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Construction

The mod p {\displaystyle p} Adams resolution ( X s , g s ) {\displaystyle (X_{s},g_{s})} for a spectrum X {\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 spectra²pg 43. By this, we start by considering the map

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

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

K = ⋁ k = 1 ∞ ⋁ I k Σ k H Z / p {\displaystyle K=\bigvee _{k=1}^{\infty }\bigvee _{I_{k}}\Sigma ^{k}H\mathbb {Z} /p}

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

X 0 ← X 1 ↓ K 0 {\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

X 0 ← X 1 ← X 2 ← ⋯ ↓ ↓ ↓ K 0 K 1 K 2 {\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 ( X s , g s ) {\displaystyle (X_{s},g_{s})} . This means

X s = Hofiber ( f s − 1 : X s − 1 → K s − 1 ) {\displaystyle X_{s}={\text{Hofiber}}(f_{s-1}:X_{s-1}\to K_{s-1})}

is the homotopy fiber of f s − 1 {\displaystyle f_{s-1}} and g s : X s → X s − 1 {\displaystyle 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 {\displaystyle {\mathcal {A}}_{p}} -resolution of the cohomology H ∗ ( X ) {\displaystyle H^{*}(X)} of a spectrum X {\displaystyle X} . From the Adams resolution, there are short exact sequences

0 ← H ∗ ( X s ) ← H ∗ ( K s ) ← H ∗ ( Σ X s + 1 ) ← 0 {\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

0 ← H ∗ ( X ) ← H ∗ ( K 0 ) ← H ∗ ( Σ K 1 ) ← H ∗ ( Σ 2 K 2 ) ← ⋯ {\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 H ∗ ( X ) {\displaystyle H^{*}(X)} as an A p {\displaystyle {\mathcal {A}}_{p}} -module.

E*-Adams resolution

Because there are technical difficulties with studying the cohomology ring E ∗ ( E ) {\displaystyle E^{*}(E)} in general³pg 280, we restrict to the case of considering the homology coalgebra E ∗ ( E ) {\displaystyle E_{*}(E)} (of co-operations). Note for the case E = H F p {\displaystyle E=H\mathbb {F} _{p}} , H F p ∗ ( H F p ) = A ∗ {\displaystyle H\mathbb {F} _{p*}(H\mathbb {F} _{p})={\mathcal {A}}_{*}} is the dual Steenrod algebra. Since E ∗ ( X ) {\displaystyle E_{*}(X)} is an E ∗ ( E ) {\displaystyle E_{*}(E)} -comodule, we can form the bigraded group

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

which contains the E 2 {\displaystyle E_{2}} -page of the Adams–Novikov spectral sequence for X {\displaystyle X} satisfying a list of technical conditions⁴pg 50. To get this page, we must construct the E ∗ {\displaystyle E_{*}} -Adams resolution⁵pg 49, which is somewhat analogous to the cohomological resolution above. We say a diagram of the form

X 0 ← g 0 X 1 ← g 1 X 2 ← ⋯ ↓ ↓ ↓ K 0 K 1 K 2 {\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 f s : X s → K s {\displaystyle f_{s}:X_{s}\to K_{s}} is an E ∗ {\displaystyle E_{*}} -Adams resolution if

1. X s + 1 = Hofiber ( f s ) {\displaystyle X_{s+1}={\text{Hofiber}}(f_{s})} is the homotopy fiber of f s {\displaystyle f_{s}}
2. E ∧ X s {\displaystyle E\wedge X_{s}} is a retract of E ∧ K s {\displaystyle E\wedge K_{s}} , hence E ∗ ( f s ) {\displaystyle E_{*}(f_{s})} is a monomorphism. By retract, we mean there is a map h s : E ∧ K s → E ∧ X s {\displaystyle h_{s}:E\wedge K_{s}\to E\wedge X_{s}} such that h s ( E ∧ f s ) = i d E ∧ X s {\displaystyle h_{s}(E\wedge f_{s})=id_{E\wedge X_{s}}}
3. K s {\displaystyle K_{s}} is a retract of E ∧ K s {\displaystyle E\wedge K_{s}}
4. Ext t , u ( E ∗ ( S ) , E ∗ ( K s ) ) = π u ( K s ) {\displaystyle {\text{Ext}}^{t,u}(E_{*}(\mathbb {S} ),E_{*}(K_{s}))=\pi _{u}(K_{s})} if t = 0 {\displaystyle t=0} , otherwise it is 0 {\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 E ∗ {\displaystyle 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 ∗ {\displaystyle E_{*}} -Adams resolution is rather simple to state in comparison to the previous resolution for any associative, commutative, connective ring spectrum E {\displaystyle E} satisfying some additional hypotheses. These include E ∗ ( E ) {\displaystyle E_{*}(E)} being flat over π ∗ ( E ) {\displaystyle \pi _{*}(E)} , μ ∗ {\displaystyle \mu _{*}} on π 0 {\displaystyle \pi _{0}} being an isomorphism, and H r ( E ; A ) {\displaystyle H_{r}(E;A)} with Z ⊂ A ⊂ Q {\displaystyle \mathbb {Z} \subset A\subset \mathbb {Q} } being finitely generated for which the unique ring map

θ : Z → π 0 ( E ) {\displaystyle \theta :\mathbb {Z} \to \pi _{0}(E)}

extends maximally. If we set

K s = E ∧ F s {\displaystyle K_{s}=E\wedge F_{s}}

and let

f s : X s → K s {\displaystyle f_{s}:X_{s}\to K_{s}}

be the canonical map, we can set

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

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

E ∗ ( K s ) = E ∗ ( E ) ⊗ π ∗ ( E ) E ∗ ( X s ) {\displaystyle E_{*}(K_{s})=E_{*}(E)\otimes _{\pi _{*}(E)}E_{*}(X_{s})}

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

Relation to cobar complex

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

See also

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

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. 978-0-08-087440-1 ↩

2. Ravenel, Douglas C. (1986). Complex cobordism and stable homotopy groups of spheres. Orlando: Academic Press. ISBN 978-0-08-087440-1. OCLC 316566772. 978-0-08-087440-1 ↩

3. Adams, J. Frank (John Frank) (1974). Stable homotopy and generalised homology. Chicago: University of Chicago Press. ISBN 0-226-00523-2. OCLC 1083550. 0-226-00523-2 ↩

4. Ravenel, Douglas C. (1986). Complex cobordism and stable homotopy groups of spheres. Orlando: Academic Press. ISBN 978-0-08-087440-1. OCLC 316566772. 978-0-08-087440-1 ↩

5. Ravenel, Douglas C. (1986). Complex cobordism and stable homotopy groups of spheres. Orlando: Academic Press. ISBN 978-0-08-087440-1. OCLC 316566772. 978-0-08-087440-1 ↩