nLab Adams resolution

Definition

• Classical definition

• Via injective resolutions

Classical definition

For $X$ a spectrum and $E^\bullet$ a generalized cohomology theory represented by a spectrum $E$, then an $E$-Adams resolution of $X$ is a diagram of the form

$$\array{ \vdots \\ \downarrow \\ F_2 &\stackrel{f_2}{\longrightarrow}& K_2 \\ \downarrow \\ F_1 &\stackrel{f_1}{\longrightarrow}& K_1 \\ \downarrow \\ X &\stackrel{f_0}{\longrightarrow}& K_0 }$$

where

• each $K_i$ is a wedge of suspensions of $E$;

• each $F_{n+1} \to F_n \to K_n$ is a homotopy fiber sequence;

• each $f_n$ is a surjection on cohomology.

The original and default case is that where $E = H \mathbb{F}_p$ is an Eilenberg-MacLane spectrum with mod $p$ coefficients, in which case $E^\bullet$ is ordinary cohomology with these coefficients. In this case the $K_i$ are generalized Eilenberg-MacLane spectra.

The long exact sequences of homotopy groups for all the homotopy fibers in this diagram arrange into a diagram of the form

$$\array{ \vdots \\ \downarrow & \nwarrow \\ \pi_\bullet(F_2) &\stackrel{\pi_\bullet(f_2)}{\longrightarrow}& \pi_\bullet(K_2) \\ \downarrow & \nwarrow^{\mathrlap{\pi_\bullet(\partial_2)}} \\ \pi_\bullet(F_1) &\stackrel{\pi_\bullet(f_1)}{\longrightarrow}& \pi_\bullet(K_1) \\ \downarrow & \nwarrow^{\mathrlap{\pi_\bullet(\partial_1)}} \\ \pi_\bullet(X) &\stackrel{\pi_\bullet(f_0)}{\longrightarrow}& \pi_\bullet(K_0) } \,,$$

where the diagonal maps are the images of the connecting homomorphisms and hence decrease degree in $\pi_\bullet$ by one. This is an (unrolled) exact couple. The corresponding spectral sequence is the Adams spectral sequence induced by the given Adams resolution.

In the case of $E = H \mathbb{F}_p$, applying cohomology $H^\bullet(-, \mathbb{F}_p)$ to the original diagram yields a free resolution of the cohomology ring $H^\bullet(X,\mathbb{Z}_p)$ by a chain complex of free modules over the Steenrod algebra $A_p$.

$$\array{ H^\bullet(K_0) &\leftarrow& H^\bullet(\Sigma K_1) &\leftarrow& H^\bullet(\Sigma^2 K_2) &\leftarrow& \cdots \\ \downarrow && \downarrow && \downarrow \\ H^\bullet(X) &\leftarrow& 0 &\leftarrow& 0 &\leftarrow& \cdots }$$

The computation of the cohomology of $X$ by means of this resolution is given by the Adams spectral sequence.

Via injective resolutions

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

Write $HoSpectra$ for the stable homotopy category and write

$$[-,-] \;\colon\; HoSpectra^{op} \times HoSpectra \longrightarrow Ab$$

for the hom-functor with values in abelian groups.

Definition

For $S \in HoSpectra$, the homotopy functor it represents is the representable functor

$$[S,-] \;\colon\; HoSpectra \longrightarrow Ab$$

(as opposed to the other, contravariant, functor).

Example

For $S = \Sigma^\infty S^n \simeq \Sigma^n \mathbb{S}$ we have

$$[\Sigma^\infty S^n ,- ]\simeq \pi_n$$

is the $n$th stable homotopy group-functor.

Throughout, let $E$ be a ring spectrum.

$E$-Injective spectra

First we consider a concept of $E$-injective objects in Spectra.

Definition

Say that

1. a sequence of spectra

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

   is

   1. a (long) exact sequence if the induced sequence of homotopy functors, def. , is a long exact sequence in $[HoSpectra,Ab]$;

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

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

      is (long) exact;

2. a morphism $A \longrightarrow B$ is

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

   2. an epimorphism if $A \longrightarrow B \longrightarrow 0$ is an exact sequence.

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

Example

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

Remark/Warning

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

Lemma

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

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

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

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

Definition

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

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

Lemma

If $S$ is $E$-injective in the sense of def. , then there exists a spectrum $X$ such that $S$ is a retract in HoSpectra of $E \wedge X$.

$E$-Adams resolutions

Definition

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

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

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

Lemma

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

Lemma

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

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

Example

(standard resolution)

Consider the augmented cosimplicial which is the $\mathbb{S} \to E$-Amitsur complex smashed with $X$:

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

Its corresponding Moore complex (the sequence whose maps are the alternating sum of the above coface maps) is an $E$-Adams resolution, def. .

$E$-Adams towers

Definition

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

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

such that

1. each hook is a homotopy fiber sequence (hence it is a tower of homotopy fibers);

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

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

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

   $$0 \to X \stackrel{i_0}{\to} I_0 \stackrel{i_2}{\to} I_2 \stackrel{}{\to} \cdots \,.$$

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

The associated inverse sequence is

$$X = X_0 \stackrel{\gamma_0}{\longleftarrow} \Omega C_1 \stackrel{\gamma_1}{\longleftarrow} C_2 \longleftarrow \cdots$$

where $C_{k+1} \coloneqq hocofib(i_k)$.

(In (Ravenel) it is is the associated inverse sequence that is called the associated resolution.)

=–

Example

Every $E$-Adams resolution of $X$, def. , induces an $E$-Adams tower, def. of which it is the associated $E$-Adams resolution.

Related concepts

• Whitehead tower, Postnikov tower

• E-nilpotent completion

• Adams spectral sequence

References

Reviews include

• Doug Ravenel, around Chapter 2, def. 2.1.3 of Complex cobordism and stable homotopy groups of spheres

• Stanley Kochmann, section 3.6 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996

A streamlined presentation close in spirit to constructions in homological algebra was given in

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

and is reproduced and expanded on in

• Mike Hopkins, section 5 of Complex oriented cohomology theories and the language of stacks, course notes 1999 (pdf)

• Nersés Aramian, The Adams spectral sequence (pdf)