(also nonabelian homological algebra)
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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
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
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 $A_p$.
The computation of the cohomology of $X$ by means of this resolution is given by the Adams spectral sequence.
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
for the hom-functor with values in abelian groups.
For $S \in HoSpectra$, the homotopy functor it represents is the representable functor
(as opposed to the other, contravariant, functor).
For $S = \Sigma^\infty S^n \simeq \Sigma^n \mathbb{S}$ we have
is the $n$th stable homotopy group-functor.
Throughout, let $E$ be a ring spectrum.
First we consider a concept of $E$-injective objects in Spectra.
Say that
a sequence of spectra
is
a (long) exact sequence if the induced sequence of homotopy functors, def. , is a long exact sequence in $[HoSpectra,Ab]$;
(for $n = 2$) a short exact sequence if
is (long) exact;
a morphism $A \longrightarrow B$ is
a monomorphism if $0 \longrightarrow A \longrightarrow B$ is an exact sequence;
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 $E$.
Every homotopy cofiber sequence of spectra is exact in the sense of def. .
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. .
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
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
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
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 $E \wedge X$.
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
Any two consecutive maps in an $E$-Adams resolution compose to the zero morphism.
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
(standard resolution)
Consider the augmented cosimplicial which is the $\mathbb{S} \to E$-Amitsur complex smashed with $X$:
Its corresponding Moore complex (the sequence whose maps are the alternating sum of the above coface maps) is an $E$-Adams resolution, def. .
An $E$-Adams tower of a spectrum $X$ is a commuting diagram in HoSpectra of the form
such that
each hook is a homotopy fiber sequence (hence it is a tower of homotopy fibers);
the composition of the $(\Sigma \dashv \Omega)$-adjuncts of $\Sigma_{p_{n-1}}$ with $\Sigma^n \kappa_n$
constitute an $E$-Adams resolution of $X$, def. :
Call this the associated $E$-Adams resolution of the $E$-Adams tower.
The associated inverse sequence is
where $C_{k+1} \coloneqq hocofib(i_k)$.
(In (Ravenel) it is is the associated inverse sequence that is called the associated resolution.)
=–
Every $E$-Adams resolution of $X$, def. , induces an $E$-Adams tower, def. of which it is the associated $E$-Adams resolution.
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
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 December 10, 2020 at 13:31:17. See the history of this page for a list of all contributions to it.