Stable Homotopy theory
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 and a ring spectrum , then under mild assumptions the Adams spectral sequence converges to the homotopy groups of the -nilpotent completion of , while under stronger assumptions the latter is the -Bousfield localization of spectra. The second page of the spectral sequence is given by the -homology of as modules over the dual -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 is ordinary homology mod , while the Adams-Novikov spectral sequence (Novikov 67) is the case where MU is complex cobordism cohomology theory or BP, Brown-Peterson theory.
Generally, for a suitable E-infinity algebra there is a corresponding -Adams(-Novikov) spectral sequence whose second page is given by -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 . 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 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
in E-∞ geometry/spectral geometry.
Moreover, a spectrum is equivalently a quasicoherent sheaf on and is accordingly the Sweedler coring that expresses the descent property of pullled back along the cover , dually the -localization of . The Adams spectral sequence may then be seen to be the computation of the homotopy groups of the -localization of in terms of its restriction to that cover.
In general, notably for , the 1-image of is smaller than and therefore this process computes not all of , but just the restriction to that one image (for instance just the -local component). Examples of ring spectra which are “complete” with respect to the sphere spectrum in that the above 1-image coincides with notably includes the complex cobordism cohomology spectrum MU (Hopkins 99, p. 70).
That explains the relevance of the Adams-Novikov spectral sequence (noticing that the wedge summands of are the BP-spectra) and the close interplay between the ANSS and chromatic homotopy theory.
Via injective resolutions
As derived descent in higher algebra
The -Adams spectral sequence
We here discuss Adams spectral sequences for computation of -localization of mapping spectra for by a general commutative ring spectrum which is flat in a certain sense (def. 8 below).
The classical Adams spectral sequence is the special case with and A, discussed below.
The Adams-Novikov spectral sequence is the special case with and MU, discussed below.
A streamlined discussion of -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 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, -Adams resolutions in the stable homotopy category are the direct analog of injective resolutions in an abelian category.
For the sphere spectrum then the homotopical functor that it co-represents according to def. 1
is the stable homotopy group-functor.
Throughout, is a ring spectrum.
a sequence of spectra
a (long) exact sequence if the induced sequence of homotopy functors according to def. 1, is a long exact sequence in ;
(for ) a short exact sequence if
is (long) exact in the above sense;
a morphism is
a monomorphism if is an exact sequence in the above sense;
an epimorphism if is an exact sequence in the above sense.
For a ring spectrum, then a sequence of spectra is called (long/short) -exact and a morphism is epi/mono, respectively, if becomes long/short exact or epi/mono, respectively, after taking smash product with .
The suspension functor preserves exact sequences in the sense of def. 2.
By the suspension/looping adjunction-isomorphism (prop.) and so the statement follows from the assumption that is long exact.
If a morphism, has a retraction in Ho(Spectra) then it is a monomorphism in the sense of def. 2.
We need to check that for every the morphism is surjective. By retraction, given , then is a preimage.
For any spectrum the morphism
is an -monomorphism in the sense of def. 2.
We need to check that is a monomorphism in the sense of def. 2. Observe that this morphism has a retraction given by . Therefore it is a monomorphism by example 4.
If is a monomorphism in the sense of def. 2, then there exists a morphism such that the wedge sum morphism is a weak homotopy equivalence
In particular, every morphism in Ho(Spectra) has an extension along a monomorphism in this sense.
If is an epimorpimsm in the sense of def. 2, then there exists a homotopy section , i.e. , together with a morphism such that the wedge sum morphism is a weak homotopy equivalence
Given a monomorphism , consider the correspondiing homotopy cofiber sequence
We first observe that the connecting homomorphism is equivalent to the zero morphism . This follows because by example 2 the sequence
is an exact sequence (of homotopy groups) for every , while by example 3 the morphism on the right is epi, so that .
Now since is also a homotopy fiber sequence, the pasting law identifies :
For a ring spectrum, say that a spectrum is -injective if for each morphism and each -monomorphism in the sense of def. 2, there is a diagram in HoSpectra of the form
A spectrum is -injective in the sense of def. 3, precisely if it is a retract in HoSpectra of a free -modules, hence of for some spectrum .
In one direction, assume that is -injective and consider the diagram
By example 5 here the vertical morphism is an -monomorphism, and so by assumption there is a lift
which exhibits as a retract of .
In the other direction, given a retraction we show that there exist extensions in
whenever the vertical morphism is an -monomorphism. To see this, complete the extension problem to the following commuting diagram
Now, since is assumed to be an -monomorphism, the morphism on the right is a monomorphism in the sense of def. 2, and so by lemma 1 there exists an extension in
By composition and commutativity, this gives the required extension of along .
For a ring spectrum, then an -Adams resolution of an spectrum is a long exact sequence, in the sense of def. 2, of the form
such that each is -injective, def. 3.
Any two consecutive maps in an -Adams resolution, def. 4, compose to the zero morphism.
The following lemma says that -Adams resolutions may be extended along morphisms.
For an -Adams resolution, def. 4, and for any morphism, then there exists an -Adams resolution and a commuting diagram
There are two -Adams resolutions that we will consider. Following (Hopkins 99) we call them the “normalized resolution” and the “standard resolution”. But beware that what all the traditional literature (Adams 74, Bousfield 79, Ravenel 86, …) considers (and somewhat implicitly) is the “normalized” resolution, not the standard resolution. The standard resolution is standard only from the more recent perspective of E-∞ geometry: it is the Amitsur complex of the -algebra , exhibiting the formal dual of the Cech nerve of .
(normalized -Adams resolution)
Let denote the homotopy fiber of the unit of the ring spectrum , fitting into a homotopy fiber sequence
For a spectrum, its normalized -Adams resolution is the top row of
(e.g. Hopkins 99, corollary 5.3).
(standard -Adams resolution)
Any ring spectrum gives rise to an augmented cosimplicial spectrum (its bar construction)
The corresponding Amitsur complex is given by forming alternating sums of the coface maps
Given any spectrum , then forming the smash product with this sequence yields a sequence of the form
This is called the standard -Adams resolution of .
(e.g. Hopkins 99, def. 5.4).
The standard resolution of example 7 is indeed an -Adams resolution of in the sense of def. 4.
As generally for bar resolutions, one checks that the alternating sum of the codegeneracy maps constitute homotopy operators that give contracting homotopies when commuted with the alternating sum of the coface maps. This gives that the sequence is -exact. Moreover, the terms in the sequence are all -injective by lemma 2.
An -Adams tower of a spectrum is a commuting diagram in the stable homotopy category of the form
each hook is a homotopy fiber sequence;
the composition of the -adjuncts of with
constitute an -Adams resolution of , def. 4:
Call this the associated -Adams resolution of the -Adams tower.
(Hopkins 99, def. 4.10)
The following is the main statement of the above little theory of -injective spectra.
Every -Adams resolution (def. 4) induces an -Adams tower, def. 5 of which it is the associated -Adams resolution.
Given an -Adams resolution
consider the induced diagram
constructed inductively as follows:
To start with, is the homotopy cofiber of , and is the morphism universally induced from this by the fact that , by lemma 3. Observe that is an -monomorphism and is an -epimorphism in the sense of def. 2.
Then assume that an -epi/mono factorization
has been constructed. Let now be its homotopy cofiber. Since is -epi, the equivalence from lemma 3 implies that already . With this, the universal property of the homotopy cofiber induces a morphism . As before, is -epi and is -mono, and so the induction proceeds.
Using this, we now construct an -Adams tower as follows (…).
There is another tower associated with an -Adams resolutions:
Given an -Adams resolutions (def. 4), its associated inverse sequence is
with the as in the proof of prop. 2 and .
Let be a normalized -Adams resolution according to example 6. Then its associated inverse sequence according to def. 6 is
This is the tower of spectra considered in the original texts (Adams 74, p. 318) and (Bousfield 79, p. 271).
-Adams spectral sequences
Given spectra and , and given an -Adams resolution of , def. 4, or equivalently (by prop. 2) an -Adams tower over , def. 5,
then the corresponding -Adams spectral sequence for the mapping spectrum is the associated spectral sequence of a tower of fibrations of the image of that tower of fibrations under the mapping spectrum operation :
More in detail, the associated exact couple of the tower is
The -Adams spectral sequence of the -Adams tower is the spectral sequence induced by this exact couple.
Given two -Adams towers, def. 5, for some , then the corresponding two -Adams spectral sequences, def. 7, are isomorphic from the -page on.
The first page
Due to prop. 3, for understanding the -page of any -Adams spectral sequence, def. 7, it is sufficient to understand the -page of the -Adams spectral sequence that is induced by the standard -resolution of example 7. By construction, that page is
with the differentials being the image under of the alternating sum of the morphisms that insert unit elements.
We discuss now how, under favorable conditions, these homotopy groups of mapping spectra of the form may alternatively be computed as morphisms of -homology equipped with suitable comodule structure over a Hopf algebroid structure on the dual -Steenrod operations . Then below we discuss that, as a result, the -homology of the -page is seen to compute the Ext-groups from the -homology of to the -homology of , regarded as -comodules. This re-formulation of the -page is the one that makes it be useful for computations.
The first condition needed for this to work is the following.
Call the commutative ring spectrum flat if one, equivalently both, of the morphisms
is a flat morphism.
Examples of ring spectra that are not flat in the sense of def. 8 include HZ, and .
The key consequence of the assumption that is flat in the sense of def. 8 is the following.
If is flat, def. 8, then for all spectra there is a natural isomorphisms
and hence for all there are isomorphisms
(e.g. Adams 74, part III, lemma 12.5, Schwede 12, prop. 6.20)
The desired natural homomorphism
is given on and by .
To see that this is an isomorphism, observe that by flatness of , the assignment is a generalized homology functor, hence represented by some spectrum. The above morphism, natural in , thus constitutes a homomorphism of generalized homology theories. By the Whitehead theorem for generalized homology for this to be an isomorphism it is sufficient to check that it induces isomorphisms on the point. This is manifestly the case.
Finally we get the claimed isomorphisms for all by induction:
Commutative Hopf algebroids
(e.g. Ravenel 86, def. A1.1.1)
Writing out the formally dual axioms of an internal groupoid as in remark 5 yields the following equivalent but maybe more explicit definition of commutative Hopf algebroids, def. 9
A commutative Hopf algebroid is
two commutative rings, and ;
the universally induced satifies
Comodules and cotensor product
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:
Given a commutative Hopf algebroid over , def. 10, then a left comodule over is
an -module ;
an -module homomorphism (co-action)
A homomorphism between comodules is a homomorphism of underlying -modules making commuting diagrams with the co-action morphism. Write
for the resulting category of left comodules over . Analogously for right comodules.
Given a commutative Hopf algebroid over , def. 10, then itself becomes a left -comodule (def. 12) with coaction given by
and a right -comodule with coaction given by
Consider a commutative Hopf algebroid over , def. 10. Any left comodule over (def. 12) becomes a right comodule via the coaction
where the isomorphism in the middle the is braiding in and where is the conjugation map of .
Dually, a right comodule becoomes a left comodule with the coaction
Given a commutative Hopf algebroid over , def. 10, and given a right -comodule and a left comodule (def. 12), then their cotensor product is the kernel of the difference of the two coaction morphisms:
If both and are left comodules, then their cotensor product is the cotensor product of with regarded as a right comodule via prop. 7.
e.g. (Ravenel 86, def. A1.1.4).
Given a commutative Hopf algebroid over , (def.), and given a left -comodule (def.). Regard itself canonically as a right -comodule via example 11. Then the cotensor product
is called the primitive elements of :
(e.g. Ravenel 86, prop. A1.1.5)
Given a commutative Hopf algebroid over , def. 10, and given two left -comodules (def. 12), such that is projective as an -module, then
gives the structure of a right -comodule;
The cotensor product (def. 13) with respect to this right comodule structure is isomorphic to the hom of -comodules:
Hence in particular
(e.g. Ravenel 86, lemma A1.1.6)
The Hopf algebroid of dual -Steenrod operations
Now we identify the commutative Hopf algebroids arising in the -Adams spectral sequence:
If is flat according to def. 8, then, via the isomorphism of proposition 5, the cosimplicial spectrum (the -standard resolution of from example 7) exhibits:
for : Hopf algebroid-structure, def. 9, remark 5, on over – called the dual -Steenrod algebra;
for general : comodule-structure on over the dual -Steenrod algebra.
(e.g. Baker-Lazarev 01, theorem 1.1)
Via prop. 5, the image under of the cosimplicial spectrum is identified as on the right of the following diagram
Analogously the coaction is induced as on the right of the following diagram
The key use of the Hopf coalgebroid structure of prop. 9 for the present purpose is that it is extra structure inherited from maps of spectra under smashing with :
For any two spectra, the morphism (of -graded abelian groups) given by smash product with
factors through -comodule homomorphisms over the dual -Steenrod algebra:
In order to put all this together, we need to invoke a universal coefficient theorem in the following form.
If is among the examples S, HR for , MO, MU, MSp, KO, KU, then for all -module spectra with action the morphism of -graded abelian groups
(from the stable homotopy group of the mapping spectrum to the hom groups of -modules)
is an isomorphism.
This is the universal coefficient theorem of (Adams 74, chapter III, prop. 13.5), see also (Schwede 12, chapter II, prop. 6.20), and see at Kronecker pairing – Universal coefficient theorem.
With this we finally get the following statement, which serves to identity maps of certain spectra with their induced maps on -homology:
If the assumptions of prop. 10 hold, then for any two spectra, the morphism of -graded abelian groups from example 14 of the form
is an isomorphism.
(Adams 74, part III, page 323)
By the general formula for expressing adjuncts, the morphism fits into the following commuting diagram
the right vertical map comes from the isomorphism of prop. 5;
the bottom isomorphism is the cofree/forgetful adjunction isomorphism of prop. 6;
the the left vertical morphism is an isomorphism by prop. 10.
Therefore also the top morphism is an iso.
For spectra, and for a commutative ring spectrum from the list in example 9, then the -page of the -Adams spectral sequence, def. 7, for , induced by the standard -Adams resolution for from example 7, is of the form
The next step is to identify the chain homology of this with the comodule Ext-groups.
The second page
If is flat, def. 8, and satisfies the conditions of prop. 10, and a projective module over , then the entries of the -page of any -Adams spectral sequence, def. 7, for are the Ext-groups of commutative Hopf algebroid-comodules for the commutative Hopf algebroid structure on -operations from prop. 9:
In the special case that , then (by prop. 5) these are equivalently Cotor-groups
By prop. 3 it is sufficient to show this for the standard -Adams resolution of prop. 1. For that case the page is given by prop. 12, and so by the standard theory of derived functors in homological algebra (see the section Via acyclic resolutions), it is now sufficient to see that:
the category is an abelian category;
the graded chain complex of prop. 12 is the image under the hom-functor of an -acyclic resolution of .
These two statements are prop. 13 and prop. 15 below.
We now discuss the relevant general aspects of homological algebra in categories of comodules over commutative Hopf algebroids needed for the proof of theorem 1 from prop. 12.
(e.g. Ravenel 86, theorem A1.1.3)
It is clear that, without any condition the Hopf algebroid, is an additive category.
We need to show that with the assumption that is flat over , then this is also a pre-abelian category in that kernels and cokernels exist. Let be a morphism of comodules, hence a commuting diagram in Mod of the form
Consider the kernel of in Mod and its image under
By the assumption that is a flat module over , also is a kernel. By its universal property this induces uniquely a morphism as shown on the left, making the above diagram commute. This means that the -module uniquely inherits the structure of a -comodule such as to make a comodule homomorphism. By the same universal property it follows that with this comodule structure is in fact the kernel of in .
The argument for the existence of cokernels proceeds formally dually. Therefore it follows that the comparison morphism
formed in has underlying it the corresponding comparison morphism in . There this is an isomorphism, hence it is an isomorphism also in , and so the latter is not just a pre-abelian category but in fact an abelian category itself.
(e.g. Ravenel 86, lemma A1.2.2)
First of all, assuming the axiom of choice, then the category of modules has enough injectives (see this proposition). Now by prop. 6 we have the adjunction
Observe that the left adjoint is a faithful functor (being a forgetful functor) and that, by the proof of prop. 13, it is an exact functor. With this a standard lemma applies (here) which says that
with an injective module, then the co-free comodule is an injective object in ;
for any object, and for a monomorphism of -modules into an injective -module, then the adjunct is a monomorphism in (and into an injective comodule).
Let be a commutative Hopf algebroid over , def. 9, 10, such that is a flat morphism, Let be a Hopf comodule, def. 12, such that the underlying -module is a projective module (a projective object in Mod).
Then (assuming the axiom of choice) every co-free comodule, prop. 6, is an -acyclic object for the hom functor .
We need to show that the derived functors vanish in positive degree on all co-free comodules, hence on , for .
To that end, let be an injective resolution of in . By prop. 14 then is a sequence of injective objects in and by the assumption that is flat over it is an injective resolution of in . Therefore the derived functor in question is given by
Here the second equivalence is the cofree/forgetful adjunction isomorphism of prop. 6, while the last equality then follows from the assumption that the -module underlying 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 prop. 15 the proof of theorem 1 is completed.
For a spectrum and a ring spectrum, consider the inverse sequence
associated to the normalized -Adams resolution of , as in example 8. The E-nilpotent completion of is the homotopy limit over the sequence of homotopy cofibers of this tower:
This exists and comes with a canonical morphism .
(Bousfield 79, prop. 5.5, recalled as Ravenel 84, theorem 1.13)
(Ravenel 84, example 1.16
We consider now conditions for this morphism to be an equivalence.
For a ring, its core is the equalizer in
Let be a connective ring spectrum such that the core of , def. 20, is either of
the localization of the integers at a set of primes, ;
Then the map in remark 11 is an equivalence
the Eilenberg-MacLane spectrum of a prime field. For a connective spectrum, its -nilpotent completion is its p-completion
(where denotes the Moore spectrum of the abelian group ).
MU. Every spectrum is alreay -nilpotently complete
BP at prime . For every spectrum its -nilpotent completion is its p-localization
(where is the result of inverting all primes different from ).
For more discussion of E-infinity (derived) formal completions via totalizations of Amitsur complexes, see (Carlsson 07).
For a spectrum and a ring spectrum, consider the -Adams spectral sequence of (def. 7, prop. 3, prop. 4). If for each there is such that
then the -Adams spectral sequence converges strongly (def.) to the stable homotopy groups of the E-nilpotent completion of (def. 14):
(Bousfield 79, recalled as Ravenel 84, theorem 1.15)
For and , then theorem 1 and theorem 2 with example 1 gives a spectral sequence
This is the classical Adams spectral sequence.
For and MU, then theorem 1 and theorem 2 with example 1 gives a spectral sequence
This is the Adams-Novikov spectral sequence.
As derived descent in higher algebra
We discuss the general definition of -Adams-Novikov spectral sequences for suitable E-∞ rings expressed in higher algebra, as in (Lurie, Higher Algebra). We follow Lurie 10, a nice exposition is 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 over and -∞-module a spectral sequence converging to the homotopy groups of the -localization of , and this is
The quick idea is this: Given an -module , regard it as a quasicoherent sheaf on . Choose a map . This is a cover of its 1-image , which is the derived formal completion of at the image of . Restrict attention then to the restriction of to that formal completion . (So if was already an atlas, hence was already complete, we stick with the original ). Then pull back to the Cech nerve of the cover . The realization of this Cech nerve reproduces the completed image, and hence the canonical filtration on the Cech nerve gives a filtration spectral sequence for .
Spectral sequences computing homotopy groups of filtered objects
Let thoughout be a stable (∞,1)-category equipped with a t-structure such that its heart is an abelian category.
A generalized filtered object in is simply a sequential diagram
Or rather, the object being filtered is the homotopy limit
and the sequential diagram exhibits the filtering.
This appears as (Higher Algebra, def. 184.108.40.206).
For a generalized filtered object , def. 16, write
for the homotopy fiber of the th structure map, for all , 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.
This is due to (Higher Algebra, prop. 220.127.116.11). 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.
Homotopy groups of cosimplicial totalizations filtered by coskeleta
Given an cosimplicial object
its totalization is filtered, def. 16, by the totalizations of its coskeleta
The filtration spectral sequence, prop. 17, applied to the filtration of a totalization by coskeleta as in def. 18, has as -term the cohomology groups of the Moore complex associated with the cosimplicial object
This is (Higher Algebra, remark 18.104.22.168). Review is around (Wilson 13, theorem 1.2.4).
Canonical cosimplicial resolution of -algebras
We discuss now the special case of coskeletally filtered totalizations coming from the canonical cosimplicial objects induced from E-∞ algebras (“Amitsur complexes”, “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 86, ch. 3, prop. 3.1.2).
If is such that the self-generalized homology (the dual -Steenrod operations) is such that as a module over it is a flat module, then there is a natural equivalence
The condition in prop. 19 is satisfied for
It is NOT satisfied for
Here the homotopy groups are expressed by Ext-groups using a universal coefficient theorem for generalized cohomology (Adams 74, III.13).
(e.g. Wilson 13, theorem 1.3.5, based on Bousfield 79)
We consider now conditions for this morphism to be an equivalence.
For a ring, its core is the equalizer in
Let be a connective E-∞ ring such that the core of , def. 20, is either of
the localization of the integers at a set of primes, ;
Then the map in remark 11 is an equivalence
The -Adams-Novikov spectral sequence
Summing this up yields the general -Adams(-Novikov) spectral sequence
Let a connective E-∞ ring that satisfies the conditions of prop. 20. Then by prop. 17 and prop. 20 there is a strongly convergent multiplicative spectral sequence
converging to the homotopy groups of the -localization of . If moreover the dual -Steenrod algebra is flat as a module over , then, by prop. 18 and remark 10, the -term of this spectral sequence is given by the Ext-groups over the -Steenrod Hopf algebroid.
The original sources are
Convergence was notably discussed in
Aldridge Bousfield, The localization of spectra with respect to homology, Topology 18 (1979), no. 4, 257–281. (pdf)
Douglas Ravenel, Localization with respect to certain periodic homology theories, American Journal of Mathematics, Vol. 106, No. 2, (Apr., 1984), pp. 351-414 (pdf)
The spectrum-level discussion of the ASS goes back to around
- R. M. F. Moss, On the composition pairing of Adams spectral sequences, Proceedings of the London Mathematical Society 3.1 (1968): 179-192.
A streamlined presentation of this which is 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)
further highlighted in
and worked out in some more detail in
For full details of some of the steps involved see also (Schwede 12).
Doug Ravenel, Complex cobordism and stable homotopy groups of spheres, 1986 onwards
Stanley Kochman, section 3.6 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
John McCleary, chapter 9 of A user’s guide to spectral sequences, Cambridge University Press, 2001
Doug Ravenel, A Novice’s guide to the Adams-Novikov spectral sequence, in Geometric Applications of Homotopy Theory II Volume 658 of the series Lecture Notes in Mathematics pp 404-475, 2006(?)
Alan Hatcher, Spectral sequences in algebraic topology II: The Adams spectral sequence (pdf)
Robert Bruner, An Adams spectral sequence primer, 2009 (pdf)
Stefan Schwede, chapter II, section 10.3 of Symmetric spectra, 2012
John Rognes, The Adams spectral sequence (following Bruner 09), 2012 (pdf)
The modern point of view of higher algebra is in
and nicely surveyed in
Hopf algebroid -structure on
Further examples with more general coefficients
Mark Behrens, The Adams spectral sequence for (pdf)
Michael Hill, The 3-local -homology of , Proceedings of the AMS, 2007 (pdf)