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
This construction can be generalized using a spectrum
, or the complex cobordism spectrum
, and is used in the construction of the Adams–Novikov spectral sequence[1]pg 49.
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 spectra[1]pg 43.
is an Eilenberg–Maclane spectrum representing the generators of
, and the map comes from the properties of Eilenberg–Maclane spectra.
Then, we can form a commutative diagram
Recursively iterating through this construction yields a commutative diagram
comes from the universal properties of the homotopy fiber.
Now, we can use the Adams resolution to construct a free
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
Because there are technical difficulties with studying the cohomology ring
in general[2]pg 280, we restrict to the case of considering the homology coalgebra
is the dual Steenrod algebra.
-comodule, we can form the bigraded group
satisfying a list of technical conditions[1]pg 50.
-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
-Adams resolution since we no longer need to take a wedge sum of spectra for every generator.
-Adams resolution is rather simple to state in comparison to the previous resolution for any associative, commutative, connective ring spectrum
satisfying some additional hypotheses.
being finitely generated for which the unique ring map
be the canonical map, we can set
from its ring spectrum structure, hence
-term of the associated Adams–Novikov spectral sequence is then cobar complex