,-) \colon \Gamma CoMod \longrightarrow A Mod$. Therefore it computes the Ext-functor. Using that forming co-free comodules is right adjoint to forgetting $\Gamma$-comodule structure over $A$ (prop. ), this yields:

$$\begin{aligned} Ext^\bullet_\Gamma(A,A) & \simeq H_\bullet(Hom_\Gamma(A, \Gamma \otimes_A A[\{y_i\}_{i \in I}] ), d) \\ & \simeq H_\bullet(Hom_A(A, A[\{y_i\}_{i \in I}] ), d= 0 ) \\ & \simeq Hom_A(A, A[\{y_i\}_{i \in I}] ) \\ & \simeq A[\{x_i\}_{i \in I}] \end{aligned} \,.$$

Lemma

If $(\Gamma,A)$ as above is equipped with a filtering, then there is a spectral sequence

$$\mathcal{E}_1 \;=\; Ext_{gr_\bullet \Gamma}(gr_\bullet A, gr_\bullet A) \;\Rightarrow\; Ext_{\Gamma}(A, A)$$

converging to the Ext over $\Gamma$ from $A$ to itself, whose first page is the $Ext$ over the associated graded Hopf algebra $gr_\bullet \Gamma$.

(Ravenel 86, lemma 3.1.9, Kochman 96, prop. 3.7.5)

Proof

The filtering induces a filtering on the cobar complex (def. ) which computes $Ext_\Gamma$ (prop. ). The spectral sequence in question is the corresponding spectral sequence of a filtered complex. Its first page is the homology of the associated graded complex (by this prop.), which hence is the homology of the cobar complex (def. ) of the associated graded Hopf algebra $gr_\bullet \Gamma$. By prop. this is the Ext-groups as shown.

Let now $A \coloneqq \mathbb{F}_2$, $\Gamma \coloneqq \mathcal{A}^\bullet_{2}$ be the mod 2 dual Steenrod algebra. By Milnor's theorem (prop. ), as an $\mathbb{F}_2$-algebra this is

$$\mathcal{A}^\bullet_{2} = Sym_{\mathbb{F}_2}(\xi_1, \xi_2, \cdots) \,.$$

and the coproduct is given by

$$\Psi(\xi_n) = \underoverset{k = 0}{i}{\sum} \xi_{i - k}^{2^k} \otimes \xi_k \,,$$

where we set $\xi_0 \coloneqq 1$.

Definition

Introduce new generators

$$h_{i,n} \coloneqq \left\{ \array{ \xi_i^{2^n} & for \; i \geq 1, \, k \geq 0 \\ 1 & for \; i = 0 } \right.$$

Remark

By binary expansion of powers, there is a unique way to express every monomial in $\mathbb{F}_2[\xi_1, \xi_2, \cdots]$ as a product of the new generators in def. such that each such element appears at most once in the product. E.g.

$$\begin{aligned} \xi_i^5 \xi_j^7 & = \xi_i^{2^0 + 2^2} \xi_j^{2^0 + 2^1 + 2^2} \\ & = h_{i,0} h_{i,1} h_{j,0} h_{j,1} h_{j,2} \end{aligned} \,.$$

Proposition

In terms of the generators $\{h_{i,n}\}$ from def. , the coproduct on the dual Steenrod algebra $\mathcal{A}^\ast_{2}$ takes the following simple form

$$\Psi(h_{i,n}) \;=\; \underoverset{k = 0}{i}{\sum} h_{i-k,n+k}\otimes h_{k,n} \,.$$

Proof

Using that the coproduct of a bialgebra is a homomorphism for the algebra structure and using freshman's dream arithmetic over $\mathbb{F}_2$, one computes:

$$\begin{aligned} \Psi(h_{i,n}) & = \Psi\left(\xi_i^{2^n}\right) \\ & = (\Psi(\xi_i))^{2^n} \\ & = \left(\underoverset{k = 0}{i}{\sum} \xi_{i-k}^{2^k} \otimes \xi_k\right)^{2^n} \\ & = \underoverset{k = 0}{i}{\sum} \left(\xi_{i-k}^{2^k}\right)^{2^n} \otimes \xi^{2^n}_k \\ & = \underoverset{k = 0}{i}{\sum} \xi_{i-k}^{2^k \cdot 2^n} \otimes \xi^{2^n}_k \\ & = \underoverset{k = 0}{i}{\sum} \xi_{i-k}^{2^{(k+n)}} \otimes \xi^{2^n}_k \\ & = \underoverset{k = 0}{i}{\sum} h_{i-k,n+k}\otimes h_{k,n} \end{aligned} \,.$$

Proposition

There exists a converging spectral sequence of graded $\mathbb{F}_2$-vector spaces of the form

$$E_1^{s,t,p} = \mathbb{F}_2[ \{ h_{i,n} \}_{{i \geq 1,} \atop {n \geq 0}}] \;\Rightarrow\; Ext^{s,t}_{\mathcal{A}_2^\ast}(\mathbb{F}_2, \mathbb{F}_2) \,,$$

called the May spectral sequence (where $s$ and $t$ are from the bigrading of the spectral sequence itself, while the index $p$ is that of the graded $\mathbb{F}_2$-vector spaces), with

1. $h_{i,n} \in E_1^{1, 2^{2^{i+n} - 2^n - 1, 2i - 1 }}$

2. first differential given by

   $$d_1 (h_{i,n}) = \underoverset{k = 0}{i}{\sum} h_{i-k,n+k}\otimes h_{k,n} \,;$$

3. higher differentials of the form

   $$d_r \;\colon\; E_r^{s,t, p} \longrightarrow E_r^{s+1, t-1, p-2r+1} \,,$$

   where the filtration is by maximal degree.

Notice that since everything is $\mathbb{F}_2$-linear, the extension problem of this spectral sequence is trivial.

(Kochman 96, prop. 5.3.1)

Proof

Define a grading on the dual Steenrod algebra $\mathcal{A}^\bullet_{2}$ (theorem ) by taking the degree of the generators from def. to be (this idea is due to (Ravenel 86, p.69))

$${\vert h_{i,n} \vert} \coloneqq 2i-1$$

and extending this additively to monomials, via the unique decomposition of remark .

For example

$$\begin{aligned} \vert \xi_i^5 \xi_j^7\vert & = {\vert h_{i,0} h_{i,1} h_{j,0} h_{j,1} h_{j,2} \vert} \\ & = 2(2i-1) + 3(2j-1) \end{aligned} \,.$$

Consider the corresponding increasing filtration

$$\cdots \subset F_p \mathcal{A}^\ast_{2} \subset F_{p+1} \mathcal{A}^\ast_{2} \subset \cdots \subset \mathcal{A}^\ast_{2}$$

with filtering stage $p$ containing all elements of total degree $\leq p$.

Observe via prop. that

$$\begin{aligned} \Psi(h_{i,n}) & = \underset{deg = 2i-1}{\underbrace{h_{i,n} \otimes 1}} + \underoverset{0 \lt k \lt i}{}{\sum} \underset{deg = 2i-2}{\underbrace{h_{i-k,n+k} \otimes h_{k,n}}} + \underset{deg = 2i-1}{\underbrace{1 \otimes h_{i,n} }} \end{aligned} \,.$$

This means that after projection to the associated graded Hopf algebra

$$F_\bullet \mathcal{A}^\ast_{2} \longrightarrow gr_\bullet \mathcal{A}^\ast_{2} \coloneqq F_\bullet( \mathcal{A}^\ast_2)/F_{\bullet-1}( \mathcal{A}^\ast_2 )$$

all the generators $h_{i,n}$ become primitive elements:

$$\begin{aligned} \Psi(h_{i,n}) & = h_{i,n}\otimes 1 + 1 \otimes h_{i,n} \;\;\;\;\; \in gr_\bullet \mathcal{A}^\ast_{2} \otimes gr_\bullet \mathcal{A}^\ast_{2} \end{aligned} \,.$$

Hence lemma applies and says that the $Ext$ from $\mathbb{F}_2$ to itself over the associated graded Hopf algebra is the polynomial algebra in these generators:

$$Ext_{gr_\bullet \mathcal{A}^\ast_{2}}(\mathbb{F}_2,\mathbb{F}_2) \simeq \mathbb{F}_2[\{h_{i,n}\}_{{i \geq 1,} \atop {n \geq 0}}] \,.$$

Moreover, lemma says that this is the first page of a spectral sequence that converges to the $Ext$ over the original Hopf algebra:

$$\mathcal{E}_1 = \mathbb{F}_2[\{h_{i,n}\}_{{i \geq 1} \atop {n \geq 0}}] \;\Rightarrow\; Ext_{\mathcal{A}^\ast_{2}}(\mathbb{F}_2,\mathbb{F}_2) \,.$$

Moreover, again by lemma , the differentials on any $r$-page are the restriction of the differentials of the bar complex to the $r$-almost cycles (prop.). Now the differential of the cobar complex is the alternating sum of the coproduct on $\mathcal{A}^\ast_{2}$, hence by prop. this is:

$$d_1 (h_{i,n}) = \underoverset{k = 0}{i}{\sum} h_{i-k,n+k}\otimes h_{k,n} \,.$$

The second page

Now we use the May spectral sequence (prop. ) to compute the second page and in fact the stable page of the classical Adams spectral sequence (cor. ) in low internal degrees $t-s$.

Lemma

(terms on the second page of May spectral sequence)

In the range $t - s \leq 13$, the second page of the May spectral sequence for $Ext_{\mathbb{A}^\ast_{\mathbb{F}_2}}(\mathbb{F}_2,\mathbb{F}_2)$ has as generators all the

• $h_n$

• $b_{i,n} \coloneqq (h_{i,n})^2$

as well as the element

• $x_7 \coloneqq h_{2,0} h_{2,1} + h_{1,1} h_{3,0}$

subject to the relations

• $h_n h_{n+1} = 0$

• $h_2 b_{2,0} = h_0 x_7$

• $h_2 x_7 = h_0 b_{2,1}$.

e.g. (Ravenel 86, lemma 3.2.8 and lemma 3.2.10, Kochman 96, lemma 5.3.2)

Proof

Remember that the differential in the cobar complex (def. ) lands not in $\Gamma = \mathcal{A}^\ast_2$ itself, but in the unit coideal $\overline{\Gamma} \coloneqq coker(\eta)$ where the generator $h_{0,n} = \xi_0 = 1$ disappears.

Using this we find for the differential $d_1$ of the generators in low degree on the first page of the May spectral sequence (prop. ) via the formula for the differential from prop. , the following expressions:

$$\begin{aligned} d_1(h_n) & \coloneqq d_1(h_{1,n}) \\ & = \overline{\Psi}(h_{1,n}) \\ & = h_{1,n} \otimes \underset{= 0}{\underbrace{ h_{0,n} }} + \underset{= 0}{\underbrace{ h_{0,n+1} }} \otimes h_{1,n} \\ & = 0 \end{aligned}$$

and hence all the elements $h_n$ are cocycles on the first page of the May spectral sequence.

Also, since $d_1$ is a derivation (by definition of the cobar complex, def. ) and since the product of the image of the cobar complex in the first page of the May spectral sequence is graded commutative, we have for all $n,k$ that

$$\begin{aligned} d_1 (h_{n,k})^2 & = 2 h_{n,k} (d_1 (h_{n,k})) \\ & = 0 \end{aligned}$$

(since $2 = 0 \; mod \; 2$).

Similarly we compute $d_1$ on the other generators. These terms do not vanish, but so they impose relations on products in the cobar complex:

$$\begin{aligned} d_1(h_{2,0}) & = h_{1,1} \otimes h_{1,0} \\ d_1( h_{2,1} ) & = h_{1,2} \otimes h_{1,1} \\ d_1( h_{2,2} ) & = h_{1,3} \otimes h_{1,2} \\ d_1( h_{2,3} ) & = h_{1,4} \otimes h_{1,3} \\ d_1( h_{3,0} ) & = h_{2,1} \otimes h_{1,0} + h_{1,2} \otimes h_{2,0} \end{aligned}$$

This shows that $h_n h_{n+1} = 0$ in the given range.

The remaining statements follow similarly.

Remark

With lemma , so far we see the following picture in low degrees.

$$\array{ \vdots & \vdots \\ 3 & h_0^4 & & & {h_1^3},\; {h_0^2 h_2} \\ 2 & h_0^2 & & h_1^2 & h_0 h_2 \\ 1& h_0 & h_1 & & h_2 \\ & 0 & 1 & 2& 3 & 4 }$$

Here the relation

$$h_0 \otimes h_1 = 0$$

removes a vertical tower of elements above $h_1$.

So far there are two different terms in degree $(s,t-s) = (3,3)$. The next lemma shows that these become identified on the next page.

Lemma

(differentials on the second page of the May spectral sequence)

The differentials on the second page of the May spectral sequence (prop. ) relevant for internal degrees $t-s \leq 12$ are

1. $d_2(h_{n}) = 0$

2. $d_2(b_{2,n}) = h_n^2 h_{n+2} + h_{n+1}^3$

3. $d_2(x_7) = h_0 h_2^2$

4. $d_2(b_{3,0}) = h_1 b_{2,1} + h_3 b_{2,0}$

(Kochman 96, lemma 5.3.3)

Proof

The first point follows as before in lemma , in fact the $h_n$ are infinite cycles in the May spectral sequence.

We spell out the computation for the second item:

We may represent $b_{2,k}$ by $\xi_2^{2^k} \times \xi_2^{2^k}$ plus terms of lower degree. Choose the representative

$$B_{2,k} = \xi_2^{2^k} \otimes \xi_2^{2^k} \;+\; \xi_1^{2^{k + 1}} \otimes \xi_1^{2^k} \xi_2^{2^k} \;+\; \xi_1^{2^{k+1}} \xi^{2^k} \otimes \xi_1^{2^k} \,.$$

Then we compute $d B_{2,k}$, using the definition of the cobar complex (def. ), the value of the coproduct on dual generators (theorem ), remembering that the coproduct $\Psi$ on a Hopf algebra is a homomorphism for the underlying commutative ring, and using freshman's dream arithmetic to evaluate prime-2 powers of sums. For the three summands we obtain

$$\begin{aligned} d ( \xi_2^{2^k} \otimes \xi_2^{2^k}) & = \overline{\Psi} (\xi_2^{2^k}) \otimes \xi_2^{2^k} + \xi_2^{2^k} \otimes \overline{\Psi}(\xi_2^{2 k}) \\ & = \underset{c_1}{\underbrace{\xi_1^{2^{k+1}} \otimes \xi_1^{2^k} \otimes \xi_2^{2^k}}} \;+\; \underset{c_2}{\underbrace{\xi_2^{2^k} \otimes \xi_1^{2^{k+1}} \otimes \xi_1^{2^k}}} \end{aligned}$$

and

$$\begin{aligned} d ( \xi_1^{2^{k + 1}} \otimes \xi_1^{2^k} \xi_2^{2^k} ) & = \xi_1^{2^k} \otimes \overline{\Psi}( \xi_1^{2^k} \xi_2^{2^k}) \\ & = \xi_1^{2^{k + 1}} \otimes \left( \xi_1^{2^k} \otimes 1 + 1 \otimes \xi_1^{2^k} \right) \left( \xi_2^{2^k} \otimes 1 + \xi_1^{2^{k+1}} \otimes \xi_1^{2^k} + 1 \otimes \xi_2^{2^k} \right) \\ & = \underset{b}{ \underbrace{ \xi_1^{2^{k + 1}} \otimes \xi_1^{2^{k+1}+ 2^k} \otimes \xi_1^{2^k} }} \;+\; \underset{c_1}{\underbrace{ \xi_1^{2^{k+1}} \otimes \xi_1^{2^k} \otimes \xi_2^{2^k} }} \;+\; \underset{a}{\underbrace{ \xi_1^{2^{k+1}} \otimes \xi_2^{2^k} \otimes \xi_1^{2^k} }} \;+\; \xi_1^{2^{k+1}} \otimes \xi_1^{2^{k+1}} \otimes \xi_1^{2^{k+1}} \end{aligned}$$

and

$$\begin{aligned} d( \xi_1^{2^{k+1}} \xi^{2^k} \otimes \xi_1^{2^k} ) & = \overline{\Psi}( \xi_1^{2^{k+1}} \xi^{2^k} ) \otimes \xi_1^{2^k} \\ & = \left( \xi_1^{2^{k+1}} \otimes 1 + 1 \otimes \xi_1^{2^{k+1}} \right) \left( \xi_2^{2^k} \otimes 1 + \xi_1^{2^{k+1}} \otimes \xi_1^{2^k} + 1 \otimes \xi_2^{2^k} \right) \otimes \xi_1^{2^k} \\ & = \xi_1^{2^{k+2}} \otimes \xi_1^{2^k} \otimes \xi_1^{2^k} \;+\; \underset{a}{\underbrace{ \xi_1^{2^{k+1}} \otimes \xi_2^{2^k} \otimes \xi_1^{2^k} }} \;+\; \underset{c_2}{\underbrace{ \xi_2^{2^k} \otimes \xi_1^{2^{k+1}} \otimes \xi_1^{2^k} }} \;+\; \underset{b}{\underbrace{ \xi_1^{2^{k+1}} \otimes \xi_1^{2^{k+1} + 2^k} \otimes \xi_1^{2^k} }} \end{aligned} \,.$$

The labeled summands appear twice in $d B_{2,k}$ hence vanish (mod 2). The remaining terms are

$$d B_{2,k} = \xi_1^{2^{k+1}} \otimes \xi_1^{2^{k+1}} \otimes \xi_1^{2^{k+1}} \;+\; \xi_1^{2^{k+2}} \otimes \xi_1^{2^k} \otimes \xi_1^{2^k}$$

and these indeed represent the claimed elements.

Remark

With lemma the picture from remark is further refined:

For $k = 0$ the differentia $d_2(b_{2,n}) = h_n^2 h_{n+2} + h_{n+1}^3$ means that on the third page of the May spectral sequence there is an identification

$$h_1^3 = h_0^2 h_2 \,.$$

Hence where on page two we saw two distinct elements in bidegree $(s,t-s) = (3,3)$, on the next page these merge:

$$\array{ \vdots & \vdots \\ 3 & h_0^4 & & & {h_1^3} = {h_0^2 h_2} \\ 2 & h_0^2 & & h_1^2 & h_0 h_2 \\ 1& h_0 & h_1 & & h_2 \\ & 0 & 1 & 2& 3 & 4 }$$

Proceeding in this fashion, one keeps going until the 4-page of the May spectral sequence (Kochman 96, lemma 5.3.5). Inspection of degrees shows that this is sufficient, and one obtains:

Theorem

(stable page of classical Adams spectral sequence)

In internal degree $t-s \leq 12$ the infinity page (def. ) of the classical Adams spectral sequence (cor. ) is spanned by the items in the following table

Here every dot is a generator for a copy of $\mathbb{Z}/2\mathbb{Z}$. Vertical edges denote multiplication with $h_0$ and diagonal edges denotes multiplication with $h_1$.

e.g. (Ravenel 86, theorem 3.2.11, Kochman 96, prop. 5.3.6), graphics taken from (Schwede 12))

The first dozen stable stems

Theorem gives the stable page of the classical Adams spectral sequence in low degree. By corollary and def. we have that a vertical sequence of dots encodes an 2-primary part of the stable homotopy groups of spheres according to the graphical calculus of remark (the rules for determining group extensions there is just the solution to the extension problem (rmk.) in view of def. ):

$k =$	0	1	2	3	4	5	6	7	8	9	10	11	12	13
$\pi_k(\mathbb{S}\otimes \mathbb{Z}_{(2)}) =$	$\mathbb{Z}_{(2)}$	$\mathbb{Z}/2$	$\mathbb{Z}/2$	$\mathbb{Z}/8$	$0$	$0$	$\mathbb{Z}/2$	$\mathbb{Z}/16$	$(\mathbb{Z}/2)^2$	$(\mathbb{Z}/2)^3$	$\mathbb{Z}/2$	$\mathbb{Z}/8$	$0$	$0$

The full answer in this range turns out to be this:

$k =$	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	$\cdots$
$\pi_k(\mathbb{S}) =$	$\mathbb{Z}$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	$\mathbb{Z}_{24}$	$0$	$0$	$\mathbb{Z}_2$	$\mathbb{Z}_{240}$	$(\mathbb{Z}_2)^2$	$(\mathbb{Z}_2)^3$	$\mathbb{Z}_6$	$\mathbb{Z}_{504}$	$0$	$\mathbb{Z}_3$	$(\mathbb{Z}_2)^2$	$\mathbb{Z}_{480} \oplus \mathbb{Z}_2$	$\cdots$

And expanding the range yields this :

(graphics taken from Hatcher’s website)

$E$-injective 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.

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

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. , 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. .

Example

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

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. .

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. .

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. . Observe that this morphism has a retraction given by $\mu \wedge id$. Therefore it is a monomorphism by example .

Remark/Warning

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

Lemma

1. If $f \colon B\longrightarrow A$ is a monomorphism in the sense of def. , 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. , 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 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 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. , 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. , 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 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. , and so by lemma 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. , 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. .

Lemma

Any two consecutive maps in an $E$-Adams resolution, def. , 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. , 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

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

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. follows (Bousfield 79, section 5). In (Hopkins 99, proof of corollary 5.3) the same notation is used not for the homotopy fiber of $\mathbb{S} \overset{e}{\to} E$ but for its 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 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 is indeed an $E$-Adams resolution of $X$ in the sense of def. .

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 .

$E$-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. :

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

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

(Hopkins 99, def. 4.10)

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. ) induces an $E$-Adams tower, def. 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 . Observe that $\sigma_1$ is an $E$-monomorphism and $\rho_1$ is an $E$-epimorphism in the sense of def. .

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 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. ), 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. 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 . Then its associated inverse sequence according to def. is the sequence from def.

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

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 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 resolution” for “$E$-injective resolution” as in def. , “$E$-Adams tower” for def. 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. , or equivalently (by prop. ) an $E$-Adams tower over $X$, def. ,

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

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

Proposition

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

1. the tower spectral sequence of its associated $E$-Adams tower (def. ), i.e. the spectral sequence of def. ;

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

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 – that is considered in the original sources (Adams 74, p. 318, Bousfield 79, p. 271).

As derived descent in $E_\infty$-geometry

It turns out that the E-nilpotent completion $X^\wedge_E$ according to Bousfield‘s original definition – to which the $E$-Adams spectrum seequence converges under good conditions (theorem , theorem ) – is equivalently the homotopy limit over the tower of totalizations of the canonical cosimplicial spectrum $E^\bullet \wedge X$ (prop.).

This means that the $E$-Adams spectral sequence may equivalently be regarded as computing descent of quasicoherent infinity-stacks in E-infinity geometry along the canonical morphisms $Spec(E)\longrightarrow$ Spec(S).

This perspective seems to originate in (Hopkins 99, remark 5.5 (ii)) and is greatly expanded on in (Lurie 10, lectures 8, 9). Exposition of the basic idea is in Mathew 12 and with more details in (Wilson 13). The first published proof of the equivalence of this new perspective to the Adams-Bousfield theory is (Mathew-Naumann-Noel 15, prop. 2.14).

We give a survey here.

under construction

First we recall

• Spectral sequences computing homotopy groups of filtered objects

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

• Homotopy groups of cosimplicial totalizations filtered by coskeleta.

Finally we consider specifically the examples of such given by

• Canonical cosimplicial resolutions of E-∞ algebras.

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 E-Adams-Novikov spectral sequence.

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.

Example

For instance

• $\mathcal{C} = Spec$ the stable (∞,1)-category of spectra.

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. , 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. 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. , 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. , applied to the filtration of a totalization by coskeleta as in def. , 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. is satisfied for

• $E = H \mathbb{F}_p$ an Eilenberg-MacLane spectrum with $mod\;p$ coefficients;

• $E = B P$ the Brown-Peterson spectrum;

• $E = MU$ the complex cobordism spectrum.

It is NOT satisfied for

• $E = H \mathbb{Z}$ the Eilenberg-MacLane spectrum for integer coefficients;

• $E = M S U$.

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.

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. . Then by prop. and prop. 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. and remark , the $E_2$-term of this spectral sequence is given by the Ext-groups over the $E$-Steenrod Hopf algebroid.

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

Related concepts

• d-invariant, e-invariant, f-invariant

• change of rings theorem

• Adams-Novikov spectral sequence

• Bousfield-Kan spectral sequence

• May spectral sequence, chromatic spectral sequence

• Steenrod algebra

• descent spectral sequence

• equivariant Adams spectral sequence

tower diagram/filtering	spectral sequence of a filtered stable homotopy type
filtered chain complex	spectral sequence of a filtered complex
Postnikov tower	Atiyah-Hirzebruch spectral sequence
chromatic tower	chromatic spectral sequence
skeleta of simplicial object	spectral sequence of a simplicial stable homotopy type
skeleta of Sweedler coring of E-∞ algebra	Adams spectral sequence
filtration by support	…
slice filtration	slice spectral sequence

References

General

The original sources are

• Frank Adams, On the structure and applications of the Steenrod algebra, Comm. Math. Helv. 32 (1958), 180–214.

• Sergei Novikov, The methods of algebraic topology from the viewpoint of cobordism theories, Izv. Akad. Nauk. SSSR. Ser. Mat. 31 (1967), 855–951 (Russian). (Novikov67)

• John Frank Adams, Stable homotopy and generalized homology, Chicago Lectures in mathematics, 1974

based in parts on

• John Frank Adams, Lectures on generalised cohomology, in Peter Hilton (ed.) Category Theory, Homology Theory and Their Applications III, volume 99 of Lecture Notes in Mathematics (1969), Springer-Verlag Berlin-Heidelberg-New York.

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.

The general convergence theorems are due to

• 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)

based on

• Aldridge Bousfield, Daniel Kan, The core of a ring, Journal of Pure and Applied Algebra, Volume 2, Issue 1, April 1972, Pages 73-81

A streamlined presentation of Adams resolutions and their invariance properties, close in spirit to the theory of injective resolutions 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

• Mike Hopkins, section 5 of Complex oriented cohomology theories and the language of stacks, course notes 1999 (pdf)

and worked out in some more detail in

• Nersés Aramian, The Adams spectral sequence (pdf)

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

Reviews of the traditional theory:

• Douglas Ravenel, A Novice’s guide to the Adams-Novikov spectral sequence, in: Michael Barratt, Mark Mahowald (eds.) Geometric Applications of Homotopy Theory II. Lecture Notes in Mathematics, vol 658, Springer 1978 (doi:10.1007/BFb0068728, pdf)

• Doug Ravenel, Complex cobordism and stable homotopy groups of spheres, 1986 onwards

• Stanley Kochman, section 3.6 of: Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996

• John McCleary, chapter 9 of: A user’s guide to spectral sequences, Cambridge University Press, 2001

• Paul Goerss, The Adams-Novikov spectral sequence and the Homotopy Groups of Spheres, lecture notes 2007 (pdf)

• Alan Hatcher, Spectral sequences in algebraic topology II: The Adams spectral sequence (pdf)

• Robert Bruner, An Adams spectral sequence primer, 2009 (pdf)

• Stefan Schwede, chapter II, section 10.3 of Symmetric spectra, 2012

• John Rognes, The Adams spectral sequence (following Bruner 09), 2012 (pdf)

• Introduction to the Adams Spectral Sequence

• Agnès Beaudry, Jonathan A. Campbell, §4 in: A Guide for Computing Stable Homotopy Groups, in Topology and Quantum Theory in Interaction, Contemporary Mathematics 718, Amer. Math. Soc. (2018) [arXiv:1801.07530, ams:conm/718]

  (with an eye towards functorial quantum field theory)

The modern point of view in terms of derived $E_\infty$-descent is hinted at in (Hopkins 99, remark 5.5 (ii)) and is consistently adopted in

• Jacob Lurie, Chromatic Homotopy Theory (2010)

  • 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

• Jacob Lurie, Higher Algebra

This is nicely surveyed in

• Akhil Mathew, The Adams spectral sequence as derived descent, and chromotopy, 2012

• Dylan Wilson, Spectral Sequences from Sequences of Spectra: Towards the Spectrum of the Category of Spectra, lecture at 2013 Pre-Talbot Seminar, March 2013 (pdf)

The actual proof that this new perspecive is equivalent the original Adams-Bousfield theory appears as prop. 2.14 in

• Akhil Mathew, Niko Naumann, Justin Noel, Nilpotence and descent in equivariant stable homotopy theory, Advances in Mathematics Volume 305, 10 January 2017, Pages 994-1084 (arXiv:1507.06869, doi:10.1016/j.aim.2016.09.027)

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)