nLab
Adams resolution

Contents

Context

Stable Homotopy theory

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Definition

Classical definition

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

F 2 f 2 K 2 F 1 f 1 K 1 X f 0 K 0 \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

The original and default case is that where E=H𝔽 pE = H \mathbb{F}_p is an Eilenberg-MacLane spectrum with mod pp coefficients, in which case E E^\bullet is ordinary cohomology with these coefficients. In this case the K iK_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

π (F 2) π (f 2) π (K 2) π ( 2) π (F 1) π (f 1) π (K 1) π ( 1) π (X) π (f 0) π (K 0), \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 π \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=H𝔽 pE = H \mathbb{F}_p, applying cohomology H (,𝔽 p)H^\bullet(-, \mathbb{F}_p) to the original diagram yields a free resolution of the cohomology ring H (X, p)H^\bullet(X,\mathbb{Z}_p) by a chain complex of free modules over the Steenrod algebra A pA_p.

H (K 0) H (ΣK 1) H (Σ 2K 2) H (X) 0 0 \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 computaton of the cohomology of XX by means of this resolution is given by the Adams spectral sequence.

Via 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).

Write HoSpectraHoSpectra for the stable homotopy category and write

[,]:HoSpectra op×HoSpectraAb [-,-] \;\colon\; HoSpectra^{op} \times HoSpectra \longrightarrow Ab

for the hom-functor with values in abelian groups.

Definition

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

[S,]:HoSpectraAb [S,-] \;\colon\; HoSpectra \longrightarrow Ab

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

Example

For S=Σ S nΣ nS = \Sigma^\infty S^n \simeq \Sigma^n \mathbb{N} then

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

is the nnth homotopy group-functor.

Throughout, let EE be a ring spectrum.

EE-Injective spectra

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

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, def. , is a long exact sequence in [HoSpectra,Ab][HoSpectra,Ab];

    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;

  2. a morphism ABA \longrightarrow B is

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

    2. an epimorphism if AB0A \longrightarrow B \longrightarrow 0 is an exact sequence.

For EE a ring spectrum, then a sequence of spectra is (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. .

Remark/Warning

Consecutive morphisms in an EE-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 EE-injective objects, defined below in def. .

Lemma
  1. If f:BAf \colon B\longrightarrow A is a monomorphism in the sense of def. , 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 \,.
  2. If f:ABf \colon A \longrightarrow B is an epimorpimsm in the sense of def. , 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 \,.
Definition

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

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

If SS is EE-injective in the sense of def. , then there exists a spectrum XX such that SS is a retract in HoSpectra of EXE \wedge X.

EE-Adams resolutions

Definition

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

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

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

Lemma

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

Lemma

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

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

(standard resolution)

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

XEXEEXEEEX. 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 EE-Adams resolution, def. .

EE-Adams towers

Definition

An EE-Adams tower of a spectrum XX is a commuting diagram in HoSpectra of the form

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

such that

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

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

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

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

    0Xi 0I 0i 2I 2. 0 \to X \stackrel{i_0}{\to} I_0 \stackrel{i_2}{\to} I_2 \stackrel{}{\to} \cdots \,.

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

The associated inverse sequence is

X=X 0γ 0ΩC 1γ 1C 2 X = X_0 \stackrel{\gamma_0}{\longleftarrow} \Omega C_1 \stackrel{\gamma_1}{\longleftarrow} C_2 \longleftarrow \cdots

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

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

=–

Example

Every EE-Adams resolution of XX, def. , induces an EE-Adams tower, def. of which it is the associated EE-Adams resolution.

References

Reviews include

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)

Last revised on November 28, 2018 at 07:07:06. See the history of this page for a list of all contributions to it.