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Adams resolution

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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

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

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 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

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 .

References

  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.

See also