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 $H^{\ast }(X;\mathbb{Z}/p)$ 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

$\begin{array}{c}X\\ \downarrow \\ K\end{array}$

where $K$ is an Eilenberg–Maclane spectrum representing the generators of $H^{\ast }(X)$, so it is of the form

$K=\underset{k=1}{\overset{\mathrm{\infty }}{\bigvee }}\underset{I_k}{\bigvee }{\mathrm{\Sigma }}^{k}H\mathbb{Z}/p$

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

$\begin{array}{ccc}{X}_0& \leftarrow & X_1\\ \downarrow & & \\ K_0\end{array}$

where the horizontal map is the fiber map. Recursively iterating through this construction yields a commutative diagram

$\begin{array}{cccccc}{X}_0& \leftarrow & X_1& \leftarrow & X_2& \leftarrow \cdots \\ \downarrow & & \downarrow & & \downarrow \\ K_0& & K_1& & K_2\end{array}$

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 ${𝒜}_p$-resolution of the cohomology $H^{\ast }(X)$ of a spectrum $X$. From the Adams resolution, there are short exact sequences

$0\leftarrow H^{\ast }(X_s)\leftarrow H^{\ast }(K_s)\leftarrow H^{\ast }(\mathrm{\Sigma }{X}_{s+1})\leftarrow 0$

which can be strung together to form a long exact sequence

$0\leftarrow H^{\ast }(X)\leftarrow H^{\ast }(K_0)\leftarrow H^{\ast }(\mathrm{\Sigma }{K}_1)\leftarrow H^{\ast }({\mathrm{\Sigma }}^2{K}_2)\leftarrow \cdots$

giving a free resolution of $H^{\ast }(X)$ as an ${𝒜}_p$-module.

E_*-Adams resolution

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

${\text{Ext}}_{E_{\ast }(E)}(E_{\ast }(𝕊),E_{\ast }(X))$

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 $E_{\ast }$-Adams resolution^{pg 49}, which is somewhat analogous to the cohomological resolution above. We say a diagram of the form

$\begin{array}{cccccc}{X}_0& \stackrel{g_0}{\leftarrow }& X_1& \stackrel{g_1}{\leftarrow }& X_2& \leftarrow \cdots \\ \downarrow & & \downarrow & & \downarrow \\ K_0& & K_1& & K_2\end{array}$

where the vertical arrows $f_s:X_s\to K_s$ is an $E_{\ast }$-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 $E_{\ast }(f_s)$ is a monomorphism. By retract, we mean there is a map $h_s:E\wedge K_s\to E\wedge X_s$ such that $h_s(E\wedge f_s)=i{d}_{E\wedge X_s}$
3. $K_s$ is a retract of $E\wedge K_s$
4. ${\text{Ext}}^{t,u}(E_{\ast }(𝕊),E_{\ast }(K_s))={\pi }_u(K_s)$ 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 $E_{\ast }$-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 $E_{\ast }$-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 $E_{\ast }(E)$ being flat over ${\pi }_{\ast }(E)$, ${\mu }_{\ast }$ 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

$E_{\ast }(K_s)=E_{\ast }(E){\otimes }_{{\pi }_{\ast }(E)}{E}_{\ast }(X_s)$

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 $C^{\ast }(E_{\ast }(X))$.

See also

• Adams spectral sequence
• Adams–Novikov spectral sequence
• Eilenberg–Maclane spectrum
• Hopf algebroid