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^[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

$K=\bigwedge _{k=1}^{\infty }\bigwedge _{I_{k}}\Sigma ^{k}H\mathbb {Z} /p$

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

${\begin{matrix}X_{0}&\leftarrow &X_{1}\\\downarrow &&\\K_{0}\end{matrix}}$

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

$X_{s+1}={\text{Hofiber}}(f_{s})$ is the homotopy fiber of $f_{s}$
$E\wedge X_{s}$ 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 $h_{s}(E\wedge f_{s})=id_{E\wedge X_{s}}$
$K_{s}$ is a retract of $E\wedge K_{s}$
${\text{Ext}}^{t,u}(E_{*}(\mathbb {S} ),E_{*}(K_{s}))=\pi _{u}(K_{s})$ 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