Adams operation

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#### Algebraic topology
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[[!include algebraic topology - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

Given a [[complex line bundle]] $L$ over a [[space]] $X$ its $k$th [[tensor product of vector bundles|tensor power]] $L^{\otimes k}$ is another line bundle for any $k \in \mathbb{N}$.  The line bundles define certain elements of [[complex topological K-theory]] group $K(X)$, and there is a unique [[cohomology operation]]

$$
  \psi^k \colon K(X) \to K(X)
  \,,
$$

the _$k$th Adams operation_, such that:

* $\psi^k([L]) = [L^{\otimes k}]$ if $[L]$ is the K-theory class of any line bundle,

* $\psi^k \colon K(X) \to K(X)$ is a [[group homomorphism]],

* $\psi^k$ is a [[natural transformation]]: any map $f: X \to Y$ induces a map $f^* : K(Y) \to K(X)$ on $K$-theory, and $\psi^k \circ f^* = f^* \circ \psi^k$.

More abstractly, Adams operations can be defined on any [[Lambda-ring]]. They are an example of [[power operations]].

## Definition
 {#Definition}

The Adams operations have an explicit definition in terms of the [[Lambda-ring]] structure on [[topological K-theory]], this we state as def. \ref{DefinitionInTermsOfLambdaRing} below. While explicit, this definition may look contrived on first sight. But it turns out that it satisfies a list of properties, of which two simple ones already uniquely characterize the Adams operations.  This is proposition \ref{TheBasicProperties} below.

+-- {: .num_prop #LambdaRingStructureOnKTheory}
###### Definition
**(Lambda-ring structure on topological K-theory)**

Let $X$ be a [[compact topological space]] and write $K(X)$ for its [[topological K-theory]] [[ring]]. For $E$ a [[vector bundle]] over $X$, write $[E] \in K(X)$ for its class in K-theory.  Given $E$, write

$$
  \lambda_t[E]
  \;\coloneqq\;
  \underoverset{k = 0}{\infty}{\sum}
  [\wedge^k_X E] t^k
  \;\;\in\;\;
  K(X)[ [t] ]
$$

for the [[formal power series]] with [[coefficients]] in the ring $K(X)$ being the K-theory classes of the skew-symmetrized [[tensor product of vector bundles]] of $E$ with itself.

Since the constant term of this power series is always the unit $[\wedge^0 E] = 1$, hence

$$
  \lambda_t[E] \in 1 + (t) \cdot K(X)[ [t] ]
$$

there exists a multiplicative inverse formal power series $\lambda_t[E]^{-1}$.

Then given the class of a [[virtual vector bundle]] $[E] - [F] \in K(X)$, define more generally

$$
  \lambda_t[[E- F]]
  \;\;\coloneqq\;\,
  \lambda_t[E] \cdot \lambda_t[F]^{-1}
  \;\;\in\;\;
  K(X)[ [t] ]
  \,.
$$

=--

+-- {: .num_prop #DefinitionInTermsOfLambdaRing}
###### Definition
**(explicit definition of Adams operation)**

For $E$ a [[vector bundle]] over some [[topological space]] $X$, write

$$
  \psi^0(E) \coloneqq rank(E)
$$

for the [[bundle]] which over each [[connected component]] of $X$ is the [[trivial bundle|trivial]] vector bundle of the same [[rank of a vector bundle|rank]] as $E$ over that component.

Define a [[formal power series]] with [[coefficients]] in the K-theory ring $K(X)$ by

$$
  \begin{aligned}
    \psi_t(E)
    & \coloneqq
   \underoverset{\infty}{k = 0}{\sum}
   \psi^k(E) t^k
   \\
    & \coloneqq
    \psi^0(E)
      -
     t \frac{d}{d t} log \lambda_{-t}(E)
    \;\;\in\;\;
    K(X)[ [t] ]
  \end{aligned}
  \,,
$$

where $\lambda_t$ is the [[Lambda-ring]] operation from def. \ref{LambdaRingStructureOnKTheory}.

Here the [[derivative]] of the [[logarithm]] of formal power series stands for the usual expression in terms of the [[geometric series]]:

$$
  \begin{aligned}
    \frac{d}{d t} log \lambda_{-t}(E)
    & =
    \frac{1}{\lambda_{-t}(E)}
    \frac{d}{d t}
     \lambda_{-t}(E)
     \\
     & =
     \underoverset{\infty}{k = 0}{\sum}
     \left(
        1  - \lambda_{-t}(E)
     \right)^k
    \cdot
     \frac{d}{d t} \lambda_{-t}(E)
  \end{aligned}
  \,.
$$

The **$k$th Adams operation** is the [[cohomology operation]] on [[topological K-theory]]

$$
  \psi^k \;\colon\; K(-) \longrightarrow K(-)
$$

which is the coefficient of $t^k$ in $\psi_t$.

=--

+-- {: .num_prop #TheBasicProperties}
###### Proposition
**(basic properties and characterization of Adams operations)**

The Adams operations

$$
  \psi^k \;\colon\; K(X) \longrightarrow K(X)
$$

have the following properties, for all elements $x,y \in K(X)$ and $k, l \in \mathbb{N}$ and $p \; \text{prime}$:

1. $\psi^k(x + y) = \psi^k(x) + \psi^k(y)$

   ($\psi^k$ is a [[natural transformation|natural]] [[abelian group]] [[homomorphism]])

1. $x \,\text{a line} \;\;\Rightarrow\;\; \psi^k(x) = x^k$

   (applied to a class $x \coloneqq [L]$ represented by a [[line bundle]] $L$, $\psi^k$ is the $k$th [[tensor power]])

1. $\psi^k(x \cdot y) = \psi^k(x) \cdot \psi^k(y)$

   ($\psi^k$ is in fact a [[natural transformation|natural]] [[ring]] [[homomorphism]])

1. $\psi^k(\psi^l(x)) = \psi^{k l}(x)$

1. $\psi^p(x) = x^p \, \text{mod}\, p$

1. if $x \in \tilde K(S^{2n})$ ([[reduced cohomology]]) then

   $x \in \tilde K(S^{2n}) \hookrightarrow K(S^{2n}) \;\;\Rightarrow\;\;\psi^k(x) = k^n \cdot x$.

Moreover, the first two of these already uniquely characterize the Adams operations.

=--

e.g. [Wirthmuller 12, section 11](#Wirthmuller12)

## Properties

### Basic properties

+-- {: .num_prop #AdamsOperationOnComplexTopologicalKtheoryOfnSpheres}
###### Proposition
**(Adams operations on [[complex topological K-theory]] of [[n-spheres]])**

For $n \in \mathbb{N}$, the [[Adams operations]] on the [[reduced K-theory]] of the [[n-sphere|2n-sphere]] are given by:

$$
  \array{
    \widetilde K
    \big(
      S^{2n}
    \big)
    &
    \overset{ \;\;\; \psi^k\;\;\; }{\longrightarrow}
    &
    \widetilde K
    \big(
      S^{2n}
    \big)
    \\
    V &\mapsto& k^n \cdot V
  }
$$

=--

(e.g. [Wirthmüller, Prop. on p. 45 (47 of 67)](#Wirthmuller12))

### Compatibility with complexification
 {#CompatibilityWithComplexification}

The Adams operations are compatible with the [[complexification]] map $(-) \otimes_{\mathbb{R}} \mathbb{C}$ from [[real vector bundles]] to [[complex vector bundles]], hence from [[KO-cohomology]] to [[KU-cohomology]], in that the following [[commutative diagram|diagram commutes]], for all $k$:

$$
  \array{
    KO(X)
      &\overset{ \psi^k }{\longrightarrow}&
    KO(X)
    \\
    {}^{ \mathllap{ (-) \otimes_{\mathbb{R}} \mathbb{C} } }
    \big\downarrow
      &&
    \big\downarrow
      {}^{ \mathrlap{ (-) \otimes_{\mathbb{R}} \mathbb{C} } }
    \\
    KO(X)
      &\overset{ \psi^k }{\longrightarrow}&
    KO(X)
  }
$$

([Adams 62, Thm. 5.1. (iv)](#Adams62), [Karoubi 78, Prop. IV.7.25](#Karoubi78))

### Compatibility with the Chern character
 {#CompatibilityWithTheChernCharacter}

The Adams operation are compatible with the [[Chern character map]] in the following way:

+-- {: .num_defn #AdamsLikeOperationsOnRationalCohomology}
###### Definition
**(Adams-like operations on [[rational cohomology]])**

For $X$ a [[topological space]], with [[rational cohomology]] in even degrees denoted

$$
  H^{ev}(X;\, \mathbb{Q})
  \;\coloneqq\;
  \underset{r \in \mathbb{N}}{\prod}
  H^{2 r}(X;\, \mathbb{Q})
$$

define graded [[linear maps]]

$$
  \psi^k_{H} \;\colon\; H^{ev}(X) \longrightarrow H^{ev}(X)
$$

for $k \in \mathbb{N}$ by taking their restriction to degree $2r$ to act by multiplication with $k^r$

$$
  \array{
    H^{2r}(X;\mathbb{Q})
    &\overset{\;\;\;\psi^k_H\;\;\;}{\longrightarrow}&
    H^{2r}(X;\mathbb{Q})
    \\
    \alpha_{2k} &\mapsto& k^{r} \cdot \alpha_{2r}
    \,.
  }
$$

=--

+-- {: .num_prop #AdamsOperationsAreCompatibleWithTheChernCharacter }
###### Proposition
**([[Adams operations compatible with the Chern character]])**

For $X$ a [[topological space]] with a finite [[CW-complex]]-[[mathematical structure]], the [[Chern character]] $ch$ on the [[complex topological K-theory]] of $X$ intertwines the [[Adams operations]] $\psi^n$ on K-theory with the Adams-like operations $\psi^n_H$ on [[rational cohomology]] from Def. \ref{AdamsLikeOperationsOnRationalCohomology}, for $k \geq 1$, in that the following [[commuting diagram|diagram commutes]]:

$$
  \array{
    K(X)
      &\overset{\;\;\;ch\;\;\;}{\longrightarrow}&
    H^{ev}(X;\,\mathbb{Q})
    \\
    {}^{ \mathllap{ \psi^k } }
    \big\downarrow
    &&
    \big\downarrow
    {}^{ \mathrlap{ \psi^k_H } }
    \\
    K(X)
      &\overset{\;\;\;ch\;\;\;}{\longrightarrow}&
    H^{ev}(X;\,\mathbb{Q})
    \,,
  }
$$

=--

([Adams 62, Thm. 5.1. (vi)](#Adams62), review in [Karoubi 78, Chapter V, Theorem 3.27](#Karoubi78), [Maakestad 06, Thm. 4.9](#Maakestad06))

+-- {: .proof}
###### Proof idea

Use the exponentional-formula for the [[Chern character]] with the  [[splitting principle]].

=--

\linebreak

### Adams conjecture

The [[Adams conjecture]] (a [[theorem]]) says that for all $k \in \mathbb{N}$ and $V \in K(X)$ there is $n \in \mathbb{N}$ such that the [[spherical fibration]] assigned to the [[K-theory]] class $k^n (\psi^k(V)-V)$ under the [[J-homomorphism]] is trivial, hence that

$$
  J
  \left(
     k^n \left(
       \psi^k(V) - V
     \right)
  \right)
  = 0
  \,.
$$

## Related concepts

* [[Lambda-ring]]

* [[Steenrod square]]

* [[Steenrod operation]]

* [[power operation]]

* [[Adams operation on Jacobi diagrams]]

## References

### General

The original article:

* {#Adams62} [[John Adams]], Section 5 of: _Vector fields on spheres_, Bull. Amer. Math. Soc. Volume 68, Number 1 (1962), 39-41 ([euclid:bams/1183524456](https://projecteuclid.org/euclid.bams/1183524456), [pdf](https://www.math.ens.fr/~benoist/refs/Adams.pdf))

See also:

* [[Victor M. Buchstaber]], [[Sergei P. Novikov]], _Formal groups, power systems and Adams operators_, Math. USSR-Sb.13 (1971) 80-116 [[doi:10.1070/SM1971v013n01ABEH001030](https://doi.org/10.1070/SM1971v013n01ABEH001030)]

Review:

* {#Karoubi78} [[Max Karoubi]], Section IV.7 in in: _K-Theory -- An introduction_, Grundlehren der mathematischen Wissenschaften 226, Springer 1978 ([pdf](https://webusers.imj-prg.fr/~max.karoubi/K.book/MK.book.pdf), [doi:10.1007/978-3-540-79890-3](https://link.springer.com/book/10.1007%2F978-3-540-79890-3))

* {#AguilarGitlerPrieto} Marcelo Aguilar, [[Samuel Gitler]], Carlos Prieto, Section 10 of: _Algebraic topology from a homotopical viewpoint_, Springer (2002)

* {#Maakestad06} Helge Maakestad, _Notes on the Chern-character_, Journal of Generalized Lie Theory and Applications, 2017, 11:1 ([arXiv:math/0612060](https://arxiv.org/abs/math/0612060), [doi:10.4172/1736-4337.100025](https://www.hilarispublisher.com/open-access/notes-on-the-cherncharacter-.pdf))

* [[Michael Hopkins]] (notes by [[Akhil Mathew]]), Lecture 10 in: _Spectra and stable homotopy theory_, 2012 ([pdf](http://math.uchicago.edu/~amathew/256y.pdf), [[HopkinsMathewStableHomotopyTheory.pdf:file]])

* {#Wirthmuller12} [[Klaus Wirthmüller]], Section 11 of: _Vector Bundles and K-theory_, 2012 ([[wirthmueller-vector-bundles-and-k-theory.pdf:file]])

* {#Hatcher} [[Allen Hatcher]], section 2.3 of _Vector bundles and K-theory_ ([web](https://www.math.cornell.edu/~hatcher/VBKT/VBpage.html))

* Wikipedia, _[Adams operation](http://en.wikipedia.org/wiki/Adams_operation)_

* [[Jacob Lurie]], remark 2 in: _[[Chromatic Homotopy Theory]]_, Lecture series 2010, Lecture 35 _The image of $J$_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture35.pdf))

### In representation theory
 {#ReferencesInRepresentationTheory}

Adams operations on the [[representation ring]] (the [[equivariant K-theory]] of the point) are discussed in

* {#tomDieck79} [[Tammo tom Dieck]], section 3.5 of _[[Transformation Groups and Representation Theory]]_, Lecture Notes in Mathematics 766 Springer 1979

* Robert Boltje, _A characterization of Adams operations on representation rings_, 2001 ([pdf](https://boltje.math.ucsc.edu/publications/p01v.pdf))

* [[Dai Tamaki]], [[Akira Kono]], Section 4.3 in: _Generalized Cohomology_, Translations of Mathematical Monographs, American Mathematical Society, 2006 ([ISBN: 978-0-8218-3514-2](https://bookstore.ams.org/mmono-230))

* {#tomDieck09} [[Tammo tom Dieck]], section 6.2 of  _Representation theory_, 2009 ([pdf](http://www.uni-math.gwdg.de/tammo/rep.pdf))

* [[Michael Boardman]], _Adams operations on Group representations_, 2007 ([pdf](http://www.math.jhu.edu/~jmb/note/adamrept.pdf))

* Ehud Meir, Markus Szymik, _Adams operations and symmetries of representation categories_ ([arXiv:1704.03389](https://arxiv.org/abs/1704.03389))

### In knot theory

Adams operations on [[Jacobi diagrams]] [[quotient vector space|modulo]] [[STU relations]] ([[Adams operation on Jacobi diagrams]]) are discussed in

* {#BarNatan95} [[Dror Bar-Natan]], Theorem 7, Def. 3.11 and Section 6.3.4 of:  _On the Vassiliev knot invariants_, Topology Volume 34, Issue 2, April 1995, Pages 423-472 (doi:10.1016/0040-9383(95)93237-2, [pdf](https://www.math.toronto.edu/drorbn/papers/OnVassiliev/OnVassiliev.pdf))

[[!redirects Adams operations]]


Adams operation on Jacobi diagrams

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[[!include knot theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

On the [[vector space]] $\mathcal{A}$ of [[Jacobi diagrams]] [[quotient vector space|modulo]] [[STU-relations]] ([[chord diagrams modulo 4T are Jacobi diagrams modulo STU|equivalently]] [[chord diagrams]] [[quotient vector space|modulo]] [[4T relations]])  there is a system of [[linear maps]] (for $q \in \mathbb{Z}$, $q \neq 0$)

$$
  \psi^q
  \;\colon\;
  \mathcal{A}
  \longrightarrow
  \mathcal{A}
$$

which respect the [[coalgebra]] [[mathematical structure|structure]] and satisfy

$$
  \psi^{q_2} \circ \psi^{q_1}
  \;=\;
  \psi^{q_1 \cdot q_2}
$$

and as such are (dually) analogous to the [[Adams operations]] on [[topological K-theory]].

([Bar-Natan 95, Def. 3.11 & Theorem 7](#BarNatan95))

In fact, when evaluated in [[Lie algebra weight systems]] $w_{\mathbf{N}}$ and under the identification (see [here](equivariant+K-theory#RelationToRepresentationTheory)) of the [[representation ring]] of a [[compact Lie group]] $G$ with the $G$-[[equivariant K-theory]] of the point, these Adams operations on [[Jacobi diagrams]] correspond to the [[Adams operations]] on [[equivariant K-theory]]:

$$
  w_{\mathbf{N}}(\psi^q D)
  \;=\;
  w_{\psi^q \mathbf{N}}(D)
  \,.
$$

([Bar-Natan 95, Exc. 6.24](#BarNatan95)
)

## References

* {#BarNatan95} [[Dror Bar-Natan]], _On the Vassiliev knot invariants_, Topology Volume 34, Issue 2, April 1995, Pages 423-472 (doi:10.1016/0040-9383(95)93237-2, [pdf](https://www.math.toronto.edu/drorbn/papers/OnVassiliev/OnVassiliev.pdf))

[[!redirects Adams operations on Jacobi diagrams]]


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#### Algebraic topology
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[[!include algebraic topology - contents]]
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#### Cohomology
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[[!include cohomology - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The [[Adams operations]] $\psi^k$ on [[complex topological K-theory]] are compatible with the [[Chern character map]] to [[rational cohomology]] in that the effect of $\psi^k$ on the Chern character image in degree $2r$ is multiplication by $k^r$.

## Statement

+-- {: .num_defn #AdamsLikeOperationsOnRationalCohomology}
###### Definition
**(Adams-like operations on [[rational cohomology]])**

For $X$ a [[topological space]], with [[rational cohomology]] in even degrees denoted

$$
  H^{ev}(X;\, \mathbb{Q})
  \;\colon\;
  \underset{r \in \mathbb{N}}{\prod}
  H^{2 r}(X;\, \mathbb{Q})
$$

define graded [[linear maps]]

$$
  \psi^k_{H} \;\colon\; H^{ev}(X) \longrightarrow H^{ev}(X)
$$

for $k \in \mathbb{N}$ by taking their restriction to degree $2r$ to act by multiplication with $k^r$:

\[
  \label{AdamsOperationOnOrdinaryCohomologyInDegree2r}
  \array{
    H^{2r}(X;\mathbb{Q})
    &\overset{\;\;\;\psi^k_H\;\;\;}{\longrightarrow}&
    H^{2r}(X;\mathbb{Q})
    \\
    \alpha_{2k} &\mapsto& k^{r} \cdot \alpha_{2k}
    \,.
  }
\]

=--

+-- {: .num_prop #AdamsOperationsAreCompatibleWithTheChernCharacter }
###### Proposition
**([[Adams operations compatible with the Chern character]])**

For $X$ a [[topological space]] with a finite [[CW-complex]]-[[mathematical structure]], the [[Chern character]] $ch$ on the [[complex topological K-theory]] of $X$ intertwines the [[Adams operations]] $\psi^n$ on K-theory with the Adams-like operations $\psi^n_H$ on [[rational cohomology]] from Def. \ref{AdamsLikeOperationsOnRationalCohomology}, for $k \geq 1$, in that the following [[commuting diagram|diagram commutes]]:

$$
  \array{
    K(X)
      &\overset{\;\;\;ch\;\;\;}{\longrightarrow}&
    H^{ev}(X;\,\mathbb{Q})
    \\
    {}^{ \mathllap{ \psi^k } }
    \big\downarrow
    &&
    \big\downarrow
    {}^{ \mathrlap{ \psi^k_H } }
    \\
    K(X)
      &\underset{\;\;\;ch\;\;\;}{\longrightarrow}&
    H^{ev}(X;\,\mathbb{Q})
    \,,
  }
$$

=--

([Adams 62, Thm. 5.1. (vi)](#Adams62), review in [Karoubi 78, Chapter V, Theorem 3.27](#Karoubi78), [Maakestad 06, Thm. 4.9](#Maakestad06))

+-- {: .proof}
###### Proof idea

Use the exponentional-formula for the [[Chern character]] with the  [[splitting principle]].

=--

## Related entries

* [[e-invariant]]

## References

The original statement:

* {#Adams62} [[John Adams]], Theorem 5.1 (vi) _Vector fields on spheres_, Bull. Amer. Math. Soc. Volume 68, Number 1 (1962), 39-41 ([euclid:bams/1183524456](https://projecteuclid.org/euclid.bams/1183524456), [pdf](https://www.math.ens.fr/~benoist/refs/Adams.pdf))

Textbook accounts:

* {#Karoubi78} [[Max Karoubi]], Theorem V3.27 in: _K-Theory -- An introduction_, Grundlehren der mathematischen Wissenschaften 226, Springer 1978 ([pdf](https://webusers.imj-prg.fr/~max.karoubi/K.book/MK.book.pdf), [doi:10.1007/978-3-540-79890-3](https://link.springer.com/book/10.1007%2F978-3-540-79890-3))

Review and exposition:

* {#Maakestad06} Helge Maakestad, _Notes on the Chern-character_, Journal of Generalized Lie Theory and Applications, 2017, 11:1 ([arXiv:math/0612060](https://arxiv.org/abs/math/0612060), [doi:10.4172/1736-4337.100025](https://www.hilarispublisher.com/open-access/notes-on-the-cherncharacter-.pdf))

[[!redirects Adams operations are compatible with the Chern character]]
[[!redirects the Adams operations are compatible with the Chern character map]]


Adams resolution

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###Context###
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

* [Classical definition](#ClassicalDefinition)

* [Via injective resolutions](#ViaInjectiveResolutions)

### Classical definition
 {#ClassicalDefinition}

For $X$ a [[spectrum]] and $E^\bullet$ a [[generalized cohomology theory]] [[Brown representability theorem|represented]] by a [[spectrum]] $E$, then an _$E$-Adams resolution_ of $X$ is a [[diagram]] of the form

$$
  \array{
    \vdots
    \\
    \downarrow
    \\
    F_2 &\stackrel{f_2}{\longrightarrow}& K_2
    \\
    \downarrow
    \\
    F_1 &\stackrel{f_1}{\longrightarrow}& K_1
    \\
    \downarrow
    \\
     X &\stackrel{f_0}{\longrightarrow}& K_0
  }
$$

where

* each $K_i$ is a wedge of [[suspensions]] of $E$;

* each $F_{n+1} \to F_n \to K_n$ is a [[homotopy fiber sequence]];

* each $f_n$ is a surjection on [[cohomology]].

The original and default case is that where $E = H \mathbb{F}_p$ is an [[Eilenberg-MacLane spectrum]] with mod $p$ [[coefficients]], in which case $E^\bullet$ is [[ordinary cohomology]] with these coefficients.
In this case the $K_i$ are [[generalized Eilenberg-MacLane spectra]].

The [[long exact sequences of homotopy groups]] for all the [[homotopy fibers]] in this diagram arrange into a diagram of the form

$$
  \array{
    \vdots
    \\
    \downarrow & \nwarrow
    \\
    \pi_\bullet(F_2) &\stackrel{\pi_\bullet(f_2)}{\longrightarrow}& \pi_\bullet(K_2)
    \\
    \downarrow & \nwarrow^{\mathrlap{\pi_\bullet(\partial_2)}}
    \\
    \pi_\bullet(F_1) &\stackrel{\pi_\bullet(f_1)}{\longrightarrow}& \pi_\bullet(K_1)
    \\
    \downarrow & \nwarrow^{\mathrlap{\pi_\bullet(\partial_1)}}
    \\
    \pi_\bullet(X) &\stackrel{\pi_\bullet(f_0)}{\longrightarrow}& \pi_\bullet(K_0)
  }
  \,,
$$

where the diagonal maps are the images of the [[connecting homomorphisms]] and hence decrease degree in $\pi_\bullet$ by one. This is an (unrolled) [[exact couple]]. The corresponding [[spectral sequence]] is the [[Adams spectral sequence]] induced by the given Adams resolution.

In the case of $E = H \mathbb{F}_p$, applying [[cohomology]] $H^\bullet(-, \mathbb{F}_p)$ to the original diagram
yields  a [[free resolution]] of the [[cohomology ring]] $H^\bullet(X,\mathbb{Z}_p)$ by a [[chain complex]] of [[free modules]] over the [[Steenrod algebra]] $A_p$.

$$
  \array{
    H^\bullet(K_0) &\leftarrow& H^\bullet(\Sigma K_1) &\leftarrow& H^\bullet(\Sigma^2 K_2)  &\leftarrow& \cdots
    \\
    \downarrow && \downarrow && \downarrow
    \\
    H^\bullet(X)
    &\leftarrow& 0
    &\leftarrow& 0
   &\leftarrow&  \cdots
  }
$$

The computation of the [[cohomology]] of $X$ by means of this resolution is given by the [[Adams spectral sequence]].

### Via injective resolutions
 {#ViaInjectiveResolutions}

A streamlined discussion of $E$-Adams resolutions in close analogy to [[injective resolutions]] in [[homological algebra]] was given in ([Miller 81](#Miller81)), advertized in ([Hopkins 99](#Hopkins99)) and worked out in more detail in ([Aramian](#Aramian)).

Write $HoSpectra$ for the [[stable homotopy category]] and write

$$
  [-,-] \;\colon\; HoSpectra^{op} \times HoSpectra \longrightarrow Ab
$$

for the [[hom-functor]] with values in [[abelian groups]].

+-- {: .num_defn #HomotopyFunctor}
###### Definition

For $S \in HoSpectra$, the _homotopy functor it represents_ is the [[representable functor]]

$$
  [S,-] \;\colon\; HoSpectra \longrightarrow Ab
$$

(as opposed to the other, contravariant, functor).

=--

+-- {: .num_example}
###### Example

For $S = \Sigma^\infty S^n \simeq \Sigma^n \mathbb{S}$ we have

$$
  [\Sigma^\infty S^n ,- ]\simeq \pi_n
$$

is the $n$th [[stable homotopy group]]-[[functor]].

=--

Throughout, let $E$ be a [[ring spectrum]].

#### $E$-Injective spectra

First we consider a concept of $E$-[[injective objects]] in [[Spectra]].

+-- {: .num_defn #ExactSequences}
###### Definition

Say that

1. a sequence of spectra

   $$
     A_1 \longrightarrow A_2 \longrightarrow \cdots \longrightarrow A_n
   $$

   is

   1. a (long) _exact sequence_ if the induced sequence of homotopy functors, def. \ref{HomotopyFunctor}, is a [[long exact sequence]] in $[HoSpectra,Ab]$;

   2. (for $n = 2$) a _short exact sequence_ if

      $$
        0 \longrightarrow A_1 \longrightarrow A_2 \longrightarrow A_3 \longrightarrow 0
      $$

      is (long) exact;

1. a morphism $A \longrightarrow B$  is

   1. a _monomorphism_ if $0 \longrightarrow A \longrightarrow B$ is an exact sequence;

   1. an _epimorphism_ if $A \longrightarrow B \longrightarrow 0$ is an exact sequence.

For $E$ a [[ring spectrum]], then a sequence of spectra is (long/short) _$E$-exact_ and a morphism is epi/mono, respectively, if becomes long/short exact or epi/mono, respectively, after taking [[smash product of spectra|smash product]] with $E$.

=--

+-- {: .num_example}
###### Example

Every [[homotopy cofiber sequence]] of spectra is exact in the sense of def. \ref{ExactSequences}.

=--

+-- {: .num_remark}
###### Remark/Warning

Consecutive morphisms in an $E$-exact sequence according to def. \ref{ExactSequences} in general need not compose up to homotopy, to the [[zero morphism]]. But this does become true for sequences of $E$-injective objects, defined below in def. \ref{EInjective}.

=--

+-- {: .num_lemma}
###### Lemma

1. If $f \colon B\longrightarrow A$ is a monomorphism in the sense of def. \ref{ExactSequences}, then there exists a morphism $g \colon C \longrightarrow A$ such that the [[wedge sum]] morphism is a [[weak homotopy equivalence]]

   $$
     f \vee g \;\colon\; B \wedge C \stackrel{\simeq}{\longrightarrow} A
     \,.
   $$

1. If $f \colon A \longrightarrow B$ is an epimorpimsm in the sense of def. \ref{ExactSequences}, then there exists a homotopy [[section]] $s \colon B\to A$, i.e. $f\circ s\simeq Id$, together with a morphism $g \colon C \to A$ such that the [[wedge sum]] morphism is a [[weak homotopy equivalence]]

   $$
     s \vee f \colon B\vee C \stackrel{\simeq}{\longrightarrow} A
     \,.
   $$

=--

+-- {: .num_defn #EInjective}
###### Definition

For $E$ a [[ring spectrum]], say that a spectrum $S$ is _$E$-injective_ if for each morphism $A \longrightarrow S$ and  each $E$-monomorphism $f \colon A \longrightarrow B$ in the sense of def. \ref{ExactSequences}, there is a [[diagram]] in [[HoSpectra]] of the form

$$
  \array{
    A &\longrightarrow & S
    \\
    \downarrow & \nearrow_{\mathrlap{\exists}}
    \\
    B
  }
  \,.
$$

=--

+-- {: .num_lemma}
###### Lemma

If $S$ is $E$-injective in the sense of def. \ref{EInjective}, then there exists a spectrum $X$ such that $S$ is a [[retract]] in [[HoSpectra]] of $E \wedge X$.

=--

#### $E$-Adams resolutions

+-- {: .num_defn #EAdamsResolution}
###### Definition

For $E$ a [[ring spectrum]], then an _$E$-Adams resolution_ of an spectrum $S$ is a long exact sequence, in the sense of def. \ref{ExactSequences}, of the form

$$
  0 \longrightarrow S \longrightarrow I_0 \longrightarrow I_1 \longrightarrow I_2 \longrightarrow \cdots
$$

such that each $I_j$ is $E$-injective, def. \ref{EInjective}.

=--

+-- {: .num_lemma}
###### Lemma

Any two consecutive maps in an $E$-Adams resolution compose to the [[zero morphism]].

=--

+-- {: .num_lemma}
###### Lemma

For $X \to X_\bullet$ an $E$-Adams resolution, def. \ref{EAdamsResolution}, and for $X \longrightarrow Y$ any morphism, then there exists an $E$-Adams resolution $Y \to J_\bullet$ and a [[commuting diagram]]

$$
  \array{
     X &\longrightarrow& I_\bullet
     \\
     \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{g_\bullet}}
     \\
     Y &\longrightarrow& J_\bullet
  }
  \,.
$$

=--

+-- {: .num_example}
###### Example
**(standard resolution)**

Consider the augmented [[cosimplicial object|cosimplicial]] which is the $\mathbb{S} \to E$-[[Amitsur complex]] [[smash product of spectra|smashed]] with $X$:

$$
  X \longrightarrow E \wedge X \stackrel{\longrightarrow}{\longrightarrow} E \wedge E \wedge X \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}
  E \wedge E \wedge E \wedge X
 \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}}
 \cdots
  \,.
$$

Its corresponding [[Moore complex]] (the sequence whose maps are the alternating sum of the above coface maps) is an $E$-Adams resolution, def. \ref{EAdamsResolution}.

=--

#### $E$-Adams towers

+-- {: .num_defn #EAdamsTower}
###### Definition

An _$E$-Adams tower_ of a spectrum $X$ is a [[commuting diagram]] in [[HoSpectra]] of the form

$$
  \array{
    && \vdots
    \\
    && \downarrow^{\mathrlap{p_2}}
    \\
    && X_2 &\stackrel{\kappa_2}{\longrightarrow}& \Omega^2 I_3
    \\
    &\nearrow& \downarrow^{\mathrlap{p_1}}
    \\
    && X_1 &\stackrel{\kappa_1}{\longrightarrow}& \Omega I_2
    \\
    &\nearrow& \downarrow^{\mathrlap{p_0}}
    \\
    X
    &\underset{}{\longrightarrow}&
    X_0 = I_0
    &\stackrel{\kappa_0}{\longrightarrow}&
    I_1
  }
$$

such that

1. each hook is a [[homotopy fiber sequence]] (hence it is a [[tower of homotopy fibers]]);

1. the [[composition]] of the $(\Sigma \dashv \Omega)$-[[adjuncts]] of $\Sigma_{p_{n-1}}$ with $\Sigma^n \kappa_n$

   $$
     i_{n+1} \;\colon\; I_n \stackrel{\widetilde {\Sigma p_{n-1}}}{\longrightarrow}
     \Sigma^n X_n \stackrel{\Sigma^{n}\kappa_n}{\longrightarrow} I_{n+1}
   $$

   constitute an $E$-Adams resolution of $X$, def. \ref{EAdamsResolution}:

   $$
     0 \to X \stackrel{i_0}{\to} I_0 \stackrel{i_2}{\to} I_2 \stackrel{}{\to} \cdots
     \,.
   $$

Call this the _associated $E$-Adams resolution_ of the $E$-Adams tower.

The _associated inverse sequence_ is

$$
  X = X_0 \stackrel{\gamma_0}{\longleftarrow} \Omega C_1 \stackrel{\gamma_1}{\longleftarrow} C_2 \longleftarrow \cdots
$$

where $C_{k+1} \coloneqq hocofib(i_k)$.

=--

(In ([Ravenel](#Ravenel)) it is is the associated inverse sequence that is called the associated resolution.)

=--
+-- {: .num_example}
###### Example

Every $E$-Adams resolution of $X$, def. \ref{EAdamsResolution}, induces an $E$-Adams tower, def. \ref{EAdamsTower} of which it is the associated $E$-Adams resolution.

=--

## Related concepts

* [[Whitehead tower]], [[Postnikov tower]]

* [[E-nilpotent completion]]

* [[Adams spectral sequence]]

## References

Reviews include

* {#Ravenel} [[Doug Ravenel]], around Chapter 2, def. 2.1.3 of _[[Complex cobordism and stable homotopy groups of spheres]]_

* {#Kochmann96} [[Stanley Kochmann]], section 3.6 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

A streamlined presentation close in spirit to constructions in [[homological algebra]] was given in

* {#Miller81} [[Haynes Miller]], _On relations between Adams spectral sequences, with an application to the stable homotopy of a Moore space_, J. Pure Appl. Algebra 20 (1981) ([pdf](http://math.mit.edu/~hrm/papers/miller-relations-between-adams-spectral-sequences.pdf))

and is reproduced and expanded on in

* {#Hopkins99} [[Mike Hopkins]], section 5 of _Complex oriented cohomology theories and the language of stacks_, course notes 1999 ([pdf](http://www.math.rochester.edu/u/faculty/doug/otherpapers/coctalos.pdf))

* {#Aramian} [[Nersés Aramian]], _The Adams spectral sequence_ ([[AramianANSS.pdf:file]])

[[!redirects Adams resolutions]]

[[!redirects Adams tower]]
[[!redirects Adams towers]]