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Adams spectral sequence

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Idea

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

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

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

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

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

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

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

in E-∞ geometry/spectral geometry.

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

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

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

Details

  1. Via the canonical resolution

  2. Via injective resolutions

  3. As derived descent in higher algebra

The EE-Adams spectral sequence

We here discuss Adams spectral sequences for computation of EE-localization of mapping spectra [Y,X][Y,X] for EE a commutative ring spectrum which is flat in a certain sense (def. 11 below), via the “canonical” EE-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 EE-Adams spectral sequences given by setting Y=X=𝕊Y = X = \mathbb{S} the sphere spectrum and E=E = H𝔽 p\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=𝕊Y = X = \mathbb{S} and E=E = MU, discussed below.

The spectral sequence

Filtered spectra

We introduce the types of spectral sequences of which the EE-Adams spectral sequences (def. 9 below) are an example.

Definition

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

Y 3f 2Y 2f 1Y 1f 0Y 0Y. \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 XX in (stable or not) homotopy theory is a \mathbb{Z}-graded sequence X X_\bullet such that XX is the homotopy colimit XlimX X\simeq \underset{\longrightarrow}{\lim} X_\bullet. But for the present purpose we stick with the simpler special case of def. 1.

Remark

There is no condition on the morphisms in def. 1. In particular, they are not required to be n-monomorphisms or n-epimorphisms for any nn.

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. 1 as equivalently being a tower of fibrations over that spectrum.

The following definition 2 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 XHo(Spectra)X \in Ho(Spectra) and a filtered spectrum Y Y_\bullet as in def. 1.

Write A kA_k for the homotopy cofiber of its kkth stage, such as to obtain the diagram

Y 3 f 2 Y 2 f 1 Y 1 f 0 Y g 3 g 2 g 1 g 0 A 3 A 2 A 1 A 0 \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

Y k+1 f k Y k g k A k \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+1f kY kg kA kh kΣY k+1 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,] [X,-]_\bullet (def.) to the above tower. This yields a diagram of \mathbb{Z}-graded abelian groups of the form

[X,Y 3] [X,f 2] ) [X,Y 2] [X,f 1] [X,Y 1] [X,f 0] [X,Y 0] [X,g 3] [X,g 2] [X,g 1] [X,g 0] [X,A 3] [X,A 2] [X,A 1] [X,A 0] , \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 kk extends to a long exact sequence of homotopy groups (prop.) via connecting homomorphisms [X,h k] [X,h_k]_\bullet

[X,A k] +1[X,h k] +1[X,Y k+1] [X,f k] [X,Y k] [X,g k] [X,A k] [X,h k] [X,Y k+1] 1. \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][X,h_k] as a morphism of degree -1, then all this information fits into one diagram of the form

[X,Y 3] [X,f 2] [X,Y 2] [X,f 1] [X,Y 1] [X,f 0] [X,Y 0] [X,h 2] [X,h 1] [X,h 0] [X,g 0] [X,A 3] [X,A 2] [X,A 1] [X,A 0] , \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 ] [X,Y_\bullet]_{\bullet} and [X,A ] [X,A_\bullet]_{\bullet}, then this information may further be rolled-up to a single diagram of the form

[X,Y ] [X,f ] [X,Y ] [X,h ] [X,g ] [X,A ] . \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

D s,t(X,Y) [X,Y s] ts E s,t(X,Y) [X,A s] ts. \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

morphismbidegree
[X,f][X,f](1,1)(-1,-1)
[X,g][X,g](0,0)(0,0)
[X,h][X,h](1,0)(1,0)

This way tt counts the cycles of going around the triangles:

D s+1,t+1(X,Y)[X,f]D s,t(X,Y)[X,g]E s,t(X,Y)[X,h]D s+1,t(X,Y) \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. 4 below.

Definition

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

𝒟 3, i 2 𝒟 2, i 1 𝒟 1, i 0 𝒟 0, k 2 j 2 k 1 j 1 k 0 j 0 3, 2, 1, 0, \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

𝒟 s+1,t+1i s𝒟 s,tj s s,tk s𝒟 s+1,t. \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

𝒟 i 𝒟fto k j , \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. 2 is inter-locking, in that every [X,Y s] ts[X,Y_s]_{t-s} appears twice:

[X,Y s+1] ts1 [X,h] [X,g] [X,A s] ts d 1 [X,A s+1] ts1 d 1 [X,A s+2] ts2 [X,h] [X,g] [X,Y s+2] ts2 \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 1d_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(d_1)^2 = 0. Hence there is a cochain complex:

[X,A s] ts d 1 [X,A S+1] ts1 d 1 [X,A s+2] ts2 . \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[X,A s] tsc \in [X,A_s]_{t-s} lifts to an element c^[X,Y s+2] ts1\hat c \in[X,Y_{s+2}]_{t-s-1} precisely if d 1c=0d_1 c = 0:

c^ [X,Y s+2] ts1 [X,f] [X,Y s+1] ts1 [X,h] [X,g] c [X,A s] ts d 1 [X,A s+1] ts1 \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 2, the differential

d 1[X,g][X,h] d_1 \;\coloneqq \; [X,g] \circ [X,h]

is the composite

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

Some terminology:

Definition

Given an exact couple, def. 4,

𝒟 , i 𝒟 , k j , \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

djk d \coloneqq j \circ k

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

dd =jkj=0k =0. \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

( ,,djk). (\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. 4, then its derived exact couple is the diagram

𝒟˜ i˜ 𝒟˜ k˜ j˜ ˜im(i) i im(i) k ji 1 H(,jk) \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. ˜ker(d)/im(d)\tilde{\mathcal{E}} \coloneqq ker(d)/im(d) (with djkd \coloneqq j \circ k from def. 5);

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

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

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

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

Lemma

The derived exact couple in def. 6 is well defined and is itself an exact couple, def. 4.

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 i˜\tilde i is well-defined.

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

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

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

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

That the sequence of maps is again exact:

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

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

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

Definition

Given an exact couple, def. 4, then the induced spectral sequence of the exact couple is the sequence of pages, def. 5, of the induced sequence of derived exact couples, def. 6, lemma 1.

The rrth page of the spectral sequence is the page (def. 5) of the rrth exact couple, denoted

{ r,d r}. \{\mathcal{E}_r, d_r\} \,.
Remark

So the spectral sequence of an exact couple (def. 7) is a sequence of cochain complexes ( r,d r)(\mathcal{E}_r, d_r), where the cohomology of one is the terms of the next one:

r+1H( r,d r). \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. 7 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. 1,

X 3 f 2 X 2 f 1 X 1 f 0 X g 3 g 2 g 1 g 0 A 3 A 2 A 1 A 0 \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 XHo(Spectra)X \in Ho(Spectra), the induced spectral sequence of a filtered spectrum is the spectral sequence that is induced, by def. 7 from the exact couple (def. 4) given by def. 2:

𝒟 i 𝒟 k j s,tD s,t(X,Y) [X,f] s,tD s,t(X,Y) [X,h] [X,g] s,tE s,t(X,Y)s,t[X,Y s] ts [X,f] s,t[X,Y s] ts [X,h] [X,g] [X,A s] ts \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=𝒟 (1,1) 𝒟 (1,0) (0,0) . 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

1 s,t=[X,A s] ts \mathcal{E}^{s,t}_1 = [X,A_s]_{t-s}
d 1=[X,gh]. d_1 = [X, g \circ h] \,.

As we pass to derived exact couples, by def. 6, the bidegree of ii and kk is preserved, but that of jj increases by (1,1)(1,1) with each page, since (by def. 5)

deg(j˜) =deg(ji 1) =deg(j)deg(i) =deg(j)+(1,1). \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

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

and so the differentials on the rrth page are of the form

d r: r s,t r s+r,t+r1. 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 ss increases vertically and upwards and tst-s increases horizontally and to the right, so that d rd_r goes up rr 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

  • ss is the filtration degree;

  • tst-s is the total degree;

  • tt 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

𝒟 s<0,0, s<0,0 \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.

EE-Adams filtrations

Given a homotopy commutative ring spectrum (E,μ,e)(E,\mu,e), then an EE-Adams spectral sequence is a spectral sequence as in def. 8, where each cofiber is induced from the unit morphism e:𝕊Ee \;\colon\; \mathbb{S} \longrightarrow E:

Definition

Let X,YHo(Spectra)X,Y \in Ho(Spectra) be two spectra (def.), and let (E,μ,e)CMon(Ho(Spectra),,𝕊)(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),,𝕊)(Ho(Spectra), \wedge, \mathbb{S}) (prop.).

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

[X,Y] [X,Σ Y] [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. 8) of the image under [X,] [X,-]_\bullet of the tower

f 0 Y 3 g 3 EY 3=A 3 f 0 Y 2 g 2 EY 2=A 2 f 0 Y 1 g 1 EY 1=A 1 f 0 Y= Y 0 g 0 EY 0=A 0, \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+1f nY ng nA nh nΣY n+1 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 nEY n A_n \coloneqq E \wedge Y_n

is the derived smash product of spectra (corollary) of EE with the stage Y nY_n (cor.), and where

g n:Y n Y n 1𝕊Y neidEY n 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 nY_n (cor.) with the derived smash product of spectra of the unit ee of EE and the identity on Y nY_n.

Hence, by def 8, the first page is

E 1 s,t(X,Y)[X,A s] ts E_1^{s,t}(X,Y) \;\coloneqq\; [X, A_s ]_{t-s}
d 1=[X,gh] d_1 = [X, g \circ h]

and the differentials are of the form

d r:E r s,t(X,Y)E r s+r,t+r1(X,Y). d_r \;\colon\; E_r^{s,t}(X,Y) \longrightarrow E_r^{s+r, t+r-1}(X,Y) \,.

A priori E r ,(X,Y)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<0,(X,Y)0 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]:[X,Y k] [X,eid Y k][X,EY k] [X,g_k] \;\colon\; [X,Y_k]_\bullet \overset{[X,e \wedge id_{Y_k}] }{\longrightarrow} [X, E \wedge Y_k]_\bullet

in def. 9 is sometimes called the Boardman homomorphism (Adams 74, p. 58).

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

π (g k):π (Y k)E (Y k). \pi_\bullet(g_k) \;\colon\; \pi_\bullet(Y_k) \longrightarrow E_\bullet(Y_k) \,.

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

This way one may think of the EE-Adams filtration on YY in def. 9 as the result of filtering any spectrum YY by iteratively projecting out all its EE-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 EE-Adams spectral sequences (def. 9):

Definition

For (E,μ,e)CMon(Ho(Spectra),,𝕊)(E,\mu,e) \in CMon(Ho(Spectra),\wedge, \mathbb{S}) a homotopy commutative ring spectrum (def.), write E¯\overline{E} for the homotopy fiber of its unit e:𝕊Ee \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)Ho(Spectra) of the form

E¯𝕊eE, \overline{E} \longrightarrow \mathbb{S} \overset{e}{\longrightarrow} E \,,

equivalently an exact triangle (prop.) of the form

E¯𝕊eEΣE¯. \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 “E¯\overline{E}” not for the homotopy fiber of 𝕊eE\mathbb{S} \overset{e}{\to} E but for its homotopy cofiber, hence for what is ΣE¯\Sigma \overline{E} according to (Adams 74).

Lemma

In terms of def. 10, the spectra entering the definition of the EE-Adams spectral sequence in def. 9 are equivalently

Y pE¯ pY Y_p \;\simeq\; \overline{E}^p \wedge Y

and

A pEE¯ pY. A_p \;\simeq\; E \wedge \overline{E}^p \wedge Y \,.

where we write

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

Hence the first page of the EE-Adams spectral sequence reads equivalently

E 1 s,t(X,Y)[X,EE¯ sY] ts. 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=0p = 0. Assume then by induction that it holds for some p0p \geq 0. Since the smash product of spectra-functor ()E¯ pY(-) \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

E¯𝕊eE \overline{E} \longrightarrow \mathbb{S} \overset{e}{\longrightarrow} E

from def. 10 yields another homotopy cofiber sequence, now of the form

E¯ p+1YE¯ pYg pEE¯ pY \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 pg_p by the induction assumption, hence A p+1EE¯ pYA_{p+1}\simeq E \wedge \overline{E}^p \wedge Y. Since Y p+1Y_{p+1} is defined to be the homotopy fiber of g pg_p, it follows that Y p+1E¯ p+1YY_{p+1} \simeq \overline{E}^{p+1} \wedge Y.

Remark

Terminology differs across authors. The filtration in def. 9 in the rewriting by lemma 2 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 “EE-injective resolutions” that we discuss below and uses “Adams tower” for yet another concept, def. 43 below. See also remark 22.

We proceed now to analyze the first two pages and then the convergence properties of EE-Adams spectral sequences of def. 9.

The first page

By lemma 2 the first page of an EE-Adams spectral sequence (def. 9) looks like

E 1 s,t(X,Y) [X,EE¯ sY] st. \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 EE-homology equipped with suitable comodule structure over a Hopf algebroid structure on the dual EE-Steenrod operations E (E)E_\bullet(E) (The EE-generalized homology of EE (rmk.)). Then below we discuss that, as a result, the d 1d_1-cohomology of the first page computes the Ext-groups from the EE-homology of YY to the EE-homology of XX, regarded as E (E)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,μ,e)(E,\mu,e) (def.) flat if the canonical right π (E)\pi_\bullet(E)-module structure on E (E)E_\bullet(E) (prop.) (equivalently the canonical left module struture, see prop. 2 below) is a flat module.

The key consequence of the assumption that EE is flat in the sense of def. 11 is the following.

Proposition

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

E (E) π (E)E (X)[𝕊,EEX] =π (EEX) 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 ()E_\bullet(-) the canonical π (E)\pi_\bullet(E)-modules from this prop.) given on elements

Σ n 1𝕊α 1EE,Σ n 2𝕊α 2EX \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

α 1α 2:Σ n 1+n 2𝕊Σ n 1𝕊Σ n 2𝕊α 1α 2EEEXid Eμid XEEX. \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 (E)E_\bullet(E) is a flat module over π (E)\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 (E)×E (X)E_\bullet(E) \times E_\bullet(X) to E (E) π (E)E (X)E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(X)) follows from the associativity of μ\mu.

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

First consider the case that XΣ n𝕊X \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 (X)=E (Σ nX)π n(E) E_\bullet(X) = E_\bullet(\Sigma^n X) \simeq \pi_{\bullet-n}(E)

and

π (EEX)=π (EEΣ n𝕊)E n(E) \pi_\bullet(E \wedge E \wedge X) = \pi_\bullet(E \wedge E \wedge \Sigma^n \mathbb{S}) \simeq E_{\bullet-n}(E)

and

E (E) π (E)π n(E)E n(E) 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 EE.

For general XX, we may without restriction assume that XX 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 XX is a CW-spectrum with finitely many cells. Consider the homotopy cofiber sequence of the (k+1)(k+1)st cell attachment (by that remark):

Σ n k1𝕊 X k X k+1 Σ n k𝕊 ΣX k \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 (E) π (E)E ()π ([𝕊,EE()])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

E (E) π (E)E (Σ n k1𝕊) E (E) π (E)E (X k) E (E) π (E)E (X k+1) E (E) π (E)E (Σ n k𝕊) E (E) π (E)E (ΣX k) [𝕊,EEΣ n k1𝕊] [𝕊,EEX k] [𝕊,EEX k+1] [𝕊,EEΣ n k𝕊] [𝕊,EEΣX k] . \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 EE()E \wedge E \wedge (-) preserves homotopy cofiber sequences (by this lemma, part of the tensor triangulated structure on Ho(Spectra)Ho(Spectra) prop.), and since [𝕊,] π ()[\mathbb{S},-]_\bullet \simeq \pi_\bullet(-) sends homotopy cofiber sequences to long exact sequences (prop.). By the same reasoning, E ()E_\bullet(-) of the homotopy cofiber sequence is long exact; and by the assumption that E (E)E_\bullet(E) is flat, the functor E (E) π (E)()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 XX, 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. [𝕊,] π ()[\mathbb{S},-]_\bullet \simeq \pi_\bullet(-) commutes over filtered colimits of CW-spectrum inclusions (by this lemma, since spheres are compact);

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

Using prop. 1, we find below (theorem 1) that the first page of the EE-Adams spectral sequence may be equivalently rewritten as hom-groups of comodules over E (E)E_\bullet(E) regarded as a graded commutative Hopf algebroid. We now first discuss what this means.

The EE-Steenrod algebra

We discuss here all the extra structure that exists on the EE-self homology E (E)E_\bullet(E) of a flat homotopy commutative ring spectrum. For E=H𝔽 pE = 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 𝒜 p *\mathcal{A}^\ast_p. Therefore one may generally speak of E (E)E_\bullet(E) as being the dual EE-Steenrod algebra.

Without the qualifier “dual” then “EE-Steenrod algebra” refers to the EE-self-cohomology E (E)E^\bullet(E). For E=H𝔽 pE = H \mathbb{F}_p this Steenrod algebra 𝒜 p\mathcal{A}_p (without “dual”) is traditionally considered first, and the classical Adams spectral sequence was originally formulated in terms of 𝒜 p\mathcal{A}_p instead of 𝒜 p *\mathcal{A}_p^\ast. But one observes (Adams 74, p. 280) that the “dual” Steenrod algebra E (E)E_\bullet(E) is much better behaved, at least as long as EE is flat in the sense of def. 11.

Moreover, the dual EE-Steenrod algebra E (E)E_\bullet(E) is more fundamental in that it reflects a stacky geometry secretly underlying the EE-Adams spectral sequence (Hopkins 99). This is the content of the concept of “commutative Hopf algebroid” (def. 14 below) which is equivalently the formal dual of a groupoid internal to affine schemes, def. 13.

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

For XX a finite set consider

X×X×X =(pr 1,pr 3) X×X s=pr 1 t=pr 2 X \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 XX 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 [X]\mathbb{Z}[X], regarded as commutative rings under pointwise multiplication.

Since

[X×X][X][X] \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

([X][X]) [X]([X][X]) [X][X] [X] \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 [X][X][X]\mathbb{Z}[X] \rightrightarrows \mathbb{Z}[X] \otimes \mathbb{Z}[X] are ffef \mapsto f \otimes e and feff \mapsto e \otimes f, respectively, where ee denotes the unit element in [X]\mathbb{Z}[X];

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

  3. the morphism

    [X][X][X][C][C]([X][X]) [X]([X][X]) \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 fgfegf \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 [X]\mathbb{Z}[X] with any flat homotopy commutative ring spectrum EE,

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

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

Definition

Let (E,μ,e)(E, \mu, e) be a homotopy commutative ring spectrum (def.) which is flat according to def. 11.

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

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

(rmk.) equipped with the following structure:

  1. the graded commutative ring structure

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

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

  2. the graded commutative ring structure

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

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

  3. the homomorphism of graded commutative rings

    Ψ:E (E)E (E) π (E)E (E) \Psi \;\colon\; E_\bullet(E) \longrightarrow E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(E)

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

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

    via prop. 1;

  4. the homomorphisms of graded commutative rings

    η L:π (E)E (E) \eta_L \;\colon\; \pi_\bullet(E) \longrightarrow E_\bullet(E)

    and

    η R:π (E)E (E) \eta_R \;\colon\; \pi_\bullet(E) \longrightarrow E_\bullet(E)

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

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

    and

    E E 1𝕊EideEE, 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

    ϵ:E (E)π (E) \epsilon \;\colon\; E_\bullet(E) \longrightarrow \pi_\bullet(E)

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

    μ:EEE \mu \;\colon\; E \wedge E \longrightarrow E

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

  6. the homomorphisms graded commutative rings

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

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

    τ E,E:EEEE \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,μ,e)(E, \mu, e) a homotopy commutative ring spectrum (def.), consider E (E)E_\bullet(E) with its canonical left and right π (E)\pi_\bullet(E)-module structure as in this prop.. These module structures coincide with those induced by the ring homomorphisms η L\eta_L and η R\eta_R from def. 12.

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

Proposition

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

E (E)π (EE)π (τ E,E)π (EE)E (E) 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)

EEEμidEE, 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

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

E 1E 2E 3 idτ E 2,E 3 E 1E 3E 2 id τ E 1,E 3E 2 E 3E 2E 1 μid idμ EE 3 τ E,E 3 E 3E. \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,μ,e)(E,\mu,e) is a commutative monoid (def.), it satisfies μ=μτ\mu = \mu \circ \tau so that we may factor this diagram as follows:

E 1E 2E 3 idτ E 2,E 3 E 1E 3E 2 τ E 1,E 2id τ E 1,E 3E 2 E 2E 1E 3 τ E 2E 1,E 3 E 3E 2E 1 μid idμ EE 3 τ E,E 3 E 3E. \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 EE-Steenrod algebras of def. 12 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 opgCRing^{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. 13. For RgCRingR \in gCRing, write Spec(R)Spec(R) for the same object, but regarded as an object in gCRing opgCRing^{op}.

An internal category in gCRing opgCRing^{op} is a diagram in gCRing opgCRing^{op} of the form

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

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

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

acting as assigning inverses with respect to \circ.

The key basic fact to use 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 opCRing^{op}, see at CRing – Properties – Cocartesian comonoidal structure:

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

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

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

as well as

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

and satisfying formally dual conditions, spelled out as def. 14 below. Here

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

  • ϵ\epsilon is called the co-unit;

  • Ψ\Psi is called the comultiplication;

  • cc is called the antipode or conjugation

Remark

Generally, in a commutative Hopf algebroid, def. 13, the two morphisms η L,η R:AΓ\eta_L, \eta_R\colon A \to \Gamma from remark 7 need not coincide, they make Γ\Gamma genuinely into a bimodule over AA, and it is the tensor product of bimodules that appears in remark 7. But it may happen that they coincide:

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

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

Writing out the formally dual axioms of an internal groupoid as in remark 7 yields the following equivalent but maybe more explicit definition of commutative Hopf algebroids, def. 13

Definition

A commutative Hopf algebroid is

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

  2. ring homomorphisms

    1. (left/right unit)

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

    2. (comultiplication)

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

    3. (counit)

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

    4. (conjugation)

      c:ΓΓc \colon \Gamma \longrightarrow \Gamma

such that

  1. (co-unitality)

    1. (identity morphisms respect source and target)

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

    2. (identity morphisms are units for composition)

      (id Γ Aϵ)Ψ=(ϵ Aid Γ)Ψ=id Γ(id_\Gamma \otimes_A \epsilon) \circ \Psi = (\epsilon \otimes_A id_\Gamma) \circ \Psi = id_\Gamma;

    3. (composition respects source and target)

      1. Ψη R=(id Aη R)η R\Psi \circ \eta_R = (id \otimes_A \eta_R) \circ \eta_R;

      2. Ψη L=(η L Aid)η L\Psi \circ \eta_L = (\eta_L \otimes_A id) \circ \eta_L

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

  3. (inverses)

    1. (inverting twice is the identity)

      cc=id Γc \circ c = id_\Gamma;

    2. (inversion swaps source and target)

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

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

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

      ΓAΓ ΓΓ coeq Γ AΓ ()c() α Γ \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

      ΓAΓ ΓΓ coeq Γ AΓ c()() β Γ \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

      αΨ=η Lϵ\alpha \circ \Psi = \eta_L \circ \epsilon

      and

      βΨ=η Rϵ\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 RR a commutative ring, then RRR \otimes R becomes a commutative Hopf algebroid over RR, formally dual (via def. 13) to the pair groupoid on Spec(R)CRing opSpec(R) \in CRing^{op}.

For XX a finite set and R=[X]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,μ,e)(E, \mu, e) be a homotopy commutative ring spectrum (def.) which is flat according to def. 11, then the dual EE-Steenrod algebra (E (E),π (E)) (E_\bullet(E), \pi_\bullet(E)) with the structure maps (η L,η R,ϵ,c,Ψ)(\eta_L, \eta_R, \epsilon, c, \Psi) from prop. 12 is a graded commutative Hopf algebroid according to def. 14:

(E (E),π (E))CommHopfAlgd (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 EEE \wedge E satisfies the axioms of a commutative Hopf algebroid object in homotopy commutative ring spectra, over EE, by direct analogy to example 1 (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 π ()\pi_\bullet(-) that forms stable homotopy groups to all structure morphisms of EEE \wedge E yields the claimed structure morphisms of E (E)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 opgCRing^{op} (def. 13, remark 7)

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

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

Spec(Gamma)AAtsSpec(A)equSpec(A) /. 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. 14)

AAAη Rη LΓ. A \underoverset {\underset{\eta_R}{\longrightarrow}} {\overset{\eta_L}{\longrightarrow}} {\phantom{AA}} \Gamma \,.

By example 1 every commutative ring gives rise to a commutative Hopf algebroid RRR \otimes R over RR. The core of RR is the formal dual of the corresponding affine scheme of isomorphism classes according to def. 15:

Definition

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

cRRRR. 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 RR is solid (def. 16):

ccRcR. c c R \simeq c R \,.

(Bousfield-Kan 72, prop. 2.2)

Proposition

The following is the complete list of solid rings (def. 16) up to isomorphism:

  1. The localization of the ring of integers at a set JJ of prime numbers (def. 22)

    [J 1]; \mathbb{Z}[J^{-1}] \,;
  2. the cyclic rings

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

    for n2n \geq 2;

  3. the product rings

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

    for n2n \geq 2 such that each prime factor of nn is contained in the set of primes JJ;

  4. the ring cores of product rings

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

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

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

Comodules over the EE-Steenrod algebra
Definition

Let (E,μ,e)(E, \mu, e) be a homotopy commutative ring spectrum (def.) which is flat according to def. 11.

For XHo(Spectra)X \in Ho(Spectra) any spectrum, say that the comodule structure on E (X)E_\bullet(X) (rmk.)) over the dual EE-Steenrod algebra (def. 12) is

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

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

    Ψ E (X):E (X)E (E) π (E)E (X) \Psi_{E_\bullet(X)} \;\colon\; E_\bullet(X) \longrightarrow E_\bullet(E) \otimes_{\pi_\bullet(E)} E_\bullet(X)

    induced under π ()\pi_\bullet(-) and via prop. 1 from the morphism of spectra

    EXE𝕊XideidEEX. 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 Γ\Gamma over AA as in def. 14, hence an internal groupoid in gCRing opgCRing^{op}, then a comodule ring over it is an action in CRing opCRing^{op} of that internal groupoid.

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

Definition

Given a \mathbb{Z}-graded commutative Hopf algebroid Γ\Gamma over AA (def. 14) then a left comodule over Γ\Gamma is

  1. a \mathbb{Z}-graded AA-module NN;

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

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

such that

  1. (co-unitality)

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

  2. (co-action property)

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

A homomorphism between graded comodules f:N 1N 2f \colon N_1 \to N_2 is a homomorphism of underlying graded AA-modules such that the following diagram commutes

N 1 f N 1 Ψ N 1 Ψ N 2 Γ AN 1 id Af Γ AN 2. \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

ΓCoMod \Gamma CoMod

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

Hom Γ(,)Hom ΓCoMod(,). 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:N 1N 2f\colon N_1 \to N_2 the condition that the underlying morphism be grading-preserving. Say that ff has degree nn if it increases degree by nn. This gives a \mathbb{Z}-graded hom-group

Hom Γ (,). Hom^\bullet_\Gamma(-,-) \,.
Example

For (Γ,A)(\Gamma,A) a commutative Hopf algebroid, then AA becomes a left Γ\Gamma-comodule (def. 19) with coaction given by the right unit

Aη RΓΓ AA. A \overset{\eta_R}{\longrightarrow} \Gamma \simeq \Gamma \otimes_A A \,.
Proof

The required co-unitality property is the dual condition in def. 14

ϵη R=id A \epsilon \circ \eta_R = id_A

of the fact in def. 13 that identity morphisms respect sources:

id:Aη RΓΓ AAϵ AidA AAA 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

Ψη R=(id Aη R)η R \Psi \circ \eta_R = (id \otimes_A \eta_R) \circ \eta_R

of the fact in def. 13 that composition of morphisms in a groupoid respects sources

A η R Γ η R Ψ ΓΓ AA id Aη R Γ AΓ. \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,μ,e)(E, \mu, e) be a homotopy commutative ring spectrum (def.) which is flat according to def. 11, and for XHo(Spectra)X \in Ho(Spectra) any spectrum, then the morphism Ψ E (X)\Psi_{E_\bullet(X)} from def. 17 makes E (X)E_\bullet(X) into a comodule (def. 19) over the dual EE-Steenrod algebra (def. 12)

E (X)E (E)CoMod. 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 Γ\Gamma over AA, def. 14, then AA itself becomes a left Γ\Gamma-comodule (def. 19) with coaction given by

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

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

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

More generally:

Proposition

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

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

between the category of Γ\Gamma-comodules, def. 19 and the category of modules over AA, where the cofree functor is right adjoint.

Moreover:

  1. The co-free Γ\Gamma-comodule on an AA-module CC is Γ AC\Gamma \otimes_A C equipped with the coaction induced by the comultiplication Ψ\Psi in Γ\Gamma.

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

    NfΓ AC N \overset{f}{\longrightarrow} \Gamma \otimes_A C

    is its composite with the counit ϵ\epsilon of Γ\Gamma

    f˜:NfΓ ACϵ AidA ACC. \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 Γ\Gamma-comodule is a morphism of AA-modules of the form

f:NΓ AC f \;\colon\; N \longrightarrow \Gamma \otimes_A C

making the following diagram commute

N f Γ AC Ψ N Ψ Aid Γ AN id Af Γ AΓ AC. \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

f˜:NfΓ ACϵ AidA ACC, \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 ff along the counit of Γ\Gamma. By definition this makes the following square commute

Γ AN id Af Γ AΓ AC = id Aϵ Aid Γ AN id Af˜ Γ AC. \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

N f Γ AC Ψ N Ψ Aid Γ AN id Af Γ AΓ AC = id Aϵ Aid Γ AN id Af˜ Γ AC. \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 Γ AC\Gamma \otimes_A C. But this means by the commutativity of the outer rectangle that ff is uniquely fixed in terms of f˜\tilde f by the relation

f=(id Af)Ψ. f = (id \otimes_A f) \circ \Psi \,.

This establishes a natural bijection

NfΓ ACNf˜C \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 Γ\Gamma over AA, def. 14. Any left comodule NN over Γ\Gamma (def. 19) becomes a right comodule via the coaction

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

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

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

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

Given a commutative Hopf algebroid Γ\Gamma over AA, def. 14, and given N 1N_1 a right Γ\Gamma-comodule and N 2N_2 a left comodule (def. 19), then their cotensor product N 1 ΓN 2N_1 \Box_\Gamma N_2 is the kernel of the difference of the two coaction morphisms:

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

If both N 1N_1 and N 2N_2 are left comodules, then their cotensor product is the cotensor product of N 2N_2 with N 1N_1 regarded as a right comodule via prop. 8.

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

Example

Given a commutative Hopf algebroid Γ\Gamma over AA, (def.), and given NN a left Γ\Gamma-comodule (def.). Regard AA itself canonically as a right Γ\Gamma-comodule via example 3. Then the cotensor product

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

is called the primitive elements of NN:

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

Given a commutative Hopf algebroid Γ\Gamma over AA, def. 14, and given N 1,N 2N_1, N_2 two left Γ\Gamma-comodules (def. 19), then their cotensor product (def. 20) is commutative, in that there is an isomorphism

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

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

Lemma

Given a commutative Hopf algebroid Γ\Gamma over AA, def. 14, and given N 1,N 2N_1, N_2 two left Γ\Gamma-comodules (def. 19), such that N 1N_1 is projective as an AA-module, then

  1. The morphism

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

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

  2. The cotensor product (def. 20) with respect to this right comodule structure is isomorphic to the hom of Γ\Gamma-comodules:

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

    Hence in particular

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

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

Remark

In computing the second page of EE-Adams spectral sequences, the second statement in lemma 4 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. 12 for the present purpose is that it is extra structure inherited by morphisms in EE-homology from morphisms of spectra. Namely forming EE-homology f *:E (X)E (Y)f_\ast \colon E_\bullet(X) \to E_\bullet(Y) of a morphism of a spectra f:XYf \colon X \to Y does not just produce a morphism of EE-homology groups

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

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

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

This is the statement of lemma 5 below. The point is that E (E)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:XYf \colon X \to Y.

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

One such case is exhibited by prop. 11 below. This allows to equivalently re-write the first page of the EE-Adams spectral sequence in terms of EE-homology homomorphisms in theorem 1 below.

Lemma

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

π (E) : [X,Y] Hom Ab (E (X),E (Y)) (XfY) (E (X)f *E (Y)) \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 (E)E_\bullet(E)-comodule homomorphisms (def. 19) over the dual EE-Steenrod algebra (def. 12):

Hom E (E) (E (X),E (Y)) forget [X,N] π (E) Hom Ab (E (X),E (Y)), \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 (X)E_\bullet(X) and E (Y)E_\bullet(Y) are regarded as EE-Steenrod comodules according to def. 19, prop. 6.

Proof

By def. 19 we need to show that for XfYX \overset{f}{\longrightarrow} Y a morphism in Ho(Spectra)Ho(Spectra) then the following diagram commutes

E (X) f * E (Y) Ψ E (X) Ψ E (Y) E (E) π (E)E (X) id π (E)f * E (E) π (E)E (Y). \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. 19 and prop. 6 this is the image under foming stable homotopy groups π ()\pi_\bullet(-) of the following diagram in Ho(Spectra)Ho(Spectra):

EX idf EY E𝕊X E𝕊Y ideid ideid EEX ididf EEY. \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)×Ho(Spectra)Ho(Spectra) \times Ho(Spectra).

Proposition

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

Then the homomorphism of graded abelian groups

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

given by

(XfEY)π (EXidfEEYμidEY) (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

β:[X,EY] Hom EMod (EX,EY)π Hom π (E) (E (X),E (Y)), \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. EMod=EMod(Ho(Spectra),,𝕊)E Mod = E Mod(Ho(Spectra), \wedge, \mathbb{S}) denotes the category of homotopy module spectra over EE (def.)

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

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

This is because (by this prop.) the first map is given by first smashing with EE and then postcomposing with the EE-action on the free module EXE \wedge X, which is the pairing EEμEE \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 2N_1, N_2 any two EE-homotopy module spectra (def.) such that π (N 1)\pi_\bullet(N_1) is a projective module over π (E)\pi_\bullet(E), then

Hom EMod (N 1,N 2)π Hom π (E) (π (N 1),π (N 2)) 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 π (N 1)\pi_\bullet(N_1) is in fact a π (E)\pi_\bullet(E)-free module.

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

iIΣ |x i|EN 1 \underset{i \in I}{\vee} \Sigma^{\vert x_i\vert} E \longrightarrow N_1

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

Observe that this sits in a commuting diagram of the form

Hom EMod (iIΣ |x i|E,N 2) π Hom π (E) (π (iIΣ |x i|E),π (N 2)) iI[Σ |x i|𝕊,N 2] iIπ +|x i|(N 2) \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 π (E)\pi_\bullet(E).

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

Now consider the general case that π (N 1)\pi_\bullet(N_1) is a projective module over π (E)\pi_\bullet(E). Since (graded) projective modules are precisely the retracts of (graded) free modules (prop.), there exists a diagram of π (E)\pi_\bullet(E)-modules of the form

id:π (N 1)π (iIΣ |x i|E)π (N 1) 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 π (E)\pi_\bullet(E)-modules

π (iIΣ |x i|E)π (N 1)π (iIΣ |x i|E). \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 π \pi_\bullet of an idempotent of homotopy ring spectra

e :iIΣ |x i|EiIΣ |x i|E 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)Ho(Spectra) all idempotents split (prop.), hence there exists a diagram of spectra of the form

e:iIΣ |x i|EXiIΣ |x i|E 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 π (e)=e \pi_\bullet(e) = e_\bullet.

Consider the composite

XiIΣ |x i|EN 1. X \longrightarrow \underset{i \in I}{\vee} \Sigma^{\vert x_i \vert} E \longrightarrow N_1 \,.

Since π (e)=e \pi_\bullet(e) = e_\bullet it follows that under π \pi_\bullet this is an isomorphism, then that XN 1X \simeq N_1 in the stable homotopy category.

In conclusion this exhibits N 1N_1 as a retract of an free EE-homotopy module spectrum

id:N 1iIΣ |x i|EN 1, 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 1N_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. 10, with the free homotopy EE-module spectrum EYE \wedge Y replaced by any homotopy EE-module spectrum FF, is considered in (Adams 74, chapter III, prop. 13.5) (“universal coefficient theorem”). Strong conditions are considered that ensure that

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

is an isomormphism (expressing the FF-cohomology of XX as the π (E)\pi_\bullet(E)-linear dual of the EE-homology of XX).

For the following we need only the weaker but much more general statement of prop. 10, 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 EE-homology:

Proposition

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

  1. EE is flat according to def. 11;

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

Then the morphism from lemma 5

[X,EY] π (E)Hom E (E) (E (X),E (EY)))Hom E (E) (E (X),E (E) π (E)E (Y))) [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. 1).

(Adams 74, part III, page 323)

Proof

Observe that the following diagram commutes:

[X,EY] π (E) Hom E (E) (E (X),E (E) π (E)E (Y))) ϕ UC ϵid() Hom π (E) (E (X),E (Y)), \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 5;

  2. the right vertical morphism is the adjunction isomorphism from prop. 7;

  3. the left diagonal morphism is the one from prop. 10.

To see that this indeed commutes, notice that

  1. the top morphism sends (XfEY)(X \overset{f}{\to} E \wedge Y) to E (X)E (f)E (EY)π (EEY)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 (X)E (f)π (EEY)π (μid)π (EY)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. 7 (which says that the map is given by postcomposition with the counit of E (E)E_\bullet(E)) and def. 12 (which says that this counit is represented by μ\mu);

  3. by prop. 10 this is the same as the action of the left diagonal morphism.

But now

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

  2. the left diagonal morphism is an isomorphism by prop. 10

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

In conclusion:

Theorem

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

  1. EE is flat according to def. 11;

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

Then the first page of the EE-Adams spectral sequence, def. 9, for [Y,X] [Y,X]_\bullet is isomorphic to the following chain complex of graded homs of comodules (def. 19) over the dual EE-Steenrod algebra (E (E),π (E))(E_\bullet(E), \pi_\bullet(E)) (prop. 12):

E 1 s,t(X,Y)Hom E (E) t(E (X),E s(A s)),d 1=Hom E (E)(E (X),E (gh)) 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 ))

\,

0Hom E (E) t(E (X),E (A 0))d 1Hom E (E) t(E (X),E 1(A 1))d 1Hom E (E) t(E (X),E 2(A 2))d 1. 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. 11 applied to def. 9:

E 1 s,t(X,Y) =[X,EY sA s] ts Hom E (E) ts(E (X),E (EY sA s)) Hom E (E) t(E (X),E s(A s)) \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,μ,e)(E, \mu, e) be a homotopy commutative ring spectrum (def.), and let X,YHo(Spectra)X, Y \in Ho(Spectra) be two spectra such that

  1. EE is flat according to def. 11;

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

Then the entries of the second page of the EE-Adams spectral sequence for [X,Y] [X,Y]_\bullet (def. 9) are the Ext-groups of commutative Hopf algebroid-comodules (def. 19) over the commutative Hopf algebroid structure on the dual EE-Steenrod algebra E (E)E_\bullet(E) from prop. 12:

E 2 s,t(X,Y)Ext E (E) s,t(E (X),E (Y)). E_2^{s,t}(X,Y) \simeq Ext^{s,t}_{E_\bullet(E)}(E_\bullet(X), E_\bullet(Y)) \,.

(On the right ss is the degree that goes with any Ext-functor, and the “internal degree” tt is the additional degree of morphisms between graded modules from def. 19.)

In the special case that X=𝕊X = \mathbb{S} is the sphere spectrum, then (by prop. 4) these are equivalently Cotor-groups

E 2 s,t(X,Y)Cotor E (E) s,t(π (E),E (Y)). 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 1, under the given assumptions the first page reads

E 1 s,t(X,Y)Hom E (E) t(E (X),E s(A s)),d 1=Hom E (E)(E (X),E (gh)) 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 ))

\,

0Hom E (E) t(E (X),E (A 0))d 1Hom E (E) t(E (X),E 1(A 1))d 1Hom E (E) t(E (X),E 2(A 2))d 1. 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 4 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 (E)CoModE_\bullet(E) CoMod (def. 19, prop. 3) is an abelian category with enough injectives (so that all right derived functors on E (E)CoModE_\bullet(E) CoMod exist);

  2. the first page graded chain complex (E 1 ,t(X,Y),d 1)(E^{\bullet,t}_1(X,Y), d_1) is the image under the hom-functor FHom E (E)(E (Y),)F \coloneqq Hom_{E_\bullet(E)}(E_\bullet(Y),-) of an FF-acyclic resolution of E (X)E_\bullet(X) (so that its cohomology indeed computes the Ext-derived functor (theorem)).

That E (E)CoModE_\bullet(E) CoMod is an abelian category is lemma 7 below, and that it has enough injectives is lemma 8.

Lemma 6 below shows that E (A )E_\bullet(A_\bullet) is a resolution of E (Y)E_\bullet(Y) in E (E)CoModE_\bullet(E) CoMod. By prop. 1 it is a resolution by cofree comodules (def. 7). That these are FF-acyclic is lemma 9 below.

EE-Adams resolutions

We discuss that the first page of the EE-Adams spectral sequence indeed exhibits a resolution as required by the proof of theorem 2.

Lemma

Given an EE-Adams spectral sequence (E r s,t(X,Y),d r)(E^{s,t}_r(X,Y),d_r) as in def. 9, then the sequences of morphisms

0E (Y p)E (g p)E (A p)E (h p)E 1(Y p+1)0 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

0 E (Y) E (g 0) E (A 0) E 1(A 1) E 2(A 2) E (h 0) E (g 1) E (h 1) E (g 2) E 1(Y 1) E 2(Y 2) \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 (A ),)(E_\bullet(A_\bullet), \partial) as a resolution of E (Y)E_\bullet(Y).

(Adams 74, theorem 15.1, page 322)

Proof

Consider the image of the defining homotopy cofiber sequence

Y pg pEY ph pΣY p+1 Y_p \overset{g_p}{\longrightarrow} E \wedge Y_p \overset{h_p}{\longrightarrow} \Sigma Y_{p+1}

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

EY pEg pEEY pEh pΣEY p+1 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 π ()[𝕊,] \pi_\bullet(-) \simeq [\mathbb{S},-]_\bullet (lemma) to this yields a long exact sequence (prop.)

E (Y p+1)E (f p)E (Y p)E (g p)E (A p)E (h p)E 1(Y p+1)E 1(f p)E 1(Y p)E 1(g p)E 1(A p). \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,μ,e)(E,\mu,e), the product operation μ\mu on the homotopy commutative ring spectrum EE is a left inverse to g pg_p in that

id:EY pEg pEEY pμidEY p. 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 (g p)E_\bullet(g_p) is a monomorphism, hence its kernel is trivial, and so by exactness E (f p)=0E_\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. 19) over commutative Hopf algebroids (def. 13), needed in the proof of theorem 2.

Lemma

Let (Γ,A)(\Gamma, A) be a commutative Hopf algebroid Γ\Gamma over AA (def. 13, 14), such that the right AA-module structure on Γ\Gamma induced by η R\eta_R is a flat module.

Then the category ΓCoMod\Gamma CoMod of comodules over Γ\Gamma (def. 19) is an abelian category.

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

Proof

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

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

To that end, let f:(N 1,Ψ N 1)(N 2,Ψ N 2)f \colon (N_1,\Psi_{N_1}) \longrightarrow (N_2,\Psi_{N_2}) be a morphism of comodules, hence a commuting diagram in AAMod of the form

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

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

ker(f) N 1 f N 2 Ψ N 1 Ψ N 2 Γ Aker(f) Γ AN 1 id Γ Af Γ AN 2. \array{ ker(f) &\longrightarrow& N_1 &\stackrel{f}{\longrightarrow}& N_2 \\ \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 Γ\Gamma is a flat module over AA, also Γ Aker(f)ker(Γ Af)\Gamma \otimes_A ker(f) \simeq ker(\Gamma \otimes_A f) is a kernel. 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 AA-module ker(f)ker(f) uniquely inherits the structure of a Γ\Gamma-comodule such as to make ker(f)N 1ker(f) \to N_1 a comodule homomorphism. By the same universal property it follows that ker(f)ker(f) with this comodule structure is in fact the kernel of ff in ΓCoMod\Gamma CoMod.

The argument for the existence of cokernels proceeds formally dually. Hence ΓCoMod\Gamma CoMod is a pre-abelian category.

But it also follows from this construction that the comparison morphism

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

formed in ΓCoMod\Gamma CoMod has underlying it the corresponding comparison morphism in AModA Mod. There this is an isomorphism by the fact that the category of modules AModA Mod is an abelian category, hence it is an isomorphism also in ΓCoMod\Gamma CoMod. So the latter is in fact an abelian category itself.

Lemma

Let (Γ,A)(\Gamma, A) be a commutative Hopf algebroid Γ\Gamma over AA (def. 13, 14), such that the right AA-module structure on Γ\Gamma induced by η R\eta_R is a flat module.

Then

  1. every co-free Γ\Gamma-comodule (def. 7) on an injective module over AA is an injective object in ΓCoMod\Gamma CoMod;

  2. ΓCoMod\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 AModA Mod has enough injectives (by this proposition).

Now by prop. 7 we have the adjunction

AModcofreeforgetΓCoMod. 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 7, it is an exact functor. This implies that

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

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

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

Lemma

Let (Γ,A)(\Gamma, A) be a commutative Hopf algebroid Γ\Gamma over AA (def. 13, 14), such that the right AA-module structure on Γ\Gamma induced by η R\eta_R is a flat module. Let NΓCoModN \in \Gamma CoMod be a Γ\Gamma-comodule (def. 19) such that the underlying AA-module is a projective module (a projective object in AAMod).

Then (assuming the axiom of choice in the ambient set theory) every co-free comodule (prop. 7) is an FF-acyclic object for FF the hom functor Hom ΓCoMod(N,)Hom_{\Gamma CoMod}(N,-).

Proof

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

To that end, let I I^\bullet be an injective resolution of KK in AModA Mod. By lemma 8 then Γ AI \Gamma \otimes_A I^\bullet is a sequence of injective objects in ΓCoMod\Gamma CoMod and by the assumption that Γ\Gamma is flat over AA it is an injective resolution of Γ AK\Gamma \otimes_A K in ΓCoMod\Gamma CoMod. Therefore the derived functor in question is given by

1Hom Γ(N,Γ AK) H 1(Hom Γ(N,Γ AI )) H 1(Hom A(N,I )) 0. \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. 7, while the last equality then follows from the assumption that the AA-module underlying NN is a projective module (since hom functors out of projective objects are exact functors (here) and since derived functors of exact functors vanish in positive degree (here)).

With lemma 9 the proof of theorem 2 is completed.

Convergence

We discuss the convergence of EE-Adams spectral sequences (def. 9), i.e. the identification of the “limit”, in an appropriate sense, of the terms E r s,t(X,Y)E_r^{s,t}(X,Y) on the rrth page of the spectral sequence as rr goes to infinity.

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

If one knows that [X,Y] q[X,Y]_q is a finitely generated abelian group (as is the case notably for π q s=[𝕊,𝕊] q\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. 3 below) these groups are direct sums of powers n\mathbb{Z}^n of the infinite cyclic group with finite cyclic groups of the form /p k\mathbb{Z}/p^k \mathbb{Z}, and so all one needs to compute is the powers kk “one prime pp at a time”. This we review below in Primary decomposition of abelian groups.

The deeper reason that EE-Adams spectral sequences tend to converge to localizations of the abelian groups [X,Y] [X,Y]_\bullet of morphisms of spectra, is that they really converges to the actual homotopy groups but 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 EE-Adams spectral sequences below in Convergence statements.

Primary decomposition of abelian groups

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

Theorem

(fundamental theorem of finitely generated abelian groups)

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

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

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

fundamental theorem of finite abelian groups:

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

fundamental theorem of cyclic groups:

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

/ni/p i k i \mathbb{Z}/n\mathbb{Z} \simeq \underset{i}{\bigoplus} \mathbb{Z}/ p_i^{k_i} \mathbb{Z}

where all the p ip_i are distinct and k ik_i is the maximal power of the prime factor p ip_i in the prime decomposition of nn.

Specifically, for each natural number dd dividing nn it contains /d\mathbb{Z}/d\mathbb{Z} as the subgroup generated by n/d/nn/d \in \mathbb{Z}\to \mathbb{Z}/n\mathbb{Z}. In fact the lattice of subgroups of /n\mathbb{Z}/n\mathbb{Z} is the formal dual of the lattice of natural numbers n\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 pp a prime number, there are, up to isomorphism, two abelian groups of order p 2p^2, namely

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

and

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

For p 1p_1 and p 2p_2 two distinct prime numbers, p 1p 2p_1 \neq p_2, then there is, up to isomorphism, precisely one abelian group of order p 1p 2p_1 p_2, namely

/p 1/p 2. \mathbb{Z}/p_1 \mathbb{Z} \oplus \mathbb{Z}/p_2 \mathbb{Z} \,.

This is equivalently the cyclic group

/p 1p 2/p 1/p 2. \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 11 to (p 2,p 1)(p_2,p_1).

Example

Moving up, for two distinct prime numbers p 1p_1 and p 2p_2, there are exactly two abelian groups of order p 1 2p 2p_1^2 p_2, namely (/p 1)(/p 1)(/p 2)(\mathbb{Z}/p_1 \mathbb{Z})\oplus (\mathbb{Z}/p_1 \mathbb{Z}) \oplus (\mathbb{Z}/p_2 \mathbb{Z}) and (/p 1 2)(/p 2)(\mathbb{Z}/p_1^2 \mathbb{Z})\oplus (\mathbb{Z}/p_2 \mathbb{Z}). The latter is the cyclic group of order p 1 2p 2p_1^2 p_2. For instance, /12(/4)(/3)\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 2p 2 2p_1^2 p_2^2, three abelian groups of order p 1 3p 2p_1^3 p_2, and so on.

More generally, theorem 3 may be used to compute exactly how many abelian groups there are of any finite order nn (up to isomorphism): write down its prime factorization, and then for each prime power p kp^k appearing therein, consider how many ways it can be written as a product of positive powers of pp. That is, each partition of kk yields an abelian group of order p kp^k. Since the choices can be made independently for each pp, the numbers of such partitions for each pp are then multiplied.

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

Remark

(graphical representation of pp-primary decomposition)

Theorem 3 says that for any prime number pp, the p-primary part of any finitely generated abelian group is determined uniquely up to isomorphism by

  • a total number kk \in \mathbb{N} of powers of pp;

  • a partition k=k 1+k 2++k qk = k_1 + k_2 + \cdots + k_q.

The corresponding p-primary group is

i=1q/p k i. \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

  • kk dots;

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

For example the graphical representation of the pp-primary group

/p/p/p 3/p 4 \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 /p\mathbb{Z}/p\mathbb{Z};

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

  • a vertical line between two dots means that the the generator of the group corresponding to the upper dot is, regarded after inclusion into the group extension, the product by pp 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 AA which forms a group extension of the form

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

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

This means that of the two possible choices of extensions (by example 5) AA corresponds to the non-trivial extension A=/p 2A = \mathbb{Z}/p^2\mathbb{Z}. Because then in

/p /p 2 /p \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=p1p = p \cdot 1.

Conversely, the notation

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

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

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

such that multiplication by pp 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 5) AA corresponds to the trivial extension A=/p/pA = \mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z}. Because then in

/p /p/p /p \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)(1,0) in the middle group, and multiplication of that by pp is (0,0)(0,0) instead of (0,1)(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

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

such that pp-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=/p 3A = \mathbb{Z}/p^3 \mathbb{Z}.

And so on.

For example the stable page of the 𝔽 2\mathbb{F}_2-classical Adams spectral sequence for computation of the 2-primary part of the stable homotopy groups of spheres π ts(𝕊)\pi_{t-s}(\mathbb{S}) has in (“internal”) degree ts13t-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 1h_1 that encodes the additional ring structure on π (𝕊)\pi_\bullet(\mathbb{S}) which here we are not concerned with) and applying the above prescription, we read off for instance that π 3(𝕊)/8\pi_3(\mathbb{S}) \simeq \mathbb{Z}/8\mathbb{Z} (because all three dots are connected) while π 8(𝕊)/2/2\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=k =012345678910111213
π k(𝕊) (2)=\pi_k(\mathbb{S})_{(2)} = (2)\mathbb{Z}_{(2)}/2\mathbb{Z}/2/2\mathbb{Z}/2/8\mathbb{Z}/80000/2\mathbb{Z}/2/16\mathbb{Z}/16(/2) 2(\mathbb{Z}/2)^2(/2) 3(\mathbb{Z}/2)^3/2\mathbb{Z}/2/8\mathbb{Z}/80000

Here the only entry that needs further explanation is the one for k=0k = 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 (2)\mathbb{Z}_{(2)} refers to the 2-adic integers (def. 10), i.e. for the limit of abelian groups

(2)=lim n1/2 n \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. 24) to finitely generated abelian groups as in theorem 3. 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 AA such that it is an extension of /p\mathbb{Z}/p\mathbb{Z} by itself of the form

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

(Because it is supposed to encode an extension of /p\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 11 below we have a short exact sequence

0 (p)p() (p)/p0. 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 (A,B)\Ext^\bullet(A,B) between abelian groups A,BA, B \in Ab are concentrated in degrees 0 and 1 (prop.). Since

Ext 0(A,B)Hom(A,B) 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 1Ext^1, therefore often abbreviated to just “ExtExt”:

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

Let KK be an abelian group.

Then an abelian group AA is called KK-local if all the Ext-groups from KK to AA vanish:

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

hence equivalently (remark 13) if

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

A homomorphism of abelian groups f:BCf \colon B \longrightarrow C is called KK-local if for all KK-local groups AA the function

Hom(f,A):Hom(B,A)Hom(A,A) 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 Ext^\bullet instead of HomHom, but we do use HomHom. See (Neisendorfer 08, remark 3.2).

A homomorphism of abelian groups

η:AL KA \eta \;\colon\; A \longrightarrow L_K A

is called a KK-localization if

  1. L KAL_K A is KK-local;

  2. η\eta is a KK-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 nn \in \mathbb{N}, write /n\mathbb{Z}/n\mathbb{Z} for the cyclic group of order nn.

Lemma

For nn \in \mathbb{N} and AAbA \in Ab any abelian group, then

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

    Hom(/n,A){aA|pa=0} Hom(\mathbb{Z}/n\mathbb{Z}, A) \simeq \{ a \in A \;\vert\; p \cdot a = 0 \}
  2. the first Ext-group out of /n\mathbb{Z}/n\mathbb{Z} into AA is

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

Regarding the first item: Since /p\mathbb{Z}/p\mathbb{Z} is generated by its element 1 a morphism /pA\mathbb{Z}/p\mathbb{Z} \to A is fixed by the image aa of this element, and the only relation on 1 in /p\mathbb{Z}/p\mathbb{Z} is that p1=0p \cdot 1 = 0.

Regarding the second item:

Consider the canonical free resolution

0p()/p0. 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

0Hom(,A)Hom(n(),A)Hom(,A)Ext 1(/n,A)0, 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 An()AA \overset{n \cdot (-)}{\to} A.

Example

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

p():AA p \cdot (-) \;\colon\; A \longrightarrow A

of multiplication by pp is an isomorphism, hence if “pp is invertible in AA”.

Proof

By the first item of lemma 10 we have

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

By the second item of lemma 10 we have

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

Hence by def. 21 AA is /p\mathbb{Z}/p\mathbb{Z}-local precisely if

{aA|pa=0}0 \{ a \in A \;\vert\; p \cdot a = 0 \} \simeq 0

and if

A/pA0. A / p A \simeq 0 \,.

Both these conditions are equivalent to multiplication by pp being invertible.

Definition

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

The operation of pJ/p\underset{p \in J}{\otimes} \mathbb{Z}/p\mathbb{Z}-localization of abelian groups according to def. 21 is called inverting the primes in JJ.

Specifically

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

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

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

Definition

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

[J 1] \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 JJ.

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

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

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

Notice the parenthesis, to distinguish from the notation p\mathbb{Z}_{p} for the p-adic integers (def. 10 below).

Remark

The terminology in def. 22 is motivated by the following perspective of arithmetic geometry:

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

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

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

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

Similarly localization at pp is localization away from all points except pp.

See also at function field analogy for more on this.

Proposition

For JJ \subset \mathbb{N} a set of prime numbers, a homomorphism of abelian groups f:AlookrightarrowBf \;\colon\; A \lookrightarrow B is local (def. 21) with respect to pJ/p\underset{p \in J}{\oplus} \mathbb{Z}/p\mathbb{Z} (def. 22) if under tensor product of abelian groups with [J 1]\mathbb{Z}[J^{-1}] (def. 23) it becomes an isomorphism

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

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

AA[J 1]. 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 14 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. 21, namely localization at the p-primary component (p )\mathbb{Z}(p^\infty) of the group /\mathbb{Q}/\mathbb{Z} (def. 27 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 pp a prime number, then the p-adic completion of an abelian group AA is the abelian group given by the limit

A p lim(A/p 3AA/p 2AA/pA), 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 p lim nA/p nA A^\wedge_p \coloneqq \underset{\longleftarrow}{\lim}_n A/p^n A

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

Example

The p-adic completion (def. 24) of the integers \mathbb{Z} is called the p-adic integers, often written

p p lim n/p n, \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 pp \mathbb{Z} is a prime ideal.

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

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

Lemma

There is a short exact sequence

0 pp() p/p0. 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

/p 3 p() /p 4 /p /p 2 p() /p 3 /p /p p() /p 2 /p 0 /p /p. \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 pp-completion that is in general different from def. 24, but turns out to coincide with it in finitely generated abelian groups.

Definition

For pp a prime number, write

[1/p]lim(p()p()) \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 pp on the integers.

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

(pn)1p k+1=n1p k. (p \cdot n) \frac{1}{p^{k+1}} = n \frac{1}{p^k} \,.

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

Definition

For pp a prime number, then localization of abelian groups (def. 21) at [1/p]\mathbb{Z}[1/p] (def. 25) is called pp-completion of abelian groups.

Lemma

An abelian group AA is pp-complete according to def. 26 precisely if both the limit as well as the lim^1 of the sequence

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

vanishes:

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

and

lim 1(ApApApA)0. \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. 21 the group AA is [1/p]\mathbb{Z}[1/p]-local precisely if

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

Now use the colimit definition [1/p]=lim n\mathbb{Z}[1/p] = \underset{\longrightarrow}{\lim}_n \mathbb{Z} (def. 25) and the fact that the hom-functor sends colimits in the first argument to limits to deduce that

Hom([1/p],A) =Hom(lim n,A) lim nHom(,A) lim nA. \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 B_\bullet there is a short exact sequence of the form

0lim n 1Hom(B n,A)Ext 1(lim nB n,A)lim nExt 1(B n,A)0 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 nB_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(lim n,A)lim n 1Hom(,A)lim n 1A. 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 pp a prime number, the p-primary cyclic groups of the form /p n\mathbb{Z}/p^n \mathbb{Z} are pp-complete (def. 26).

Proof

By lemma 12 we need to check that

lim(p/p np/p np/p n)0 \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

lim 1(p/p np/p np/p n)0. \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 nn 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\lim^1 also vanishes (prop.).

Definition

For pp a prime number, write

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

(the p-primary part of /\mathbb{Q}/\mathbb{Z}), where [1/p]=lim(pp)\mathbb{Z}[1/p] = \underset{\longrightarrow}{\lim}(\mathbb{Z}\overset{p}{\to} \mathbb{Z} \overset{p}{\to} \mathbb{Z} \to \cdots ) from def. 25.

Since colimits commute over each other, this is equivalently

(p )lim(0/p/p 2). \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 pp a prime number, the [1/p]\mathbb{Z}[1/p]-localization

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

of an abelian group AA (def. 25, def. 21), hence the pp-completion of AA according to def. 26, is given by the morphism

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

into the first Ext-group into AA out of (p )\mathbb{Z}(p^\infty) (def. 27), which appears as the first connecting homomorphism δ\delta in the long exact sequence of Ext-groups

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

(def. 27).

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

Proposition

If AA is a finitely generated abelian group, then its pp-completion (def. 26, for any prime number pp) is equivalently its p-adic completion (def. 24)

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

By theorem 4 the pp-completion is Ext 1((p ),A)Ext^1(\mathbb{Z}(p^\infty),A). By def. 27 there is a colimit

(p )=lim(/p/p 2/p 3). \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

0lim 1Hom(/p n,A)X p lim nExt 1(/p n,A)0. 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 10 the lim^1 on the left is over the p np^n-torsion subgroups of AA, as nn ranges. By the assumption that AA is finitely generated, there is a maximum nn such that all torsion elements in AA are annihilated by p np^n. This means that the Mittag-Leffler condition (def.) is satisfied by the tower of pp-torsion subgroups, and hence the lim^1-term vanishes (prop.).

Therefore by exactness of the above sequence there is an isomorphism

L [1/p]X lim nExt 1(/p n,A) lim nA/p nA, \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 10.

Proposition

If AA is a pp-divisible group in that Ap()AA \overset{p \cdot (-)}{\longrightarrow} A is an isomorphism, then its pp-completion (def. 26) vanishes.

Proof

By theorem 4 the localization morphism δ\delta sits in a long exact sequence of the form

0Hom((p ),A)Hom([1/p],A)ϕHom(,A)δExt 1((p ),A). 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. 25 and since the hom-functor takes colimits in the first argument to limits, the second term is equivalently the limit

Hom([1/p],A)lim(Ap()Ap()A). 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()p \cdot (-) that the limit is over are isomorphisms, so that the limit collapses to its first term, which means that the morphism ϕ\phi in the above sequence is an isomorphism. But by exactness of the sequence this means that δ=0\delta = 0.

Corollary

Let pp be a prime number. If AA is a finite abelian group, then its pp-completion (def. 26) is equivalently its p-primary part.

Proof

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

Ai/p i k i A \simeq \underset{i}{\oplus} \mathbb{Z}/p_i^{k_i}\mathbb{Z}

of cyclic groups of ordr p i k 1p_i^{k_1} for p ip_i prime numbers and k ik_i \in \mathbb{N} (thm.).

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

Ext 1((p ),A)iExt 1((p ),/p i k 1). 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 4 the iith factor here is the pp-completion of /p i k i\mathbb{Z}/p_i^{k_i}\mathbb{Z}, and since p()p \cdot(-) is an isomorphism on /p i k i\mathbb{Z}/p_i^{k_i}\mathbb{Z} if p ipp_i \neq p (because its kernel evidently vanishes), prop. 14 says that in this case the factor vanishes, so that only the factors with p i=pp_i = p remain. On these however Ext 1((p ),)Ext^1(\mathbb{Z}(p^\infty),-) is the identity by example 11.

Localization and nilpotent completion of spectra

We discuuss

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

  1. a spectrum XX is EE-acyclic if the smash product with EE is zero, EX0E \wedge X \simeq 0;

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

  3. a spectrum XX is EE-local if the following equivalent conditions hold

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

    2. every morphism YXY \longrightarrow X out of an EE-acyclic spectrum YY is zero in Ho(Spectra)Ho(Spectra);

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

Lemma

The two conditions in the last item of def. 28 are indeed equivalent.

Proof

Notice that AHo(Spectra)A \in Ho(Spectra) being EE-acyclic means equivalently that the unique morphism 0A0 \longrightarrow A is an EE-equivalence.

Hence one direction of the claim is trivial. For the other direction we need to show that for [,X] [-,X]_\bullet to give an isomorphism on all EE-equivalences ff, it is sufficient that it gives an isomorphism on all EE-equivalences of the form 0A0 \to A.

Given a morphism f:ABf \colon A \to B, write BB/AB \longrightarrow B/A for its homotopy cofiber. Then since Ho(Spectra)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

0 A id A 0 ΣA id f id Σ 1B/A A f B B/A ΣA, \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] [-,X]_\bullet to this diagram in Ho(Spectra)Ho(Spectra) yields a diagram of graded abelian groups of the form

0 [A,X] [A,X] 0 [A,X] +1 id [f,X] id [B/A,X] +1 [A,X] [B,X] [B/A,X] [A,X] +1, \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] 0[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] [f,X]_\bullet is an isomorphism.

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

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

EAEfEBE(B/A)EΣA. 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 EB/A0E \wedge B / A \simeq 0. In detail: by the axioms we may form the morphism of homotopy cofiber sequences

EA Ef EB EB/A EΣA id (Ef) 1 id EA id EA 0 EΣA. \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,XHo(Spectra)E,X \in Ho(Spectra), then an EE-Bousfield localization of spectra of XX is

  1. an EE-local spectrum L EXL_E X

  2. an EE-equivalence XL EXX \longrightarrow L_E X.

according to def. 28.

We discuss now that EE-Localizations always exist. The key to this is the following lemma 14, which asserts that a spectrum being EE-local is equivalent to it being AA-null, for some “small” spectrum AA:

Lemma

For every spectrum EE there exists a spectrum AA such that any spectrum XX is EE-local (def. 28) precisely if it is AA-null, i.e.

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

and such that

  1. AA is EE-acyclic (def. 28);

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

  3. the class of EE-acyclic spectra (def. 28) is the class generated by AA under

    1. wedge sum

    2. the relation that if in a homotopy cofiber sequence X 1X 2X 3X_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 EHo(Spectra)E \in Ho(Spectra) any spectrum, every spectrum XX sits in a homotopy cofiber sequence of the form

G E(X)Xη XL E(X), G_E(X) \longrightarrow X \overset{\eta_X}{\longrightarrow} L_E(X) \,,

and natural in XX, such that

  1. G E(X)G_E(X) is EE-acyclic,

  2. L E(X)L_E(X) is EE-local,

according to def. 28.

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

Proof

Consider the κ\kappa-CW-spectrum spectrum AA whose existence is asserted by lemma 14. Let

I A{ACone(A)} I_A \coloneqq \{A \to Cone(A)\}

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

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

of the inclusion of AA 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 XX any fibrant sequential spectrum, and for AXA \longrightarrow X any morphism, then an extension along the cone inclusion

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

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

Moreover, due to the degreewise nature of the smash tensoring Cone(A)=AICone(A) = A \wedge I (def), the inclusion morphism ACone(A)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 κ\kappa-cell spectrum AA is κ\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 XX a factorization of the terminal morphism X*X \to \ast as an I AI_A-relative cell complex (def.) followed by an I AI_A-injective morphism (def.)

XI ACellL EXI AInj*. X \overset{I_A Cell}{\longrightarrow} L_E X \overset{I_A Inj}{\longrightarrow} \ast \,.

By the above, this means that [A,L EX]=0[A, L_E X] = 0, hence by lemma 14 that L EXL_E X is EE-local.

It remains to see that the homotopy fiber of XL EXX \to L_E X is EE-acyclic: By the tensor triangulated structure on Ho(Spectra)Ho(Spectra) (prop.) it is sufficient to show that the homotopy cofiber is EE-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 14 and applying the pasting law again, all these homotopy pushouts produce EE-acyclic objects. Hence we conclude by observing that the the transfinite composition of the morphisms between these EE-acyclic objects is EE-acyclic. Since by construction all these morphisms are relative cell complex inclusions, this follows again with the compactness of the nn-spheres (lemma).

Lemma

The morphism XL E(X)X \to L_E (X) in prop. 15 exhibits an EE-localization of XX according to def. 29

Proof

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

EG EXEXEη XEL EXEΣG EX 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 EG EX0E \wedge G_E X \simeq 0 by prop. 15, and so the axioms (def.) of the triangulated structure on Ho(Spectra)Ho(Spectra) (prop.) imply that EηE \wedge \eta is an isomorphism.

\,

Nilpotent completion of spectra

Definition

Let (E,μ,e)(E, \mu, e) be a homotopy commutative ring spectrum (def.) and YHo(Spectra)Y \in Ho(Spectra) any spectrum. Write E¯\overline{E} for the homotopy fiber of the unit 𝕊eE\mathbb{S}\overset{e}{\to} E as in def. 10 such that the EE-Adams filtration of YY (def. 9) reads (according to lemma 2)

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

For ss \in \mathbb{N}, write

E¯ s1hocof(E¯ si s𝕊) \overline{E}_{s-1} \coloneqq hocof( \overline{E}^s \overset{i^s}{\longrightarrow} \mathbb{S})

for the homotopy cofiber. Here E¯ 10\overline{E}_{-1} \simeq 0. By the tensor triangulated structure of Ho(Spectra)Ho(Spectra) (prop.), this homotopy cofiber is preserved by forming smash product with YY, and so also

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

Now let

E¯ sp s1E¯ s1 \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)Ho(Spectra) (def., prop.):

E¯ s+1 i E¯ s EE¯ s ΣE¯ s+1 = i s E¯ s+1 𝕊 E¯ s ΣE¯ s+1 p s1 E¯ s1 = E¯ s1 ΣE¯ s ΣEE¯ s. \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 YY:

Y𝕊Yp 3idE¯ 3Yp 2idE¯ 2Yp 1idE¯ 1Y 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 E Y^\wedge_E of YY is the homotopy limit over the resulting inverse sequence

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

or rather the canonical morphism into it

YY E . Y \longrightarrow Y^\wedge_E \,.

Concretely, if

Y𝕊Yp 3idE¯ 3Yp 2idE¯ 2Yp 1idE¯ 1Y 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 E 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 EE-nilpotent completion of XX (def. 30) is denoted “E XE^\wedge X”. The notation “X E X^\wedge_E” which we use here is more common among modern authors. It emphasizes the conceptual relation to p-adic completion A p A^\wedge_p of abelian groups (def. 24) and is less likely to lead to confusion with the smash product of EE with XX.

Remark

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

L EXX E L_E X \overset{}{\longrightarrow} X^\wedge_E

from the EE-Bousfield localization of spectra of XX into the EE-nilmpotent completion

(Bousfield 79, top of page 273)

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

Proposition

Let EE be a connective ring spectrum such that the core of π 0(E)\pi_0(E), def. 16, is either of

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

  • a cyclic ring cπ 0(E)/nc \pi_0(E) \simeq \mathbb{Z}/n\mathbb{Z}, for n2n \geq 2.

Then the map in remark 26 is an equivalence

L EXX E . 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 EE-Adams spectral sequence, below as theorem 5 and theorem 6.

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

  1. the infinity-page E s,t(X,Y)E_\infty^{s,t}(X,Y) (def. 31),

  2. a filtration on [X,Y E ] [X,Y^\wedge_E]_\bullet (def. 32)

  3. what it means for the former to converge to the latter (def. 33).

Broadly the statement will be that typically

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

  2. these groups are localizations of the full groups [X,Y] [X,Y]_\bullet depending on the core of π 0(E)\pi_0(E).

Literature: (Bousfield 79)

\,

Definition

Let (E,μ,e)(E, \mu, e) be a homotopy commutative ring spectrum (def.) and X,YHo(Spectra)X,Y \in Ho(Spectra) two spectra with associated EE-Adams spectral sequence {E r s,t,d r}\{E_r^{s,t}, d_r\} (def. 9).

Observe that

ifr>sthenE r+1 s,(X,Y)ker(d r| E r s,(X,Y))E r s,(X,Y) 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 rd_r on the rrth page has bidegree (r,r1)(r,r-1), and since E r s<0,(X,Y)0E_r^{s \lt 0,\bullet(X,Y)} \simeq 0, so that for r>sr \gt s the image of d rd_r in E r s,t(X,Y)E_r^{s,t}(X,Y) vanishes.

Thus define the bigraded abelian group

E s,t(X,Y)limr>sE r s,t(X,Y)=r>sE r s,t(X,Y) 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 EE-Adams spectral sequence.

Definition

Let (E,μ,e)(E, \mu, e) be a homotopy commutative ring spectrum (def.) and X,YHo(Spectra)X,Y \in Ho(Spectra) two spectra with associated EE-Adams spectral sequence {E r s,t,d r}\{E_r^{s,t}, d_r\} (def. 9) and E-nilpotent completion Y E Y^\wedge_E (def. 30).

Define a filtration

F 3[X,Y E ] F 2[X,Y E ] F 1[X,Y E ] =[X,Y E ] \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 E ] [X,Y^\wedge_E]_\bullet by

F s[X,Y E ] ker([X,Y E ] [X,Y E E¯ s1Y][X,E¯ s1Y] ), 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 E E¯ s1YY^\wedge_E \to \overline{E}_{s-1} \wedge Y is the canonical one from def. 30.

Definition

Let (E,μ,e)(E, \mu, e) be a homotopy commutative ring spectrum (def.) and X,YHo(Spectra)X,Y \in Ho(Spectra) two spectra with associated EE-Adams spectral sequence {E r s,t,d r}\{E_r^{s,t}, d_r\} (def. 9) and E-nilpotent completion Y E Y^\wedge_E (def. 30).

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

  1. [X,Y E ] lim s[X,Y E ] /F s[X,Y E ] [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. 32)

  2. F s[X,Y E ] ts/F s+1[X,Y ] tsE s,t(X,Y)s,t 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 E ] F^s [X, Y^\wedge_E]_\bullet is the filtration stage from def. 32 and E s,t(X,Y)E^{s,t}_\infty(X,Y) is the infinity-page from def. 31).

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 EE-Adams spectral sequence. It turns out that convergence is controled by the core (def. 16) of the ring π 0(E)\pi_0(E). By prop. 5 these cores are either localizations of the integers [J 1]\mathbb{Z}[J^{-1}] at a set JJ of primes (def. 22) or are cylcic rings?, or cores of products of these. We discuss the first two cases.

Theorem

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

  1. the core (def. 16) of the 0-th stable homotopy group ring of EE (prop.) is the localization of the integers at a set JJ of primes (def. 22)

    cπ 0(E)[J 1] c \pi_0(E) \simeq \mathbb{Z}[J^{-1}] \subset \mathbb{Q}
  2. XX is a CW-spectrum (def.) with a finite number of cells (rmk.);

then the EE-Adams spectral sequence for [X,Y] [X,Y]_\bullet (def. 9) converges completely (def. 33) to the localization

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

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

(Bousfield 79, theorem 6.5)

Theorem

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

  1. the core (def. 16) of the 0-th stable homotopy group ring of EE (prop.) is a prime field

    cπ 0(E)𝔽 pc \pi_0(E) \simeq \mathbb{F}_p

    for some prime number pp;

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

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

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

then the EE-Adams spectral sequence for [X,Y] [X,Y]_\bullet (def. 9) converges completely (def. 33) to the pp-adic completion (def. 24)

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

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

(Bousfield 79, theorem 6.6)

Examples

Example

Examples of commutative ring spectra EE for which the dual EE-Steenrod algebra E (E)E_\bullet(E) over π (E)\pi_\bullet(E) of def. 12 where the left and right action of π (E)\pi_\bullet(E) are not just isomorphic (via prop. 2) but actually equal according to remark 8, includes the case E=E = H𝔽 p\mathbb{F}_p.

Example

Examples of commutative ring spectra that are flat according to def. 11 include E=E =

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

Proof of the first two items

For E=𝕊E = \mathbb{S} we have 𝕊 (𝕊)π (𝕊𝕊)π (𝕊)\mathbb{S}_\bullet(\mathbb{S}) \coloneqq \pi_\bullet(\mathbb{S} \wedge \mathbb{S}) \simeq \pi_\bullet(\mathbb{S}), since the sphere spectrum 𝕊\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=H𝔽 pE = H \mathbb{F}_p we have that π (H𝔽 p)𝔽 p\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. 11 include HZ, and MSUM S U.

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

    X (H𝔽 p) X p ^lim nXM(/p n) 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 MAM A denotes the Moore spectrum of the abelian group AA).

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

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

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

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

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

Classical Adams spectral sequence (E=H𝔽 2E = H \mathbb{F}_2, X=𝕊X = \mathbb{S})

We consider now the example of the EE-Adams spectral sequence {E r s,t(X,Y),d r}\{E_r^{s,t}(X,Y), d_r\} (def. 9) for the case that

  1. E=H𝔽 pE = H \mathbb{F}_p is the Eilenberg-MacLane spectrum (def.) with coefficients in a prime field, regarded in Ho(Spectra)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=𝕊X = Y = \mathbb{S} are both the sphere spectrum.

This example is called the classical Adams spectral sequence.

The H𝔽 pH\mathbb{F}_p-dual Steenrod algebra according to the general definition 12 turns out to be the classical dual Steenrod algebra 𝒜 p *\mathcal{A}_p^\ast recalled below .

Notice that H𝔽 2H \mathbb{F}_2 satisfies the two assumptions needed to identify the second page of the H𝔽 pH\mathbb{F}_p-Adams spectral sequence according to theorem 2:

Lemma

The Eilenberg-MacLane spectrum H𝔽 pH\mathbb{F}_p is flat according to 11, and H𝔽 p(𝕊)H \mathbb{F}_p(\mathbb{S}) is a projective module over π (H𝔽 p)\pi_\bullet(H \mathbb{F}_p).

Proof

The stable homotopy groups of H𝔽 pH \mathbb{F}_p is the prime field 𝔽 p\mathbb{F}_p itself, regarded as a graded commutative ring concentrated in degree 0 (prop.)

π (H𝔽 p)𝔽 p. \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 EE-Adams spectral sequence (def. 9) for E=H𝔽 pE = H \mathbb{F}_p (def.) and X=Y=𝕊X = Y = \mathbb{S}, has on its second page the Ext-groups of classical dual Steenrod algebra comodules from 𝔽 pH𝔽 p(𝕊)\mathbb{F}_p \simeq H\mathbb{F}_p(\mathbb{S}) to itself, and converges completely (def. 33) to the p-adic completion (def. 24) of the stable homotopy groups of spheres, hence in degree 0 to the p-adic integers and in all other degrees to the pp-primary part (theorem 3)

E 2 s,t(𝕊,𝕊)=Ext 𝒜 p * s,t(𝔽 p,𝔽 p)(π (𝕊)) p. 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 16 the conditions of theorem 2 are satisfied, which implies the form of the second page.

For the convergence statement, we check the assumptions in theorem 6:

  1. By prop. 4 and prop. 5 the ring 𝔽 p=π 0(H𝔽 p)\mathbb{F}_p = \pi_0(H \mathbb{F}_p) coincides with its core: c𝔽 p𝔽 pc \mathbb{F}_p \simeq \mathbb{F}_p;

  2. 𝕊\mathbb{S} is clearly a connective spectrum;

  3. 𝕊\mathbb{S} is clearly a finite CW-spectrum;

  4. the groups π (𝕊)[𝕊,𝕊] \pi_\bullet(\mathbb{S}) \simeq [\mathbb{S},\mathbb{S}]_\bullet are degreewise finitely generated, by Serre's finiteness theorem?.

Hence theorem 6 applies and gives the convergence as stated.

Finally, by prop. 17 the dual EE-Steenrod algebra in the present case is the classical dual Steenrod algebra.

We now use the classical Adams spectral sequence from corollary 2 to compute the first dozen stable homotopy groups of spheres.

The dual Steenrod algebra
Definition

Let pp be a prime number. Write 𝔽 p\mathbb{F}_p for the corresponding prime field.

The mod pp-Steenrod algebra 𝒜 p\mathcal{A}_{p} is the graded co-commutative Hopf algebra over 𝔽 p\mathbb{F}_p which is

  • for p=2p = 2 generated by elements denoted Sq nSq^n for nn \in \mathbb{N}, n1n \geq 1;

  • for p>2p \gt 2 generated by elements denoted β\beta and P nP^n for \in \mathbb{N}, n1n \geq 1

(called the Serre-Cartan basis elements)

whose product is subject to the following relations (called the Ádem relations):

for p=2p = 2:

for 0<h<2k0 \lt h \lt 2k the

Sq hSq k=i=0[h/2](ki1 h2i)Sq h+kiSq i, 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>2p \gt 2:

for 0<h<pk0 \lt h \lt p k then

P hP k=i=0[h/p](1) h+i((p1)(ki)1 hpi)P h+kiP i 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<h<pk0 \lt h \lt p k then

P hβP k =[h/p]i=0(1) h+i((p1)(ki) hpi)βP h+kiP i +[(h1)/p]i=0(1) h+i1((p1)(ki)1 hpi1)P h+kiβP i \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 Ψ\Psi is subject to the following relations:

for p=2p = 2:

Ψ(Sq n)=k=0nSq kSq nk \Psi(Sq^n) \;=\; \underoverset{k = 0}{n}{\sum} Sq^k \otimes Sq^{n-k}

for p>2p \gt 2:

Ψ(P n)=nk=0P kP nk \Psi(P^n) \;=\; \underoverset{n}{k = 0}{\sum} P^k \otimes P^{n-k}

and

Ψ(β)=β1+1β. \Psi(\beta) = \beta \otimes 1 + 1 \otimes \beta \,.

e.g. (Kochmann 96, p. 52)

Definition

The 𝔽 p\mathbb{F}_p-linear dual of the mod pp-Steenrod algebra (def. 34) 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 𝔸 𝔽 p *\mathbb{A}_{\mathbb{F}_p}^\ast.

Proposition

There is an isomorphism

𝒜 p *H (H𝔽 p,𝔽 p)=π (H𝔽 pH𝔽 p). \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 𝒜 p *\mathcal{A}^\ast_{p} from def. 35, in terms of linear duals of the generators for 𝒜 p\mathcal{A}_{p} itself, according to def. 34.

Theorem

(Milnor’s theorem)

The dual mod 22-Steenrod algebra 𝒜 2 *\mathcal{A}^\ast_{2} (def. 35) is, as an associative algebra, the free graded commutative algebra

𝒜 p *Sym 𝔽 p(ξ 1,ξ 2,,) \mathcal{A}^\ast_{p} \simeq Sym_{\mathbb{F}_p}(\xi_1, \xi_2, \cdots, )

on generators:

  • ξ n\xi_n, n1n \geq 1 being the linear dual to Sq p n1Sq p n2Sq pSq 1Sq^{p^{n-1}} Sq^{p^{n-2}} \cdots Sq^p Sq^1,

    of degree 2 n12^n -1.

The dual mod pp-Steenrod algebra 𝒜 p *\mathcal{A}^\ast_{p} (def. 35) is, as an associative algebra, the free graded commutative algebra

𝒜 p *Sym 𝔽 p(ξ 1,ξ 2,,τ 0,τ 1,) \mathcal{A}^\ast_{p} \simeq Sym_{\mathbb{F}_p}(\xi_1, \xi_2, \cdots, \;\tau_0, \tau_1, \cdots)

on generators:

  • ξ n\xi_n, n1n \geq 1 being the linear dual to P p n1P p n2P pP 1P^{p^{n-1}} P^{p^{n-2}} \cdots P^p P^1,

    of degree 2(p n1)2(p^n-1).

  • τ n\tau_n being linear dual to P p n1P p n2P pP 1βP^{p^{n-1}} P^{p^{n-2}} \cdots P^p P^1\beta.

Moreover, the coproduct on 𝒜 p *\mathcal{A}^\ast_{p} is given on generators by

Ψ(ξ n)=k=0nξ nk p kξ k \Psi(\xi_n) = \underoverset{k = 0}{n}{\sum} \xi_{n-k}^{p^k} \otimes \xi_k

and

Ψ(τ n)=τ n1+k=0nξ nk p kξ nk p kτ k, \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 ξ 01\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 H𝔽 pH \mathbb{F}_p-Adams spectral sequence (cor. 2) we consider a suitable cochain complex whose cochain cohomology gives the relevant Ext-groups.

Definition

Let (Γ,A)(\Gamma,A) be a graded commutative Hopf algebra, hence a commutative Hopf algebroid for which the left and right units coincide η:AΓ\eta \;\colon\; A \longrightarrow \Gamma (remark 8).

Then the unit coideal of Γ\Gamma is the cokernel

Γ¯coker(AηΓ). \overline{\Gamma} \coloneqq coker( A \overset{\eta}{\longrightarrow} \Gamma) \,.
Remark

By co-unitality of graded commutative Hopf algebras (def. 14) ϵη=id A\epsilon \circ \eta = id_A the defining projection of the unit coideal (def. 36)

AηΓΓ¯ A \overset{\eta}{\longrightarrow} \Gamma \overset{}{\longrightarrow} \overline{\Gamma}

forms a split exact sequence which exhibits a direct sum decomposition

ΓAΓ¯. \Gamma \simeq A \oplus \overline{\Gamma} \,.
Lemma

Let (Γ,A)(\Gamma,A) be a commutative Hopf algebra, hence a commutative Hopf algebroid for which the left and right units coincide η:AΓ\eta \;\colon\; A \longrightarrow \Gamma.

Then the unit coideal Γ¯\overline{\Gamma} (def. 36) carries the structure of an AA-bimodule such that the projection morphism

ΓΓ¯ \Gamma \longrightarrow \overline{\Gamma}

is an AA-bimodule homomorphism. Moreover, the coproduct Ψ:ΓΓ AΓ\Psi \;\colon\; \Gamma \longrightarrow \Gamma \otimes_A \Gamma descends to a morphism Γ¯:Γ¯Γ¯ AΓ¯\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

AA Aη AΓ AΓ¯ A η Γ Γ¯, \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 η\eta is a homomorphism of left AA-modules.

Since the tensor product of abelian groups \otimes is a right exact functor it preserves cokernels, hence AΓ¯A \otimes \overline{\Gamma} is the cokernel of AAAΓA \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 Γ¯\overline{\Gamma}. Similarly the right AA-module structure is obtained.

For the second statement, consider the commuting diagram

A η Γ Γ¯ η Ψ ΓΓ AA id Aη Γ AΓ Γ¯ AΓ¯. \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 (Γ,A)(\Gamma,A), equivalently this is the co-action property of AA regarded canonically as a Γ\Gamma-comodule.

Since also the bottom morphism factors through zero, the universal property of the cokernel Γ¯\overline{\Gamma} implies the existence of the right vertical morphism as shown.

Definition

(cobar complex)

Let (Γ,A)(\Gamma,A) be a commutative Hopf algebra, hence a commutative Hopf algebroid for which the left and right units coincide AηΓA \overset{\eta}{\longrightarrow} \Gamma. Let NN be a left Γ\Gamma-comodule.

The cobar complex C Γ (N)C^\bullet_\Gamma(N) is the cochain complex of abelian groups with terms

C Γ s(N)Γ¯ A AΓ¯sfactors AN C^s_\Gamma(N) \coloneqq \underset{s\; factors}{ \underbrace{ \overline{\Gamma} \otimes_A \cdots \otimes_A \overline{\Gamma} } } \otimes_A N

(for Γ¯\overline{\Gamma} the unit coideal of def. 36, with its AA-bimodule structure via lemma 17)

and with differentials d s:C Γ s(N)C Γ s+1(N)d_s \colon C^s_\Gamma(N) \longrightarrow C^{s+1}_\Gamma(N) given by the alternating sum of the coproducts via lemma 17.

(Ravenel 86, def. A1.2.11)

Proposition

Let (Γ,A)(\Gamma,A) be a commutative Hopf algebra, hence a commutative Hopf algebroid for which the left and right units coincide AηΓA \overset{\eta}{\longrightarrow} \Gamma. Let NN be a left Γ\Gamma-comodule.

Then the cochain cohomology of the cobar complex C Γ (N)C^\bullet_\Gamma(N) (def. 37) is the Ext-groups of comodules from AA (regarded as a left comodule via def. 2) into NN

H (C Γ (N))Ext Γ (A,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 NN by co-free comodules given by the complex

D Γ (N)Γ AΓ¯ A AN 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 NN.

To see that this is indeed a resolution, one observes that a contracting homotopy is given by

s(γγ 1||γ sn)ϵ(γ)γ 1||γ sn s (\gamma \gamma_1\vert \cdots \vert \gamma_s n) \coloneqq \epsilon(\gamma) \gamma_1\vert \cdots \vert \gamma_s n

for s>0s \gt 0 and

s(γn)0. s(\gamma n) \coloneqq 0 \,.

Now from lemma 9, in view of remark 1, and since AA is trivially projective over itself, it follows that this is an FF-acyclic resolution for FHom Γ(A,)F \coloneqq Hom_\Gamma(A,-).

This means that the resolution serves to compute the Ext-functor in question and we get

Ext Γ (A,N) H (Hom Γ(A,D Γ (N))) =H (Hom Γ(A,Γ AΓ¯ A AN)) H (Hom A(A,Γ¯ A AN)) H (Γ¯ A AN),, \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

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

from prop. 7, while the last equivalence is the isomorphism of the free/forgetful adjunction

AModforgetfreeAb A Mod \underoverset {\underset{forget}{\longrightarrow}} {\overset{free}{\longleftarrow}} {\bot} Ab
The May spectral sequence

The cobar complex (def. 37) realizes the second page of the classical Adams spectral sequence (cor. 2) 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. 20 below. First we need to consider some prerequisites.

Lemma

Let (Γ,A)(\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 AA-algebra of Γ\Gamma is a free graded commutative AA-algebra on a set of generators {x i} iI\{x_i\}_{i \in I}

such that

  1. all generators x ix_i are primitive elements;

  2. AA is in degree 0;

  3. (i<j)(deg(x i)deg(x j))(i \lt j) \Rightarrow (deg(x_i) \leq deg(x_j));

  4. there are only finitely many x ix_i in a given degree,

then the Ext of Γ\Gamma-comodules from AA to itself is the free graded commutative algebra on these generators

Ext Γ(A,A)A[{x i} iI]. 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 Γ\Gamma-comodule (prop.)

Γ AA[{y i} iI] \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:x iy i d \colon x_i \mapsto y_i
d:y i0 d \colon y_i \mapsto 0

and extending as a graded derivation.

We claim that dd is a homomorphism of left comodules: Due to the assumption that all the x ix_i are primitive we have on generators that

(id,d)(Ψ(x i)) =(id,d)(x i1+1x i) =x i(d1)=0+1(dx i)=y i =Ψ(dx i) \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

(id,d)(Ψ(y i)) =(id,d)(1,y i) =(1,dy i) =0 =Ψ(0) =Ψ(dy i). \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 dd is a graded derivation on a free graded commutative algbra, and Ψ\Psi is an algebra homomorphism, this implies the statement for all other elements.

Now observe that the canonical chain map

(Γ AA[{y i} iI],d) qiA (\Gamma \otimes_A A[\{y_i\}_{i \in I}] ,\; d) \overset{\simeq_{qi}}{\longrightarrow} A

(which projects out the generators x ix_i and y iy_i and is the identity on AA), is a quasi-isomorphism, by construction. Therefore it constitutes a co-free resolution of AA in left Γ\Gamma-comodules.

Since the counit η\eta is assumed to be flat, and since A[{y i} iI]A[\{y_i\}_{i \in I}] is degreewise a free module over AA, hence in particular a projective module, prop. 9 says that the above is an acyclic resolution with respect to the functor Hom Γ(A,):ΓCoModAModHom_{\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 Γ\Gamma-comodule structure over AA (prop. 7), this yields:

Ext Γ (A,A) H (Hom Γ(A,Γ AA[{y i} iI]),d) H (Hom A(A,A[{y i} iI]),d=0) Hom A(A,A[{y i} iI]) A[{x i} iI]. \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 (Γ,A)(\Gamma,A) as above is equipped with a filtering, then there is a spectral sequence

1=Ext gr Γ(gr A,gr A)Ext Γ(A,A) \mathcal{E}_1 \;=\; Ext_{gr_\bullet \Gamma}(gr_\bullet A, gr_\bullet A) \;\Rightarrow\; Ext_{\Gamma}(A, A)

converging to the Ext over Γ\Gamma from AA to itself, whose first page is the ExtExt over the associated graded Hopf algebra gr Γgr_\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. 37) which computes Ext ΓExt_\Gamma (prop. 18). 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. 37) of the associated graded Hopf algebra gr Γgr_\bullet \Gamma. By prop. 18 this is the Ext-groups as shown.

Let now A𝔽 2A \coloneqq \mathbb{F}_2, Γ𝒜 2 \Gamma \coloneqq \mathcal{A}^\bullet_{2} be the mod 2 dual Steenrod algebra. By Milnor's theorem (prop. 7), as an 𝔽 2\mathbb{F}_2-algebra this is

𝒜 2 =Sym 𝔽 2(ξ 1,ξ 2,). \mathcal{A}^\bullet_{2} = Sym_{\mathbb{F}_2}(\xi_1, \xi_2, \cdots) \,.

and the coproduct is given by

Ψ(ξ n)=k=0iξ ik 2 kξ k, \Psi(\xi_n) = \underoverset{k = 0}{i}{\sum} \xi_{i - k}^{2^k} \otimes \xi_k \,,

where we set ξ 01\xi_0 \coloneqq 1.

Definition

Introduce new generators

h i,n{ξ i 2 n fori1,k0 1 fori=0 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 𝔽 2[ξ 1,ξ 2,]\mathbb{F}_2[\xi_1, \xi_2, \cdots] as a product of the new generators in def. 38 such that each such element appears at most once in the product. E.g.

ξ i 5ξ j 7 =ξ i 2 0+2 2ξ j 2 0+2 1+2 2 =h i,0h i,1h j,0h j,1h j,2. \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}\{h_{i,n}\} from def. 38, the coproduct on the dual Steenrod algebra 𝒜 2 *\mathcal{A}^\ast_{2} takes the following simple form

Ψ(h i,n)=k=0ih ik,n+kh k,n. \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 𝔽 2\mathbb{F}_2, one computes:

Ψ(h i,n) =Ψ(ξ i 2 n) =(Ψ(ξ i)) 2 n =(k=0iξ ik 2 kξ k) 2 n =k=0i(ξ ik 2 k) 2 nξ k 2 n =k=0iξ ik 2 k2 nξ k 2 n =k=0iξ ik 2 (k+n)ξ k 2 n =k=0ih ik,n+kh k,n. \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 𝔽 2\mathbb{F}_2-vector spaces of the form

E 1 s,t,p=𝔽 2[{h i,n} i1,n0]Ext 𝒜 2 * s,t(𝔽 2,𝔽 2), 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 ss and tt are from the bigrading of the spectral sequence itself, while the index pp is that of the graded 𝔽 2\mathbb{F}_2-vector spaces), with

  1. h i,nE 1 1,2 2 i+n2 n1,2i1h_{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)=k=0ih ik,n+kh k,n; 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:E r s,t,pE r s+1,t1,p2r+1, 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 𝔽 2\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 𝒜 2 \mathcal{A}^\bullet_{2} (theorem 7) by taking the degree of the generators from def.38 to be (this idea is due to (Ravenel 86, p.69))

|h i,n|2i1 {\vert h_{i,n} \vert} \coloneqq 2i-1

and extending this additively to monomials, via the unique decomposition of remark 17.

For example

|ξ i 5ξ j 7| =|h i,0h i,1h j,0h j,1h j,2| =2(2i1)+3(2j1). \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

F p𝒜 2 *F p+1𝒜 2 *𝒜 2 * \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 pp containing all elements of total degree p\leq p.

Observe via prop. 19 that

Ψ(h i,n) =h i,n1deg=2i1+0<k<ih ik,n+kh k,ndeg=2i2+1h i,ndeg=2i1. \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 𝒜 2 *gr 𝒜 2 *F (𝒜 2 *)/F 1(𝒜 2 *) 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,nh_{i,n} become primitive elements:

Ψ(h i,n) =h i,n1+1h i,ngr 𝒜 2 *gr 𝒜 2 *. \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 18 applies and says that the ExtExt from 𝔽 2\mathbb{F}_2 to itself over the associated graded Hopf algebra is the polynomial algebra in these generators:

Ext gr 𝒜 2 *(𝔽 2,𝔽 2)𝔽 2[{h i,n} i1,n0]. 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 19 says that this is the first page of a spectral sequence that converges to the ExtExt over the original Hopf algebra:

1=𝔽 2[{h i,n} i1n0]Ext 𝒜 2 *(𝔽 2,𝔽 2). \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 19, the differentials on any rr-page are the restriction of the differentials of the bar complex to the rr-almost cycles (prop.). Now the differential of the cobar complex is the alternating sum of the coproduct on 𝒜 2 *\mathcal{A}^\ast_{2}, hence by prop. 19 this is:

d 1(h i,n)=k=0ih ik,n+kh k,n. 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. 20) to compute the second page and in fact the stable page of the classical Adams spectral sequence (cor. 2) in low internal degrees tst-s.

Lemma

(terms on the second page of May spectral sequence)

In the range ts13t - s \leq 13, the second page of the May spectral sequence for Ext 𝔸 𝔽 2 *(𝔽 2,𝔽 2)Ext_{\mathbb{A}^\ast_{\mathbb{F}_2}}(\mathbb{F}_2,\mathbb{F}_2) has as generators all the

  • h nh_n

  • b i,n(h i,n) 2b_{i,n} \coloneqq (h_{i,n})^2

as well as the element

  • x 7h 2,0h 2,1+h 1,1h 3,0x_7 \coloneqq h_{2,0} h_{2,1} + h_{1,1} h_{3,0}

subject to the relations

  • h nh n+1=0h_n h_{n+1} = 0

  • h 2b 2,0=h 0x 7h_2 b_{2,0} = h_0 x_7

  • h 2x 7=h 0b 2,1h_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. 37) lands not in Γ=𝒜 2 *\Gamma = \mathcal{A}^\ast_2 itself, but in the unit coideal Γ¯coker(η)\overline{\Gamma} \coloneqq coker(\eta) where the generator h 0,n=ξ 0=1h_{0,n} = \xi_0 = 1 disappears.

Using this we find for the differential d 1d_1 of the generators in low degree on the first page of the May spectral sequence (prop. 20) via the formula for the differential from prop. 19, the following expressions:

d 1(h n) d 1(h 1,n) =Ψ¯(h 1,n) =h 1,nh 0,n=0+h 0,n+1=0h 1,n =0 \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 nh_n are cocycles on the first page of the May spectral sequence.

Also, since d 1d_1 is a derivation (by definition of the cobar complex, def. 37) 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,kn,k that

d 1(h n,k) 2 =2h n,k(d 1(h n,k)) =0 \begin{aligned} d_1 (h_{n,k})^2 & = 2 h_{n,k} (d_1 (h_{n,k})) \\ & = 0 \end{aligned}

(since 2=0mod22 = 0 \; mod \; 2).

Similarly we compute d 1d_1 on the other generators. These terms do not vanish, but so they impose relations on products in the cobar complex:

d 1(h 2,0) =h 1,1h 1,0 d 1(h 2,1) =h 1,2h 1,1 d 1(h 2,2) =h 1,3h 1,2 d 1(h 2,3) =h 1,4h 1,3 d 1(h 3,0) =h 2,1h 1,0+h 1,2h 2,0 \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 nh n+1=0h_n h_{n+1} = 0 in the given range.

The remaining statements follow similarly.

Remark

With lemma 20, so far we see the following picture in low degrees.

3 h 0 4 h 1 3,h 0 2h 2 2 h 0 2 h 1 2 h 0h 2 1 h 0 h 1 h 2 0 1 2 3 4 \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 0h 1=0 h_0 \otimes h_1 = 0

removes a vertical tower of elements above h 1h_1.

So far there are two different terms in degree (s,ts)=(3,3)(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. 20) relevant for internal degrees ts12t-s \leq 12 are

  1. d 2(h n)=0d_2(h_{n}) = 0

  2. d 2(b 2,n)=h n 2h n+2+h n+1 3d_2(b_{2,n}) = h_n^2 h_{n+2} + h_{n+1}^3

  3. d 2(x 7)=h 0h 2 2d_2(x_7) = h_0 h_2^2

  4. d 2(b 3,0)=h 1b 2,1+h 3b 2,0d_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 20, in fact the h nh_n are infinite cycles in the May spectral sequence.

We spell out the computation for the second item:

We may represent b 2,kb_{2,k} by ξ 2 2 k×ξ 2 2 k\xi_2^{2^k} \times \xi_2^{2^k} plus terms of lower degree. Choose the representative

B 2,k=ξ 2 2 kξ 2 2 k+ξ 1 2 k+1ξ 1 2 kξ 2 2 k+ξ 1 2 k+1ξ 2 kξ 1 2 k. 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 dB 2,kd B_{2,k}, using the definition of the cobar complex (def. 37), the value of the coproduct on dual generators (theorem 7), remembering that the coproduct Ψ\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

d(ξ 2 2 kξ 2 2 k) =Ψ¯(ξ 2 2 k)ξ 2 2 k+ξ 2 2 kΨ¯(ξ 2 2k) =ξ 1 2 k+1ξ 1 2 kξ 2 2 kc 1+ξ 2 2 kξ 1 2 k+1ξ 1 2 kc 2 \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

d(ξ 1 2 k+1ξ 1 2 kξ 2 2 k) =ξ 1 2 kΨ¯(ξ 1 2 kξ 2 2 k) =ξ 1 2 k+1(ξ 1 2 k1+1ξ 1 2 k)(ξ 2 2 k1+ξ 1 2 k+1ξ 1 2 k+1ξ 2 2 k) =ξ 1 2 k+1ξ 1 2 k+1+2 kξ 1 2 kb+ξ 1 2 k+1ξ 1 2 kξ 2 2 kc 1+ξ 1 2 k+1ξ 2 2 kξ 1 2 ka+ξ 1 2 k+1ξ 1 2 k+1ξ 1 2 k+1 \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

d(ξ 1 2 k+1ξ 2 kξ 1 2 k) =Ψ¯(ξ 1 2 k+1ξ 2 k)ξ 1 2 k =(ξ 1 2 k+11+1ξ 1 2 k+1)(ξ 2 2 k1+ξ 1 2 k+1ξ 1 2 k+1ξ 2 2 k)ξ 1 2 k =ξ 1 2 k+2ξ 1 2 kξ 1 2 k+ξ 1 2 k+1ξ 2 2 kξ 1 2 ka+ξ 2 2 kξ 1 2 k+1ξ 1 2 kc 2+ξ 1 2 k+1ξ 1 2 k+1+2 kξ 1 2 kb. \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 dB 2,kd B_{2,k} hence vanish (mod 2). The remaining terms are

dB 2,k=ξ 1 2 k+1ξ 1 2 k+1ξ 1 2 k+1+ξ 1 2 k+2ξ 1 2 kξ 1 2 k 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 21 the picture from remark 18 is further refined:

For k=0k = 0 the differentia d 2(b 2,n)=h n 2h n+2+h n+1 3d_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 2h 2. h_1^3 = h_0^2 h_2 \,.

Hence where on page two we saw two distinct elements in bidegree (s,ts)=(3,3)(s,t-s) = (3,3), on the next page these merge:

3 h 0 4 h 1 3=h 0 2h 2 2 h 0 2 h 1 2 h 0h 2 1 h 0 h 1 h 2 0 1 2 3 4 \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 ts12t-s \leq 12 the infinity page (def. 31) of the classical Adams spectral sequence (cor. 2) is spanned by the items in the following table

Here every dot is a generator for a copy of /2\mathbb{Z}/2\mathbb{Z}. Vertical edges denote multiplication with h 0h_0 and diagonal edges denotes multiplication with h 1h_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 8 gives the stable page of the classical Adams spectral sequence in low degree. By corollary 2 and def. 33 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 12 (the rules for determining group extensions there is just the solution to the extension problem (rmk.) in view of def. 33):

k=k =012345678910111213
π k(𝕊 (2))=\pi_k(\mathbb{S}\otimes \mathbb{Z}_{(2)}) = (2)\mathbb{Z}_{(2)}/2\mathbb{Z}/2/2\mathbb{Z}/2/8\mathbb{Z}/80000/2\mathbb{Z}/2/16\mathbb{Z}/16(/2) 2(\mathbb{Z}/2)^2(/2) 3(\mathbb{Z}/2)^3/2\mathbb{Z}/2/8\mathbb{Z}/80000

The full answer in this range turns out to be this:

k=k =0123456789101112131415\cdots
π k(𝕊)=\pi_k(\mathbb{S}) = \mathbb{Z} 2\mathbb{Z}_2 2\mathbb{Z}_2 24\mathbb{Z}_{24}0000 2\mathbb{Z}_2 240\mathbb{Z}_{240}( 2) 2(\mathbb{Z}_2)^2( 2) 3(\mathbb{Z}_2)^3 6\mathbb{Z}_6 504\mathbb{Z}_{504}00 3\mathbb{Z}_3( 2) 2(\mathbb{Z}_2)^2 480 2\mathbb{Z}_{480} \oplus \mathbb{Z}_2\cdots

And expanding the range yields this :

stable homotopy groups of spheres at 2

(graphics taken from Hatcher’s website)

EE-injective resolutions

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

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

Definition

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

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

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

Example

For X=𝕊X = \mathbb{S} the sphere spectrum then the homotopical functor that it co-represents according to def. 39

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

is the stable homotopy group-functor.

Throughout,EE is a ring spectrum.

Definition

Say that

  1. a sequence of spectra

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

    is

    1. a (long) exact sequence if the induced sequence of homotopy functors according to def. 39, is a long exact sequence in [HoSpectra,Ab ][HoSpectra,Ab^{\mathbb{Z}}];

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

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

      is (long) exact in the above sense;

  2. a morphism ABA \longrightarrow B is

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

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

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

Example

Every homotopy cofiber sequence of spectra is exact in the sense of def. 40.

Example

The suspension functor Σ:Ho(Spectra)Ho(Spectra)\Sigma \colon Ho(Spectra) \to Ho(Spectra) preserves exact sequences in the sense of def. 40.

Proof

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

Example

If a morphism, s:ABs \colon A \to B has a retraction r:BAr \colon B \to A in Ho(Spectra) then it is a monomorphism in the sense of def. 40.

Proof

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

Example

For any spectrum XX the morphism

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

is an EE-monomorphism in the sense of def. 40.

Proof

We need to check that EXideidEEXE \wedge X \stackrel{id \wedge e \wedge id}{\longrightarrow} E \wedge E \wedge X is a monomorphism in the sense of def. 40. Observe that this morphism has a retraction given by μid\mu \wedge id. Therefore it is a monomorphism by example 18.

Remark/Warning

Consecutive morphisms in an EE-exact sequence according to def. 40 in general need not compose up to homotopy, to the zero morphism. But this does become true (lemma 24 below) for sequences of EE-injective objects, defined below in def. 41.

Lemma
  1. If f:BAf \colon B\longrightarrow A is a monomorphism in the sense of def. 40, then there exists a morphism g:CAg \colon C \longrightarrow A such that the wedge sum morphism is a weak homotopy equivalence

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

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

  2. If f:ABf \colon A \longrightarrow B is an epimorpimsm in the sense of def. 40, then there exists a homotopy section s:BAs \colon B\to A, i.e. fsIdf\circ s\simeq Id, together with a morphism g:CAg \colon C \to A such that the wedge sum morphism is a weak homotopy equivalence

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

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

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

We first observe that the connecting homomorphism is equivalent to the zero morphism δ0\delta \simeq 0. This follows because by example 16 the sequence

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

is an exact sequence (of homotopy groups) for every XX, while by example 17 the morphism on the right is epi, so that δ *=0\delta^\ast_\bullet = 0.

Now since BrCδ0B \stackrel{r}{\longrightarrow} C \stackrel{\delta \simeq 0}{\longrightarrow} is also a