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 X and iteratively resolve with other spectra that are in the homotopy kernel of a map resolving the cohomology classes in H^*(X;\mathbb{Z}/p) using Eilenberg–MacLane spectra.

This construction can be generalized using a spectrum E, such as the Brown–Peterson spectrum BP, or the complex cobordism spectrum MU, and is used in the construction of the Adams–Novikov spectral sequence[1]pg 49.

Construction

The mod p Adams resolution (X_s,g_s) for a spectrum X 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

\begin{matrix} X \\ \downarrow \\ K \end{matrix}
where K is an Eilenberg–Maclane spectrum representing the generators of H^*(X), so it is of the form
K = \bigvee_{k=1}^\infty \bigvee_{I_k} \Sigma^kH\mathbb{Z}/p
where I_k indexes a basis of H^k(X), 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 X_1. Note, we now set X_0 = X and K_0 = K. Then, we can form a commutative diagram
\begin{matrix} X_0 & \leftarrow & X_1 \\ \downarrow & & \\ K_0 \end{matrix}
where the horizontal map is the fiber map. Recursively iterating through this construction yields a commutative diagram
\begin{matrix} X_0 & \leftarrow & X_1 & \leftarrow & X_2 & \leftarrow \cdots \\ \downarrow & & \downarrow & & \downarrow \\ K_0 & & K_1 & & K_2 \end{matrix}
giving the collection (X_s,g_s). This means
X_s = \text{Hofiber}(f_{s-1}:X_{s-1} \to K_{s-1})
is the homotopy fiber of f_{s-1} and g_s:X_s \to X_{s-1} 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 \mathcal{A}_p-resolution of the cohomology H^*(X) of a spectrum X. From the Adams resolution, there are short exact sequences

0 \leftarrow H^*(X_s) \leftarrow H^*(K_s) \leftarrow H^*(\Sigma X_{s+1}) \leftarrow 0
which can be strung together to form a long exact sequence
0 \leftarrow H^*(X) \leftarrow H^*(K_0) \leftarrow H^*(\Sigma K_1) \leftarrow H^*(\Sigma^2 K_2) \leftarrow \cdots
giving a free resolution of H^*(X) as an \mathcal{A}_p-module.

E*-Adams resolution

Because there are technical difficulties with studying the cohomology ring E^*(E) in general[2]pg 280, we restrict to the case of considering the homology coalgebra E_*(E) (of co-operations). Note for the case E = H\mathbb{F}_p, H\mathbb{F}_{p*}(H\mathbb{F}_p) =\mathcal{A}_* is the dual Steenrod algebra. Since E_*(X) is an E_*(E)-comodule, we can form the bigraded group

\text{Ext}_{E_*(E)}(E_*(\mathbb{S}), E_*(X))
which contains the E_2-page of the Adams–Novikov spectral sequence for X satisfying a list of technical conditions[1]pg 50. To get this page, we must construct the E_*-Adams resolution[1]pg 49, which is somewhat analogous to the cohomological resolution above. We say a diagram of the form
\begin{matrix} X_0 & \xleftarrow{g_0} & X_1 & \xleftarrow{g_1} & X_2 & \leftarrow \cdots \\ \downarrow & & \downarrow & & \downarrow \\ K_0 & & K_1 & & K_2 \end{matrix}
where the vertical arrows f_s: X_s \to K_s is an E_*-Adams resolution if

  1. X_{s+1} = \text{Hofiber}(f_s) is the homotopy fiber of f_s
  2. E \wedge X_s is a retract of E\wedge K_s, hence E_*(f_s) is a monomorphism. By retract, we mean there is a map h_s:E \wedge K_s \to E \wedge X_s such that h_s(E\wedge f_s) = id_{E \wedge X_s}
  3. K_s is a retract of E \wedge K_s
  4. \text{Ext}^{t,u}(E_*(\mathbb{S}), E_*(K_s)) = \pi_u(K_s) if t = 0, otherwise it is 0

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 E_*-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 E_*-Adams resolution is rather simple to state in comparison to the previous resolution for any associative, commutative, connective ring spectrum E satisfying some additional hypotheses. These include E_*(E) being flat over \pi_*(E), \mu_* on \pi_0 being an isomorphism, and H_r(E; A) with \mathbb{Z} \subset A \subset \mathbb{Q} being finitely generated for which the unique ring map

\theta:\mathbb{Z} \to \pi_0(E)
extends maximally.

If we set

K_s = E \wedge F_s
and let
f_s: X_s \to K_s
be the canonical map, we can set
X_{s+1} = \text{Hofiber}(f_s)
Note that E is a retract of E \wedge E from its ring spectrum structure, hence E \wedge X_s is a retract of E \wedge K_s = E \wedge E \wedge X_s, and similarly, K_s is a retract of E\wedge K_s. In addition
E_*(K_s) = E_*(E)\otimes_{\pi_*(E)}E_*(X_s)
which gives the desired \text{Ext} terms from the flatness.

Relation to cobar complex

It turns out the E_1-term of the associated Adams–Novikov spectral sequence is then cobar complex C^*(E_*(X)).

See also

References

  1. ^ 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