Adams resolution

In mathematics, specifically algebraic topology, there is a resolution analogous to free resolutions of spectra yielding a tool for constructing the Adams spectral sequence. Essentially, the idea is to take a connective spectrum of finite type formula_1 and iteratively resolve with other spectra that are in the homotopy kernel of a map resolving the cohomology classes in formula_2 using Eilenberg–MacLane spectra.
This construction can be generalized using a spectrum formula_3, such as the Brown–Peterson spectrum formula_4, or the complex cobordism spectrum formula_5, and is used in the construction of the Adams–Novikov spectral sequencepg 49.
Construction.
The mod formula_6 Adams resolution formula_7 for a spectrum formula_1 is a certain "chain-complex" of spectra induced from recursively looking at the fibers of maps into generalized Eilenberg–Maclane spectra giving generators for the cohomology of resolved spectrapg 43. By this, we start by considering the mapformula_9where formula_10 is an Eilenberg–Maclane spectrum representing the generators of formula_11, so it is of the formformula_12where formula_13 indexes a basis of formula_14, and the map comes from the properties of Eilenberg–Maclane spectra. Then, we can take the homotopy fiber of this map (which acts as a homotopy kernel) to get a space formula_15. Note, we now set formula_16 and formula_17. Then, we can form a commutative diagramformula_18where the horizontal map is the fiber map. Recursively iterating through this construction yields a commutative diagramformula_19giving the collection formula_7. This meansformula_21is the homotopy fiber of formula_22 and formula_23 comes from the universal properties of the homotopy fiber.
Resolution of cohomology of a spectrum.
Now, we can use the Adams resolution to construct a free formula_24-resolution of the cohomology formula_11 of a spectrum formula_1. From the Adams resolution, there are short exact sequencesformula_27which can be strung together to form a long exact sequenceformula_28giving a free resolution of formula_11 as an formula_24-module.
"E"*-Adams resolution.
Because there are technical difficulties with studying the cohomology ring formula_31 in generalpg 280, we restrict to the case of considering the homology coalgebra formula_32 (of co-operations). Note for the case formula_33, formula_34 is the dual Steenrod algebra. Since formula_35 is an formula_32-comodule, we can form the bigraded groupformula_37which contains the formula_38-page of the Adams–Novikov spectral sequence for formula_1 satisfying a list of technical conditionspg 50. To get this page, we must construct the formula_40-Adams resolutionpg 49, which is somewhat analogous to the cohomological resolution above. We say a diagram of the formformula_41where the vertical arrows formula_42 is an formula_40-Adams resolution if
Although this seems like a long laundry list of properties, they are very important in the construction of the spectral sequence. In addition, the retract properties affect the structure of construction of the formula_40-Adams resolution since we no longer need to take a wedge sum of spectra for every generator.
Construction for ring spectra.
The construction of the formula_40-Adams resolution is rather simple to state in comparison to the previous resolution for any associative, commutative, connective ring spectrum formula_3 satisfying some additional hypotheses. These include formula_32 being flat over formula_60, formula_61 on formula_62 being an isomorphism, and formula_63 with formula_64 being finitely generated for which the unique ring mapformula_65extends maximally.
If we setformula_66and letformula_42be the canonical map, we can setformula_44Note that formula_3 is a retract of formula_70 from its ring spectrum structure, hence formula_46 is a retract of formula_72, and similarly, formula_51 is a retract of formula_47. In additionformula_75which gives the desired formula_76 terms from the flatness 
Relation to cobar complex.
It turns out the formula_77-term of the associated Adams–Novikov spectral sequence is then cobar complex formula_78.