and
nonabelian homological algebra
The Adams spectral sequence (Adams 58) is a type of spectral sequences used for computations in stable homotopy theory. It computes the homotopy groups of a spectrum from its homology/cohomology, as modules/comodules over its cohomology operations. The Adams spectral sequence can be seen as a variant of the Serre spectral sequence obtained by replacing a single fibration by an “Adams resolution”.
The original Adams spectral sequence for ordinary cohomology is further refined by the Adams-Novikov spectral sequence (Novikov 67) by replacing ordinary cohomology modulo $p$ by complex cobordism cohomology theory or Brown-Peterson theory or the like. Generally, for $E$ a suitable E-infinity algebra there is a corresponding $E$-Adams(-Novikov) spectral sequence whose second page is given by $E$-generalized cohomology and which arises as the spectral sequence of a simplicial stable homotopy type of the cosimplicial object which is the Cech nerve/Sweedler coring/Amitsur complex of $E$. As such the Adams spectral sequence is an analog in stable homotopy theory of the Bousfield-Kan homotopy spectral sequence.
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 neat conceptual picture of what happens in the Adams spectral sequence has emerged long after its conception with the arrival of higher algebra in stable infinity-category theory. A nice, brief, illuminating modern (and funny) account of this is in (Wilson 13), further details are in (Lurie 10).
The Adams spectral sequence may be motivated from the strategy to compute homotopy groups from cohomology groups by subsequently applying the Hurewicz theorem to compute the lowest-degree non-trivial homotopy group from the corresponding cohomology group, then co-killing that by forming its homotopy fiber, finally applying the Serre spectral sequence to identify the next lowest non-trivial cohomology group of that fiber, and then iterating this process. The Adams spectral sequence arises when in this kind of strategy instead of co-killing only the lowest lying cohomology group, one at a time, one co-kills all nontrivial cohomology groups, then forms the corresponding homotopy fiber and so on.
This was apparently historically the way that John Adams indeed proceeded from Jean-Pierre Serre’s approach and this is still a good motivation for the whoe construction, a nice exposition is in (Wilson 13, 1.1).
We now say this again in more detail.
Given $n \in \mathbb{N}$, consider the probem of computing the homotopy groups $\pi_k(S^n) \;mod \;2$ of the $n$-sphere $S^n$. For $k \leq n$ this is clear: first for $k \}lt n$ they all vanish, and second for $k = n$ we have, by the very nature of Eilenberg-MacLane spaces $K(\mathbb{Z}_2, n)$, that the ordinary cohomology is
so that by the Hurewicz theorem it follows that also
The Hurewicz theorem does not say anything beyong the first non-vanishing cohomology group, but so to apply it again we can move up one step in the Whitehead tower of $S^n$ and hence consider the homotopy fiber
of the generator $[c_1] = 1 \in \pi_n(S^n) \simeq \mathbb{Z}_2$.
To apply the Hurewicz theorem to that fiber we need to know its lowest non-trivial cohomology group again, and this is computed via the Serre spectral sequence applied to this fiber sequence.
From here on the process repeats, and one moves higher through the Whitehead tower of $S^n$
The Adams spectral sequence arises from this strategy by co-killing not just the first non-trivial cohomology group at each stage, but all nontrivial cohomology groups at a given stage.
This is done in stable homotopy theory, so let now $X$ be a spectrum (for instance the sphere spectrum $X = \mathbb{S}$ if we still with the computation of the stable homotopy groups of spheres). Write $H \mathbb{F}_2$ for the Eilenberg-MacLane spectrum for ordinary cohomology with coefficients in $\mathbb{Z}_2$, so that an element in cohomology
is represented by the homotopy class of a homomorphism of spectra of the form
(a cocycle), where ”$\Sigma$” denotes suspension, as usual.
If $X$ is a finite spectrum then there is a finite $I$ of non-trivial cohomology classes like this, and a choice of cocycles $c_i$ for each of them gives a single map
into a generalized Eilenberg-MacLane spectrum. As before, this map classifies its homotopy fiber
which may be thought of as encoding all information about $X$ beyond its cohomology groups. Iterating this process gives the corresponding analog of the Whitehead tower, called the Adams resolution of $X$:
The Adams spectral sequence is that induced by the exact couple obtained by applying $\pi_\bullet$ to this Adams resolution.
We now say this more in detail.
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 connecting homomorphisms and hence decrease degree in $\pi_\bullet$ by one. The idea now is to compute the homotopy groups of $X$ from the decomposed information in this diagram as follows.
First, by construction the homotopy groups $\pi_\bullet(K_s)$ are known, therefore we can identify elements
if they come from elements
whose image
we understand. So the task is to understand the image of $\pi_\bullet(f_s)$ in $\pi_\bullet(K_s)$, for each $s$.
By exactness an element $\kappa_s \in \pi_\bullet(K_s)$ is in this image if its image
vanishes. Now, by construction of the resolution, “evidence” for this is that $f_{s+1}(\partial(\kappa_s)) \in \pi_{\bullet-1}(K_{s+1})$ vanishes, which in turn by exactness means equivalently that $\partial(\kappa_s)$ is the image of an element $\rho_{s+2} \in \pi_{\bullet-1}(X_{s+2}) \to \pi_{\bullet-1}(X_{s+1})$. Now again “evidence” for $\rho_{s+2}$ to vanish is that its image $f_{s+2}(\rho(s+2))$ vanishes, which again means that it comes from an element $\rho_{s+3} \in \pi_{\bullet-1}(X_{s+3}) \to \pi_{\bullet-1}(X_{s+2})$.
Proceeding by induction this way, we find that accumulated “evidence” in homotopy groups of $K_\bullet$ for an element $\kappa_s$ to represent an element in $\pi_\bullet(X)$ is that its differential $\partial \kappa_s$ factors through all the $\pi_{\bullet-1}(X_{s+k}) \to \pi_{\bullet-1}(X_s)$. This in turn means that it factors through the inverse limit $\underset{\leftarrow}{\lim}_s \pi_{\bullet-1}(X_s)$. Such an element $\kappa_s$ with
is called a permanent cycle.
In good cases, the Adams resolution is indeed a resolution which means that the inverse limit $\underset{\leftarrow}{\lim}_s X_s$ is in fact contractible. This means that all the “evidence” accumulated in a permanent cycle is indeed sufficient evidence to prove the existence of an element $\sigma_s \in \pi_\bullet(X_s)$ and hence of an element $\sigma \in \pi_\bullet(X)$.
A trivial way for this to be the case is that the original $\sigma_s$ is itself in the image under $\partial$ of some element, in which case $\kappa_s = 0$ already all by itself. These elements are called eventual boundaries. Therefore if the Adams resolution is indeed a resolution the quotient group
gives elements in $\pi_\bullet(S)$, and this quotient is what the Adams spectral sequence computes.
under construction
Given a connective spectrum $X$ such that $H^\bullet(X)$ has finite type, then for each prime number $p$ there exists a spectral sequence which converges to the homotopy groups of $X$ modulo $p$ $\pi_ast(X) \otimes \mathbb{Z}_p$ and whose $E^2$-page is
where $A$ is the Steenrod algebra.
This is built via an Adams resolution
The long exact sequence induced by this give an exact couple and the Adams spectral sequence is the corresponding spectral sequence.
This is due to (Adams 58). A review is around (Ravenel, theorem, 2.1.1, def. 2.1.8).
We discuss the general definition of $E$-Adams-Novikov spectral sequences for suitable E-∞ rings $E$ expressed in higher algebra, as in (Lurie, Higher Algebra). We follow the nice exposition in (Wilson 13).
First we recall
for the general case of filtered objects in suitable stable (∞,1)-categories. Then we consider the specialization of that to the
Finally we consider specifically the examples of such given by
In conclusion this yields for each suitable E-∞ algebra $E$ over $S$ and $S$-∞-module $X$ a spectral sequence converging to the homotopy groups of the $E$-localization of $X$, and this is
Let thoughout $\mathcal{C}$ be a stable (∞,1)-category equipped with a t-structure such that its heart is an abelian category.
A generalized filtered object in $\mathcal{C}$ is simply a sequential diagram $X \colon (\mathbb{Z}, \lt) \to \mathcal{C}$
Or rather, the object being filtered is the homotopy limit
and the sequential diagram exhibits the filtering.
This appears as (Higher Algebra, def. 1.2.2.9).
For a generalized filtered object $X_\bullet$, def. 1, write
for the homotopy fiber of the $n$th structure map, for all $n \in \mathbb{Z}$, and define an exact couple
where the maps are given by the long exact sequences of homotopy groups
We now have the spectral sequence of a filtered stable homotopy type.
Let $\mathcal{C}$ be a stable (∞,1)-category equipped with a t-structure such that its heart is an abelian category.
If $\mathcal{C}$ has sequential limits and if $X_n \simeq 0$ for all $n \gt n_0$ then the spectral sequence induced by the exact couple of def. 2 converges to the homotopy groups of the homotopy limit $\underset{\leftarrow}{\lim}_n X_n$ of the generalized filted object:
This is due to (Higher Algebra, prop. 1.2.2.14). Review is in Wilson 13, theorem 1.2.1.
For the traditional statement in the category of chain complexes see at spectral sequence of a filtered complex.
Given an cosimplicial object
its totalization $Tot Y \simeq \underset{\leftarrow}{\lim}_n Y_n$ is filtered, def. 1, by the totalizations of its coskeleta
The filtration spectral sequence, prop. 1, applied to the filtration of a totalization by coskeleta as in def. 3, has as $E_2$-term the cohomology groups of the Moore complex associated with the cosimplicial object
This is (Higher Algebra, remark 1.2.4.4). Review is around (Wilson 13, theorem 1.2.4).
We discuss now the special case of coskeletally filtered totalizations coming from the canonical cosimplicial objects induced from E-∞ algebras (“Sweedler corings”).
In this form this appears as (Lurie 10, theorem 2). A review is in (Wilson 13, 1.3). For the analog of this in the traditional formulation see (Ravenel, ch. 3, prop. 3.1.2).
Let $S$ be an E-∞ ring and let $E$ be an E-∞ algebra over $S$, hence an E-∞ ring equipped with a homomorphism
The canonical cosimplicial object associated to this (the ”$\infty$-Sweedler coring”) is that given by the iterated smash product/tensor product over $S$:
More generally, for $X$ an $S$-∞-module, the canonical cosimplicial object is
If $E$ is such that the self-generalized homology $E_\bullet(E) \coloneqq \pi_\bullet(E \wedge_S E)$ (the dual $E$-Steenrod operations) is such that as a module over $E_\bullet \coloneqq \pi_\bullet(E)$ it is a flat module, then there is a natural equivalence
This makes $(E_\bullet, E_\bullet(E))$ be the Hopf algebroid formed by the $E$-Steenrod algebra. See there for more on this.
The condition in prop. 3 is satisfied for
$E = H \mathbb{F}_p$ an Eilenberg-MacLane spectrum with $mod\;p$ coefficients;
$E = B P$ the Brown-Peterson spectrum;
$E = MU$ the complec cobordism spectrum.
It is NOT satisfied for
$E = H \mathbb{Z}$ the Eilenberg-MacLane spectrum for integers coefficients;
$E = M S U$.
Under good conditions (…), $\pi_\bullet$ of the canonical cosimplicial object provides a resolution of comodule tensor product and hence computes the Ext-groups over the Hopf algebroid:
(…)
There is a canonical map
from the $E$-Bousfield localization of spectra of $X$ into the totalization.
We consider now condition for this morphism to be an equivalence.
For $R$ a ring, its core $c R$ is the equalizer in
Let $E$ be a connective E-∞ ring such that the core or $\pi_0(E)$, def. 5 is either of
the localization of the integers at a set $J$ of primes, $c \pi_0(E) \simeq \matbb{Z}[J^{-1}]$;
$\mathbb{Z}_n$ for $n \geq 2$.
Then the map in remark 3 is an equivalence
(Bousfield 79).
Summing this up yields the general $E$-Adams(-Novikov) spectral sequence
Let $E$ a connective E-∞ ring that satisfies the conditions of prop. 4. Then by prop. 1 and prop. 4 there is a strongly convergent multiplicative spectral sequence
converging to the homotopy groups of the $c \pi_0(E)$-localization of $X$. If moreover the dual $E$-Steenrod algebra $E_\bullet(E)$ is flat as a module over $E_\bullet$, then, by prop. 2 and remark 2, the $E_2$-term of this spectral sequence is given by the Ext-groups over the $E$-Steenrod Hopf algebroid.
The $E_2$-page of the Adams spectral sequence for the $p$-component of $\pi_{n+k}(S^n)$ is the ordinary Steenrod algebra (for given prime $p$).
More generally, For $R$ an E-infinity ring such that its dual $R$-Steenrod algebra in the form of the self-homology $R_\bullet(R)$ is a Hopf algebroid over $R_\bullet = \pi_\bullet(R)$ (see at Steenrod algebra – Hopf algebroid structure), then the $E^2$-term of the $E$-Adams spectral sequence is an Ext of $E_\bullet(E)$-comodules
See the references below.
The original articles are
also
Convergence is nicely treated at the end of
A unique homological perspective is provided in
Trasditional reviews include
John Adams, part III of Stable homotopy and generalised homology, University of Chicago Press, Chicago, Ill., 1974, Chicago Lectures in Mathematics.
Doug Ravenel, Complex cobordism and stable homotopy groups of spheres, page 8 of chapter I An introduction to the homotopy groups of spheres (pdf) and section 1 of chapter 2
also Section A.6 of Ravenel’s orange book 1992
See also
Nerses Aramian, The Adams spectral sequence (pdf)
Alexander Kupers, An introduction to the Adams spectral sequence (pdf)
R. Bruner, An Adams spectral sequence primer (pdf)
Michael Adamaszek, An elementary guide to the Adams-Novikov $Ext$ (pdf)
A nice review from the modern point of view of higher algebra is in
More review along these lines is in
Jacob Lurie, Chromatic homotopy theory lecture, lecture 8, The Adams Spectral Sequence, 2010 (pdf)
Jacob Lurie, Chromatic homotopy theory lecture, lecture 9, The Adams Spectral Sequence for $MU$ (pdf)
based on