Adams resolution

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 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 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 = \bigwedge_{k=1}^\infty \bigwedge_{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 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 pg 50. To get this page, we must construct the $$E_*$$-Adams resolution 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 $$C^*(E_*(X))$$.