Adams resolution

<h2 id="construction">Construction</h2>
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 <a href="/facts/Eilenberg%E2%80%93MacLane_spectra/mrux6oMJ">Eilenberg–Maclane spectra</a> giving generators for the cohomology of resolved spectra<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>pg 43. By this, we start by considering the map<blockquote>
 
 
 
 
 
 
 
 X
 
 
 
 
 ↓
 
 
 
 
 K
 
 
 
 
 
 
 {\displaystyle {\begin{matrix}X\\\downarrow \\K\end{matrix}}}
 
</blockquote>where 
 
 
 
 K
 
 
 {\displaystyle K}
 
 is an Eilenberg–Maclane spectrum representing the generators of 
 
 
 
 
 H
 
 ∗
 
 
 (
 X
 )
 
 
 {\displaystyle H^{*}(X)}
 
, so it is of the form<blockquote>
 
 
 
 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}
 
</blockquote>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 <a href="/facts/Eilenberg%E2%80%93Maclane_spectrum/F3yELuIr">Eilenberg–Maclane spectra</a>. Then, we can take the <a href="/facts/Homotopy_fiber/hH4omFFQ">homotopy fiber</a> 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<blockquote>
 
 
 
 
 
 
 
 
 X
 
 0
 
 
 
 
 ←
 
 
 
 X
 
 1
 
 
 
 
 
 
 ↓
 
 
 
 
 
 
 
 K
 
 0
 
 
 
 
 
 
 
 
 {\displaystyle {\begin{matrix}X_{0}&\leftarrow &X_{1}\\\downarrow &&\\K_{0}\end{matrix}}}
 
</blockquote>where the horizontal map is the fiber map. Recursively iterating through this construction yields a commutative diagram<blockquote>
 
 
 
 
 
 
 
 
 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}}}
 
</blockquote>giving the collection 
 
 
 
 (
 
 X
 
 s
 
 
 ,
 
 g
 
 s
 
 
 )
 
 
 {\displaystyle (X_{s},g_{s})}
 
. This means<blockquote>
 
 
 
 
 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})}
 
</blockquote>is the <a href="/facts/Homotopy_fiber/hH4omFFQ">homotopy fiber</a> 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.
<h3>Resolution of cohomology of a spectrum</h3>
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<blockquote>
 
 
 
 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}
 
</blockquote>which can be strung together to form a long exact sequence<blockquote>
 
 
 
 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 }
 
</blockquote>giving a free resolution of 
 
 
 
 
 H
 
 ∗
 
 
 (
 X
 )
 
 
 {\displaystyle H^{*}(X)}
 
 as an 
 
 
 
 
 
 
 A
 
 
 
 p
 
 
 
 
 {\displaystyle {\mathcal {A}}_{p}}
 
-module.
<h2 id="e-adams-resolution">E*-Adams resolution</h2>
Because there are technical difficulties with studying the cohomology ring 
 
 
 
 
 E
 
 ∗
 
 
 (
 E
 )
 
 
 {\displaystyle E^{*}(E)}
 
 in general<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>pg 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 <a href="/facts/Dual_Steenrod_algebra/29c9uAmH">dual Steenrod algebra</a>. Since 
 
 
 
 
 E
 
 ∗
 
 
 (
 X
 )
 
 
 {\displaystyle E_{*}(X)}
 
 is an 
 
 
 
 
 E
 
 ∗
 
 
 (
 E
 )
 
 
 {\displaystyle E_{*}(E)}
 
-comodule, we can form the bigraded group<blockquote>
 
 
 
 
 
 Ext
 
 
 
 E
 
 ∗
 
 
 (
 E
 )
 
 
 (
 
 E
 
 ∗
 
 
 (
 
 S
 
 )
 ,
 
 E
 
 ∗
 
 
 (
 X
 )
 )
 
 
 {\displaystyle {\text{Ext}}_{E_{*}(E)}(E_{*}(\mathbb {S} ),E_{*}(X))}
 
</blockquote>which contains the 
 
 
 
 
 E
 
 2
 
 
 
 
 {\displaystyle E_{2}}
 
-page of the Adams–Novikov spectral sequence for 
 
 
 
 X
 
 
 {\displaystyle X}
 
 satisfying a list of technical conditions<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>pg 50. To get this page, we must construct the 
 
 
 
 
 E
 
 ∗
 
 
 
 
 {\displaystyle E_{*}}
 
-Adams resolution<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>pg 49, which is somewhat analogous to the cohomological resolution above. We say a diagram of the form<blockquote>
 
 
 
 
 
 
 
 
 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}}}
 
</blockquote>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
<ol><li>
 
 
 
 
 X
 
 s
 +
 1
 
 
 =
 
 Hofiber
 
 (
 
 f
 
 s
 
 
 )
 
 
 {\displaystyle X_{s+1}={\text{Hofiber}}(f_{s})}
 
 is the <a href="/facts/Homotopy_fiber/hH4omFFQ">homotopy fiber</a> of 
 
 
 
 
 f
 
 s
 
 
 
 
 {\displaystyle f_{s}}
 
</li>
<li>
 
 
 
 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 <a href="/facts/Monomorphism/Q9BrtDTs">monomorphism</a>. 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}}}
 
</li>
<li>
 
 
 
 
 K
 
 s
 
 
 
 
 {\displaystyle K_{s}}
 
 is a retract of 
 
 
 
 E
 ∧
 
 K
 
 s
 
 
 
 
 {\displaystyle E\wedge K_{s}}
 
</li>
<li>
 
 
 
 
 
 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}
 
</li></ol>
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.

<h3>Construction for ring spectra</h3>
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<blockquote>
 
 
 
 θ
 :
 
 Z
 
 →
 
 π
 
 0
 
 
 (
 E
 )
 
 
 {\displaystyle \theta :\mathbb {Z} \to \pi _{0}(E)}
 
</blockquote>extends maximally.
If we set<blockquote>
 
 
 
 
 K
 
 s
 
 
 =
 E
 ∧
 
 F
 
 s
 
 
 
 
 {\displaystyle K_{s}=E\wedge F_{s}}
 
</blockquote>and let<blockquote>
 
 
 
 
 f
 
 s
 
 
 :
 
 X
 
 s
 
 
 →
 
 K
 
 s
 
 
 
 
 {\displaystyle f_{s}:X_{s}\to K_{s}}
 
</blockquote>be the canonical map, we can set<blockquote>
 
 
 
 
 X
 
 s
 +
 1
 
 
 =
 
 Hofiber
 
 (
 
 f
 
 s
 
 
 )
 
 
 {\displaystyle X_{s+1}={\text{Hofiber}}(f_{s})}
 
</blockquote>Note that 
 
 
 
 E
 
 
 {\displaystyle E}
 
 is a retract of 
 
 
 
 E
 ∧
 E
 
 
 {\displaystyle E\wedge E}
 
 from its <a href="/facts/Ring_spectrum/fB4IaL5T">ring spectrum</a> 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<blockquote>
 
 
 
 
 E
 
 ∗
 
 
 (
 
 K
 
 s
 
 
 )
 =
 
 E
 
 ∗
 
 
 (
 E
 )
 
 ⊗
 
 
 π
 
 ∗
 
 
 (
 E
 )
 
 
 
 E
 
 ∗
 
 
 (
 
 X
 
 s
 
 
 )
 
 
 {\displaystyle E_{*}(K_{s})=E_{*}(E)\otimes _{\pi _{*}(E)}E_{*}(X_{s})}
 
</blockquote>which gives the desired 
 
 
 
 
 Ext
 
 
 
 {\displaystyle {\text{Ext}}}
 
 terms from the flatness.
<h4>Relation to cobar complex</h4>
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))}
 
.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Adams_spectral_sequence/XPqI80V4">Adams spectral sequence</a></li>
<li><a href="/facts/Adams%E2%80%93Novikov_spectral_sequence/XPqI80V4">Adams–Novikov spectral sequence</a></li>
<li><a href="/facts/Eilenberg%E2%80%93Maclane_spectrum/F3yELuIr">Eilenberg–Maclane spectrum</a></li>
<li><a href="/facts/Hopf_algebroid/56k17Ea1">Hopf algebroid</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Ravenel, Douglas C. (1986). Complex cobordism and stable homotopy groups of spheres. Orlando: Academic Press. ISBN 978-0-08-087440-1. OCLC 316566772. <a href="978-0-08-087440-1" target="_blank">978-0-08-087440-1</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Ravenel, Douglas C. (1986). Complex cobordism and stable homotopy groups of spheres. Orlando: Academic Press. ISBN 978-0-08-087440-1. OCLC 316566772. <a href="978-0-08-087440-1" target="_blank">978-0-08-087440-1</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Adams, J. Frank (John Frank) (1974). Stable homotopy and generalised homology. Chicago: University of Chicago Press. ISBN 0-226-00523-2. OCLC 1083550. <a href="0-226-00523-2" target="_blank">0-226-00523-2</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Ravenel, Douglas C. (1986). Complex cobordism and stable homotopy groups of spheres. Orlando: Academic Press. ISBN 978-0-08-087440-1. OCLC 316566772. <a href="978-0-08-087440-1" target="_blank">978-0-08-087440-1</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Ravenel, Douglas C. (1986). Complex cobordism and stable homotopy groups of spheres. Orlando: Academic Press. ISBN 978-0-08-087440-1. OCLC 316566772. <a href="978-0-08-087440-1" target="_blank">978-0-08-087440-1</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
</ol>

Adams resolution open-in-new

Adams resolution