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 and iteratively resolve with other spectra that are in the homotopy kernel of a map resolving the cohomology classes in using Eilenberg–MacLane spectra.
This construction can be generalized using a spectrum, such as the Brown–Peterson spectrum, or the complex cobordism spectrum, and is used in the construction of the Adams–Novikov spectral sequencepg 49.

Construction

The mod Adams resolution for a spectrum 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 map
where is an Eilenberg–Maclane spectrum representing the generators of, so it is of the form
where indexes a basis of, and the map comes from the properties of Eilenberg–Maclane spectra. Then, we can take the homotopy fiber of this map to get a space. Note, we now set and. Then, we can form a commutative diagram
where the horizontal map is the fiber map. Recursively iterating through this construction yields a commutative diagram
giving the collection. This means
is the homotopy fiber of and 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 -resolution of the cohomology of a spectrum. From the Adams resolution, there are short exact sequences
which can be strung together to form a long exact sequence
giving a free resolution of as an -module.

''E''*-Adams resolution

Because there are technical difficulties with studying the cohomology ring in generalpg 280, we restrict to the case of considering the homology coalgebra . Note for the case, is the dual Steenrod algebra. Since is an -comodule, we can form the bigraded group
which contains the -page of the Adams–Novikov spectral sequence for satisfying a list of technical conditionspg 50. To get this page, we must construct the -Adams resolutionpg 49, which is somewhat analogous to the cohomological resolution above. We say a diagram of the form
where the vertical arrows is an -Adams resolution if
  1. is the homotopy fiber of
  2. is a retract of, hence is a monomorphism. By retract, we mean there is a map such that
  3. is a retract of
  4. if, otherwise it is
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 -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 -Adams resolution is rather simple to state in comparison to the previous resolution for any associative, commutative, connective ring spectrum satisfying some additional hypotheses. These include being flat over, on being an isomorphism, and with being finitely generated for which the unique ring map
extends maximally.
If we set
and let
be the canonical map, we can set
Note that is a retract of from its ring spectrum structure, hence is a retract of, and similarly, is a retract of. In addition
which gives the desired terms from the flatness.

Relation to cobar complex

It turns out the -term of the associated Adams–Novikov spectral sequence is then cobar complex.