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 sequence[1]pg 49.

Construction

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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 spectra[1]pg 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 (which acts as a homotopy kernel) 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

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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

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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   (of co-operations). 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 conditions[1]pg 50. To get this page, we must construct the  -Adams resolution[1]pg 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

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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

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

See also

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References

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  1. ^ a b c d Ravenel, Douglas C. (1986). Complex cobordism and stable homotopy groups of spheres. Orlando: Academic Press. ISBN 978-0-08-087440-1. OCLC 316566772.
  2. ^ Adams, J. Frank (John Frank) (1974). Stable homotopy and generalised homology. Chicago: University of Chicago Press. ISBN 0-226-00523-2. OCLC 1083550.