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

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

The mod ${\displaystyle p}$ Adams resolution ${\displaystyle (X_{s},g_{s})}$ for a spectrum ${\displaystyle X}$ 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

${\displaystyle {\begin{matrix}X\\\downarrow \\K\end{matrix}}}$

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

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

where ${\displaystyle I_{k}}$ indexes a basis of ${\displaystyle H^{k}(X)}$, 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 ${\displaystyle X_{1}}$. Note, we now set ${\displaystyle X_{0}=X}$ and ${\displaystyle K_{0}=K}$. Then, we can form a commutative diagram

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

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

${\displaystyle {\begin{matrix}X_{0}&\leftarrow &X_{1}&\leftarrow &X_{2}&\leftarrow \cdots \\\downarrow &&\downarrow &&\downarrow \\K_{0}&&K_{1}&&K_{2}\end{matrix}}}$

giving the collection ${\displaystyle (X_{s},g_{s})}$. This means

${\displaystyle X_{s}={\text{Hofiber}}(f_{s-1}:X_{s-1}\to K_{s-1})}$

is the homotopy fiber of ${\displaystyle f_{s-1}}$ and ${\displaystyle g_{s}:X_{s}\to X_{s-1}}$ comes from the universal properties of the homotopy fiber.

Now, we can use the Adams resolution to construct a free ${\displaystyle {\mathcal {A}}_{p}}$-resolution of the cohomology ${\displaystyle H^{*}(X)}$ of a spectrum ${\displaystyle X}$. From the Adams resolution, there are short exact sequences

${\displaystyle 0\leftarrow H^{*}(X_{s})\leftarrow H^{*}(K_{s})\leftarrow H^{*}(\Sigma X_{s+1})\leftarrow 0}$

which can be strung together to form a long exact sequence

${\displaystyle 0\leftarrow H^{*}(X)\leftarrow H^{*}(K_{0})\leftarrow H^{*}(\Sigma K_{1})\leftarrow H^{*}(\Sigma ^{2}K_{2})\leftarrow \cdots }$

giving a free resolution of ${\displaystyle H^{*}(X)}$ as an ${\displaystyle {\mathcal {A}}_{p}}$-module.

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

${\displaystyle {\text{Ext}}_{E_{*}(E)}(E_{*}(\mathbb {S} ),E_{*}(X))}$

which contains the ${\displaystyle E_{2}}$-page of the Adams–Novikov spectral sequence for ${\displaystyle X}$ satisfying a list of technical conditions^{[1]pg 50}. To get this page, we must construct the ${\displaystyle E_{*}}$-Adams resolution^{[1]pg 49}, which is somewhat analogous to the cohomological resolution above. We say a diagram of the form

${\displaystyle {\begin{matrix}X_{0}&\xleftarrow {g_{0}} &X_{1}&\xleftarrow {g_{1}} &X_{2}&\leftarrow \cdots \\\downarrow &&\downarrow &&\downarrow \\K_{0}&&K_{1}&&K_{2}\end{matrix}}}$

where the vertical arrows ${\displaystyle f_{s}:X_{s}\to K_{s}}$ is an ${\displaystyle E_{*}}$-Adams resolution if

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

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 ${\displaystyle E_{*}}$-Adams resolution since we no longer need to take a wedge sum of spectra for every generator.

The construction of the ${\displaystyle E_{*}}$-Adams resolution is rather simple to state in comparison to the previous resolution for any associative, commutative, connective ring spectrum ${\displaystyle E}$ satisfying some additional hypotheses. These include ${\displaystyle E_{*}(E)}$ being flat over ${\displaystyle \pi _{*}(E)}$, ${\displaystyle \mu _{*}}$ on ${\displaystyle \pi _{0}}$ being an isomorphism, and ${\displaystyle H_{r}(E;A)}$ with ${\displaystyle \mathbb {Z} \subset A\subset \mathbb {Q} }$ being finitely generated for which the unique ring map

${\displaystyle \theta :\mathbb {Z} \to \pi _{0}(E)}$

extends maximally. If we set

${\displaystyle K_{s}=E\wedge F_{s}}$

and let

${\displaystyle f_{s}:X_{s}\to K_{s}}$

be the canonical map, we can set

${\displaystyle X_{s+1}={\text{Hofiber}}(f_{s})}$

Note that ${\displaystyle E}$ is a retract of ${\displaystyle E\wedge E}$ from its ring spectrum structure, hence ${\displaystyle E\wedge X_{s}}$ is a retract of ${\displaystyle E\wedge K_{s}=E\wedge E\wedge X_{s}}$, and similarly, ${\displaystyle K_{s}}$ is a retract of ${\displaystyle E\wedge K_{s}}$. In addition

${\displaystyle E_{*}(K_{s})=E_{*}(E)\otimes _{\pi _{*}(E)}E_{*}(X_{s})}$

which gives the desired ${\displaystyle {\text{Ext}}}$ terms from the flatness.

It turns out the ${\displaystyle E_{1}}$-term of the associated Adams–Novikov spectral sequence is then cobar complex ${\displaystyle C^{*}(E_{*}(X))}$.

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