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 EilenbergâÂÂMacLane spectra.
This construction can be generalized using a spectrum , such as the BrownâÂÂPeterson spectrum , or the complex cobordism spectrum , and is used in the construction of the AdamsâÂÂNovikov spectral sequence<sup>pg 49</sup>.
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 EilenbergâÂÂMaclane spectra giving generators for the cohomology of resolved spectra<sup>pg 43</sup>. By this, we start by considering the map<blockquote></blockquote>where is an EilenbergâÂÂMaclane spectrum representing the generators of , so it is of the form<blockquote></blockquote>where indexes a basis of , and the map comes from the properties of EilenbergâÂÂMaclane spectra. Then, we can take the homotopy fiber of this map (which acts as a homotopy kernel) to get a space . Note, we now set and . Then, we can form a commutative diagram<blockquote></blockquote>where the horizontal map is the fiber map. Recursively iterating through this construction yields a commutative diagram<blockquote></blockquote>giving the collection . This means<blockquote></blockquote>is the homotopy fiber of and comes from the universal properties of the homotopy fiber.
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<blockquote></blockquote>which can be strung together to form a long exact sequence<blockquote></blockquote>giving a free resolution of as an -module.
Because there are technical difficulties with studying the cohomology ring in general<sup>pg 280</sup>, we restrict to the case of considering the homology coalgebra (of co-operations). Note for the case , is the dual Steenrod algebra. Since is an -comodule, we can form the bigraded group<blockquote></blockquote>which contains the -page of the AdamsâÂÂNovikov spectral sequence for satisfying a list of technical conditions<sup>pg 50</sup>. To get this page, we must construct the -Adams resolution<sup>pg 49</sup>, which is somewhat analogous to the cohomological resolution above. We say a diagram of the form<blockquote></blockquote>where the vertical arrows is an -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 <u>we no longer need to take a wedge sum of spectra for every generator</u>.
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<blockquote></blockquote>extends maximally.
If we set<blockquote></blockquote>and let<blockquote></blockquote>be the canonical map, we can set<blockquote></blockquote>Note that is a retract of from its ring spectrum structure, hence is a retract of , and similarly, is a retract of . In addition<blockquote></blockquote>which gives the desired terms from the flatness.
It turns out the -term of the associated AdamsâÂÂNovikov spectral sequence is then cobar complex .