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\begin{document}

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\section*{J-homomorphism and chromatic homotopy}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{stable_homotopy_theory}{}\paragraph*{{Stable Homotopy theory}}\label{stable_homotopy_theory}

[[!include stable homotopy theory - contents]]

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

\begin{quote}%
This entry explains the [[J-homomorphism]], states how its image is the first ([[chromatic layer|chromatic]]) layer of the [[sphere spectrum]]; and then motivated by this explains some basic notions of [[chromatic homotopy theory]], notably the origin of the general $E$-[[Adams spectral sequence]].


\end{quote}
\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{i_the_jhomomorphism}{\textbf{I)} The J-homomorphism}\dotfill \pageref*{i_the_jhomomorphism} \linebreak
\noindent\hyperlink{OnGroups}{On groups}\dotfill \pageref*{OnGroups} \linebreak
\noindent\hyperlink{on_classifying_spaces}{On classifying spaces}\dotfill \pageref*{on_classifying_spaces} \linebreak
\noindent\hyperlink{ii_the_image_of_the_jhomomorphism}{\textbf{II)} The image of the J-homomorphism}\dotfill \pageref*{ii_the_image_of_the_jhomomorphism} \linebreak
\noindent\hyperlink{explicit_description}{Explicit description}\dotfill \pageref*{explicit_description} \linebreak
\noindent\hyperlink{chromatic_formulation}{Chromatic formulation}\dotfill \pageref*{chromatic_formulation} \linebreak
\noindent\hyperlink{ELocalStableHomotopyTheory}{\textbf{III)} $E$-Local stable homotopy theory}\dotfill \pageref*{ELocalStableHomotopyTheory} \linebreak
\noindent\hyperlink{bousfield_localization_of_spectra}{Bousfield localization of spectra}\dotfill \pageref*{bousfield_localization_of_spectra} \linebreak
\noindent\hyperlink{chromatic_layers}{Chromatic layers}\dotfill \pageref*{chromatic_layers} \linebreak
\noindent\hyperlink{iv_adams_spectral_sequence_for_local_homotopy_groups}{\textbf{IV)} Adams spectral sequence for $E$-local homotopy groups}\dotfill \pageref*{iv_adams_spectral_sequence_for_local_homotopy_groups} \linebreak
\noindent\hyperlink{SpectralSequencesForHomotopyGroupsOfFilteredObjects}{Spectral sequences for homotopy groups of filtered spectra}\dotfill \pageref*{SpectralSequencesForHomotopyGroupsOfFilteredObjects} \linebreak
\noindent\hyperlink{CanonicalCosimplicialResolutionOfEInfinityAlgebras}{Canonical cosimplicial resolution of $E_\infty$-algebras}\dotfill \pageref*{CanonicalCosimplicialResolutionOfEInfinityAlgebras} \linebreak
\noindent\hyperlink{TheEAdamsSpectralSequence}{The $E$-Adams-Novikov spectral sequence}\dotfill \pageref*{TheEAdamsSpectralSequence} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{i_the_jhomomorphism}{}\subsection*{{\textbf{I)} The J-homomorphism}}\label{i_the_jhomomorphism}

The \emph{[[J-homomorphism]]} is a map from the [[homotopy groups]] of the [[stable orthogonal group]] (which are completely understood) to the [[stable homotopy groups of spheres]] (which in their totality are hard to compute).

\hypertarget{OnGroups}{}\subsubsection*{{On groups}}\label{OnGroups}

\begin{defn}
\label{SphereAsCompactification}\hypertarget{SphereAsCompactification}{}
For $n \in \mathbb{N}$ regard the $n$-[[sphere]] (as a [[topological space]]) as the [[one-point compactification]] of the [[Cartesian space]] $\mathbb{R}^n$

\begin{displaymath}
S^n \simeq (\mathbb{R}^n)^\ast
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{ActionOfOrthogonalOnSphere}\hypertarget{ActionOfOrthogonalOnSphere}{}
Since the process of [[one-point compactification]] is a [[functor]] on [[proper maps]], hence on [[homeomorphisms]], via def. \ref{SphereAsCompactification} the $n$-sphere inherits from the canonical [[action]] of the [[orthogonal group]] $O(n)$ on $\mathbb{R}^n$ an [[action]]

\begin{displaymath}
O(n) \times S^n \longrightarrow S^n
\end{displaymath}
(by [[continuous maps]]) which preserves the base point (the ``point at infinity'').

\end{remark}
For definiteness we distinguish in the following notationally between

\begin{enumerate}%
\item the $n$-[[sphere]] $S^n \in Top$ regarded as a [[topological space]];


\item its [[homotopy type]] $\Pi(S^n) \in L_{whe} Top \simeq$ [[∞Grpd]] given by its [[fundamental ∞-groupoid]].



\end{enumerate}
Similarly we write $\Pi(O(n))$ for the [[homotopy type]] of the [[orthogonal group]], regarded as a [[group object in an (∞,1)-category]] in [[∞Grpd]] (using that the [[shape modality]] $\Pi$ preserves [[finite products]]).

\begin{defn}
\label{AutoequivalencesOfnSphere}\hypertarget{AutoequivalencesOfnSphere}{}
For $n \in \mathbb{N}$ write $H(n)$ for the [[automorphism ∞-group]] of homotopy self-equivalences $S^n \longrightarrow S^n$, hence

\begin{displaymath}
H(n) \coloneqq Aut_{\infty Grpd^{\ast/}}(\Pi(S^n))
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
The [[∞-group]] $H(n)$, def. \ref{AutoequivalencesOfnSphere}, constitutes the two [[connected components]] of the $n$-fold based [[loop space]] $\Omega^n S^n$ corresponding to the [[homotopy groups]] $\pm 1 \in \pi_n(S^n)$.

\end{remark}
\begin{defn}
\label{OrthogonalActionOnSphereOnHomotopyGroups}\hypertarget{OrthogonalActionOnSphereOnHomotopyGroups}{}
Via the presentation of [[∞Grpd]] by the [[cartesian closed model structure|cartesian closed]] [[model structure on topological spaces|model structure on compactly generated topological spaces]] (and using that $S^n$ and $O(n)$ and hence their product are [[compact topological spaces|compact]]) we have that for $n \in \mathbb{N}$ the [[continuous function|continuous]] [[action]] of $O(n)$ on $S^n$ of remark \ref{ActionOfOrthogonalOnSphere}, which by [[cartesian closed category|cartesian closure]] is equivalently a homomorphism of [[topological groups]] of the form

\begin{displaymath}
O(n) \longrightarrow Aut_{Top^{\ast/}}(S^n)
  \,,
\end{displaymath}
induces a homomorphism of [[∞-groups]] of the form

\begin{displaymath}
\Pi(O(n)) \longrightarrow Aut_{\infty Grpd^{\ast/}}(\Pi(S^n))
  \,.
\end{displaymath}
This in turn induces for each $i \in \mathbb{N}$ homomorphisms of [[homotopy groups]] of the form

\begin{displaymath}
\pi_i(O(n)) \longrightarrow \pi_i(\Omega^n S^n) \simeq \pi_{n+i}(S^n)
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{OrthogonalActionOnSphereOnHomotopyGroups}\hypertarget{OrthogonalActionOnSphereOnHomotopyGroups}{}
By construction, the homomorphisms of remark \ref{OrthogonalActionOnSphereOnHomotopyGroups} are compatible with [[suspension]] in that for all $n \in \mathbb{N}$ the [[diagrams]]

\begin{displaymath}
\itexarray{
    O(n) &\longrightarrow& Aut_{Top^{\ast/}}(S^n)
    \\
    \downarrow && \downarrow
    \\
    O(n+1) &\longrightarrow& Aut_{Top^{\ast/}}(S^{n+1})
  }
\end{displaymath}
in $Grp(Top)$ commute, and hence so do the diagrams

\begin{displaymath}
\itexarray{
    \Pi(O(n)) &\longrightarrow& Aut_{\infty Grpd^{\ast/}}(\Pi(S^n))
    \\
    \downarrow && \downarrow
    \\
    \Pi(O(n+1)) &\longrightarrow& Aut_{\infty Grpd^{\ast/}}(\Pi(S^{n+1}))
  }
\end{displaymath}
in $Grp(\infty Grpd)$, up to [[homotopy]].

\end{remark}
Therefore one can take the [[direct limit]] over $n$:

\begin{defn}
\label{JHom}\hypertarget{JHom}{}
By remark \ref{OrthogonalActionOnSphereOnHomotopyGroups} there is induced a homomorphism

\begin{displaymath}
J_i \;\colon\; \pi_\bullet(O) \longrightarrow \pi_\bullet(\mathbb{S})
\end{displaymath}
from the [[homotopy groups]] of the [[stable orthogonal group]] to the [[stable homotopy groups of spheres]]. This is called the \textbf{[[J-homomorphism]]}.

\end{defn}
\hypertarget{on_classifying_spaces}{}\subsubsection*{{On classifying spaces}}\label{on_classifying_spaces}

\begin{remark}
\label{DeloopedJ}\hypertarget{DeloopedJ}{}
Since the maps of def. \ref{OrthogonalActionOnSphereOnHomotopyGroups} are [[∞-group]] [[homomorphisms]], there exists their [[delooping]]

\begin{displaymath}
B J \;\colon\; B O \longrightarrow B GL_1(\mathbb{S}) = B H
  \,.
\end{displaymath}
\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
Here $GL_1(\mathbb{S})$ is the [[∞-group of units]] of the [[sphere spectrum]].

\end{remark}
This map $B J$ is the [[universal characteristic class]] of stable [[vector bundles]] with values in [[spherical fibrations]]:

\begin{defn}
\label{SphereBundleOfVectorBundle}\hypertarget{SphereBundleOfVectorBundle}{}
For $V \to X$ a [[vector bundle]], write $S^V$ for its [[fiber]]-wise [[one-point compactification]]. This is a [[sphere bundle]], a [[spherical fibration]]. Write $\mathbb{S}^V$ for the $X$-[[parameterized spectrum]] which is fiberwise the [[suspension spectrum]] of $S^V$.

\end{defn}
It is immediate that:

\begin{prop}
\label{}\hypertarget{}{}
For $V \to X$ a [[vector bundle]] classified by a map $X \to B O$, the corresponding [[spherical fibration]] $\mathbb{S}^V$, def. \ref{SphereBundleOfVectorBundle}, is classified by $X \to B O \stackrel{B J}{\longrightarrow} B GL_1(\mathbb{S})$, def. \ref{DeloopedJ}.

\end{prop}
\hypertarget{ii_the_image_of_the_jhomomorphism}{}\subsection*{{\textbf{II)} The image of the J-homomorphism}}\label{ii_the_image_of_the_jhomomorphism}

Since the [[J-homomorphism]] maps from something well-understood to something hard to understand, it is of interst to characterize its [[image]], ``the [[image of J]]''.

\hypertarget{explicit_description}{}\subsubsection*{{Explicit description}}\label{explicit_description}

The following characterization of the [[image]] of the J-homomorphism on [[homotopy groups]] derives from a statement that was first conjectured in (\href{Adams+spectrals+sequence#Adams66}{Adams 66}) -- and since called the \emph{[[Adams conjecture]]} -- and then proven in (\href{Adams+spectrals+sequence#Quillen71}{Quillen 71}, \href{Adams+spectrals+sequence#Sullivan74}{Sullivan 74}).

\begin{remark}
\label{}\hypertarget{}{}
By the discussion at \emph{\href{orthogonal%20group#HomotopyGroups}{orthogonal group -- homotopy groups}} we have that the [[homotopy groups]] of the [[stable orthogonal group]] are

\begin{tabular}{l|l|l|l|l|l|l|l|l}
$n\;mod\; 8$&0&1&2&3&4&5&6&7\\
\hline 
$\pi_n(O)$&$\mathbb{Z}_2$&$\mathbb{Z}_2$&0&$\mathbb{Z}$&0&0&0&$\mathbb{Z}$\\
\end{tabular}

Because all groups appearing here and in the following are [[cyclic groups]], we instead write down the [[order of a group|order]]

\begin{tabular}{l|l|l|l|l|l|l|l|l}
$n\;mod\; 8$&0&1&2&3&4&5&6&7\\
\hline 
${\vert\pi_n(O)\vert}$&2&2&1&$\infty$&1&1&1&$\infty$\\
\end{tabular}

\end{remark}
For the following statement it is convenient to restrict to J-homomorphism to the stable [[special orthogonal group]] $S O$, which removes the lowest degree homotopy group in the above

\begin{tabular}{l|l|l|l|l|l|l|l|l}
$n\;mod\; 8$&0&1&2&3&4&5&6&7\\
\hline 
$\pi_n(S O)$&0&$\mathbb{Z}_2$&0&$\mathbb{Z}$&0&0&0&$\mathbb{Z}$\\
\end{tabular}

\begin{tabular}{l|l|l|l|l|l|l|l|l}
$n\;mod\; 8$&0&1&2&3&4&5&6&7\\
\hline 
${\vert\pi_n(S O)\vert}$&1&2&1&$\infty$&1&1&1&$\infty$\\
\end{tabular}

\begin{theorem}
\label{AdamsQuillenTheorem}\hypertarget{AdamsQuillenTheorem}{}
The [[stable homotopy groups of spheres]] $\pi_n(\mathbb{S})$ are the [[direct sum]] of the ([[cyclic group|cyclic]]) [[image]] $im(J|_{SO})$ of the J-homomorphism, def. \ref{JHom}, applied to the [[special orthogonal group]] and the [[kernel]] of the [[Adams e-invariant]].

Moreover,

\begin{itemize}%
\item for $n = 0 \;mod \;$ and $n = 1 \;mod \; 8$ and $n$ positive the J-homomorphism $\pi_n(J) \colon \pi_n(S O) \to \pi_n(\mathbb{S})$ is [[injection|injective]], hence its image is $\mathbb{Z}_2$,


\item for $n = 3\; mod\; 8$ and $n = 7 \; mod \; 8$ hence for $n = 4 k -1$, the [[order of a group|order]] of the image is equal to the [[denominator]] of $B_{2k}/4k$ in its reduced form, where $B_{2k}$ is the [[Bernoulli number]]


\item for all other cases the image is necessarily zero.



\end{itemize}
\end{theorem}
The characterization of this image is due to (\href{Adams+spectrals+sequence#Adams66}{Adams 66}, \href{Adams+spectrals+sequence#Quillen71}{Quillen 71}, \href{Adams+spectrals+sequence#Sullivan74}{Sullivan 74}). Specifically the identification of $J(\pi_{4n-1}(S O))$ is (\href{Adams+spectrals+sequence#Adams65a}{Adams 65a, theorem 3.7} and the direct summand property is (\href{Adams+spectrals+sequence#Adams66}{Adams 66, theorems 1.1-1.6.}). That the image is a direct summand of the codomain is proven for instance in (\href{Adams+spectrals+sequence#Switzer75}{Switzer 75, end of chapter 19}).

A modern version of the proof, using methods from [[chromatic homotopy theory]], is surveyed in some detail in (\hyperlink{Lorman13}{Lorman 13}).

The statement of the theorem is recalled for instance as (\hyperlink{Ravenel}{Ravenel, chapter 1, theorem 1.1.13}). Another computation of the image of $J$ is in (\hyperlink{Ravenel}{Ravenel, chapter 5, section 3}).

\begin{remark}
\label{}\hypertarget{}{}
The order of $J(\pi_{4k-1} O)$ in theorem \ref{AdamsQuillenTheorem} is for low $k$ given by the following table

\begin{tabular}{l|l|l|l|l|l|l|l|l|l|l}
k&1&2&3&4&5&6&7&8&9&10\\
\hline 
$\vert J(\pi_{4k-1}(O))\vert$&24&240&504&480&264&65,520&24&16,320&28,728&13,200\\
\end{tabular}

\end{remark}
See for instance (\hyperlink{Ravenel}{Ravenel, Chapt. 1, p. 5}).

\begin{remark}
\label{}\hypertarget{}{}
Therefore we have in low degree the following situation

[[!include image of J -- table]]

\end{remark}
\begin{example}
\label{TableOfPPrincipalComponentsInHomotopyGroups}\hypertarget{TableOfPPrincipalComponentsInHomotopyGroups}{}
The following tables show the [[p-primary components]] of the [[stable homotopy groups of spheres]] for low values, the image of J appears as the bottom row.

Here the horizontal index is the degree $n$ of the stable homotopy group $\pi_n$. The appearance of a string of $k$ connected dots vertically above index $n$ means that there is a [[direct sum|direct summand]] [[primary group]] of [[order of a group|order]] $p^k$. See example \ref{InterpretTable} below for illustration. (These tables are taken from (\href{homotopy+groups+of+spheres#Hatcher}{Hatcher}), where in turn they were generated based on (\hyperlink{Ravenel}{Ravenel 86})).

\textbf{$p = 2$-primary component} (e.g. \hyperlink{Ravenel}{Ravenel 86, theorem 3.2.11, figure 4.4.46})



\textbf{$p = 3$-primary component}



\textbf{$p = 5$-primary component}



\end{example}
We illustrate how to read these tables:

\begin{example}
\label{InterpretTable}\hypertarget{InterpretTable}{}
The [[finite abelian group]] $\pi_3(\mathbb{S}) \simeq \mathbb{Z}_{24}$ decomposes into [[primary groups]] as $\simeq \mathbb{Z}_8 \oplus \mathbb{Z}_3$. Here $8 = 2^3$ corresponds to the three dots above $n = 3$ in the first table, and $3 = 3^1$ to the single dot over $n = 3$ in the second.

The [[finite abelian group]] $\pi_7(\mathbb{S}) \simeq \mathbb{Z}_{240}$ decomposes into [[primary groups]] as $\simeq \mathbb{Z}_{16} \oplus \mathbb{Z}_3 \oplus \mathbb{Z}_5$. Here $16 = 2^4$ corresponds to the four dots above $n = 7$ in the first table, and $3 = 3^1$ to the single dot over $n = 7$ in the second and $5 = 5^1$ to the single dot over $n = 7$ in the third table.

The [[finite abelian group]] $\pi_11(\mathbb{S}) \simeq \mathbb{Z}_{504}$ has [[primary group]]-decomposition $\cdots \simeq \mathbb{Z}_{2^3} \oplus \mathbb{Z}_{3^2} \oplus \mathbb{Z}_7$ and so this corresponds to the three connected dots over $n = 11$ in the first table and the two connected dots over $n = 11$ in the second (and there will be one dot over $n = 11$ in the fourth table for $p = 7$ not shown here).

The groups $\pi_1(\mathbb{S}) \simeq \pi_2(\mathbb{S}) \simeq \pi_6(\mathbb{S}) \simeq \pi_{10}(\mathbb{S}) \simeq \mathbb{Z}_2$ correspond to the single dots over $n = 1,2,6,10$ in the first table, respectively.

The group $\pi_8(\mathbb{S}) \simeq \mathbb{Z}_2 \oplus \mathbb{Z}_2$ corresponds to the two \emph{unconnected} dots over $n = 8$ in the first table.

Similarly the group $\pi_9(\mathbb{S}) \simeq \mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2$ corresponds to the three unconnected dots above $n = 9$ in the first table.

\end{example}
\hypertarget{chromatic_formulation}{}\subsubsection*{{Chromatic formulation}}\label{chromatic_formulation}

The above tables, example \ref{TableOfPPrincipalComponentsInHomotopyGroups}, suggest that the [[image of the J-homomorphism]] is in some sense the ``lowest order layer'' of the [[stable homotopy groups of spheres]]. This is made precise by the following characterization of the image in [[stable homotopy theory]]. We bluntly state this here and give all the relevant definitions \hyperlink{ELocalStableHomotopyTheory}{below}.

$\,$

Write $E(1)$ for the first [[Morava E-theory]] [[spectrum]] at given [[prime number]] $p$. Write $L_{E(1)}\mathbb{S}$ for the [[Bousfield localization of spectra]] of the [[sphere spectrum]] at $E(1)$.

\begin{theorem}
\label{}\hypertarget{}{}
The [[homotopy groups]] of the $E(1)$-localized sphere spectrum are

\begin{displaymath}
\pi_n L_{E(1)} \mathbb{S}
  \simeq
  \left\{
    \itexarray{
      \mathbb{Z} & if\; n = 0
       \\
       \mathbb{Q}_p/\mathbb{Z}_p & if\; n= -2
       \\
       \mathbb{Z}_{p^{k+1}} & if\; n+1 = (p-1)p^k m \;with\; m \neq 0\;mod\;p
        \\
        0 & otherwise
    }
  \right.
  \,.
\end{displaymath}
\end{theorem}
This appears as (\hyperlink{LurieLecture}{Lurie 10, theorem 6})

\begin{defn}
\label{}\hypertarget{}{}
Write $\mathbb{S}_p$ for the [[p-localization]] of the [[sphere spectrum]]. For $n \in \mathbb{Z}$, write $im(J)_n$ for the [[image]] of the $p$-localized J-homomorphism

\begin{displaymath}
J \;\colon\; \pi_n(O) \longrightarrow \pi_n(\mathbb{S}) \longrightarrow \pi_n(\mathbb{S}_{(p)})
  \,.
\end{displaymath}
\end{defn}
\begin{theorem}
\label{}\hypertarget{}{}
For $n \in \mathbb{N}$, the further [[Bousfield localization]] at [[Morava E-theory|Morava E(1)-theory]] $\mathbb{S}_{(p)} \longrightarrow L_{E(1)}\mathbb{S}$ induces a [[isomorphism]]

\begin{displaymath}
im(J)_n \stackrel{\simeq}{\longrightarrow} \pi_n (L_{E(1)} \mathbb{S})
\end{displaymath}
between the image of the $J$-homomorphism and the $E(1)$-local [[stable homotopy groups of spheres]].

\end{theorem}
In this form this appears as (\hyperlink{LurieLecture}{Lurie 10, theorem 7}). See also (\hyperlink{Behrens13}{Behrens 13, section 1}).

\begin{cor}
\label{}\hypertarget{}{}
The $E(1)$-[[Bousfield localization of spectra|localization map]] is surjective on non-negative homotopy groups:

\begin{displaymath}
\pi_n(\mathbb{S}_{(p)}) \longrightarrow \pi_n(L_{E(1)} \mathbb{S})
  \,.
\end{displaymath}
\end{cor}
For review see also (\hyperlink{Lorman13}{Lorman 13}). That $J$ factors through $L_{K(1)}\mathbb{S}$ is in (\hyperlink{Lorman13}{Lorman 13, p. 4})

\begin{remark}
\label{}\hypertarget{}{}
Hence: the image of $J$ is essentially the first [[chromatic layer]] of the [[sphere spectrum]].

\end{remark}
\hypertarget{ELocalStableHomotopyTheory}{}\subsection*{{\textbf{III)} $E$-Local stable homotopy theory}}\label{ELocalStableHomotopyTheory}

To say what all this means, we recall now [[Bousfield localization of spectra]] and then indicate the tower of localizations at the [[Morava E-theory]] spectra, the ``[[chromatic filtration]]''.

\hypertarget{bousfield_localization_of_spectra}{}\subsubsection*{{Bousfield localization of spectra}}\label{bousfield_localization_of_spectra}

\begin{defn}
\label{AcyclicAndLocal}\hypertarget{AcyclicAndLocal}{}
Let $E \in Spec$ be a [[spectrum]].

Say that another spectrum $X \in Spec$ is an \textbf{$E$-acyclic spectrum} if the [[smash product]] is [[zero object|zero]], $E \wedge X \simeq 0$.

Say that $X$ is an \textbf{$E$-local spectrum} if every [[morphism]] $Y \longrightarrow X$ out of an $E$-acyclic spectrum $Y$ is homotopic to the [[zero morphism]].

Say that a morphism $f \colon X \to Y$ is an \textbf{$E$-equivalence} if it becomes an [[equivalence]] after [[smash product]] with $E$.

\end{defn}
(e.g. \hyperlink{LurieLecture}{Lurie, Lecture 20, example 4})

\begin{prop}
\label{LocalizationCofiber}\hypertarget{LocalizationCofiber}{}
For $E$ a [[spectrum]], every other spectrum sits in an essentially unique [[homotopy cofiber sequence]]

\begin{displaymath}
G_E(X) \to X \to L_E(X)
  \,,
\end{displaymath}
where $G_E(X)$ is $E$-acyclic, and $L_E(X)$ is $E$-local, def. \ref{AcyclicAndLocal}.

Here $X \to L_E (X)$ is characterized by two properties

\begin{enumerate}%
\item $L_E(X)$ is $E$-local;


\item $X \to L_E(X)$ is an $E$-equivalence



\end{enumerate}
according to def. \ref{AcyclicAndLocal}.

\end{prop}
(e.g. \hyperlink{LurieLecture}{Lurie, Lecture 20, example 4})

\begin{defn}
\label{}\hypertarget{}{}
Given $E \in Spec$, the [[natural transformation|natural morphisms]] $X \longrightarrow L_E X$ in prop. \ref{LocalizationCofiber} exhibit the [[localization of an (infinity,1)-category]] called \textbf{Bousfield localization} at $E$.

\end{defn}
\begin{example}
\label{EModulesAreELocal}\hypertarget{EModulesAreELocal}{}
For $E$ an [[E-∞ ring]], every [[∞-module]] $X$ over $E$ is $E$-local, def. \ref{AcyclicAndLocal}.

\end{example}
(e.g. \hyperlink{LurieLecture}{Lurie, Lecture 20, example 5})

\begin{example}
\label{}\hypertarget{}{}
For $E$ an [[E-∞ algebra]] over an [[E-∞ ring]] $S$ and for $X$ an $S$-[[∞-module]], consider the dual [[Cech nerve]] [[cosimplicial object]]

\begin{displaymath}
E^{\wedge_S^{\bullet+1}}\wedge_S X \;\colon\; \Delta \longrightarrow Spectra
  \,.
\end{displaymath}
By example \ref{EModulesAreELocal} each term is $E$-local, so that the map to the [[totalization]]

\begin{displaymath}
X \longrightarrow \underset{\leftarrow}{\lim} E^{\wedge_S^{\bullet+1}} \wedge_S X
\end{displaymath}
factors through the $E$-localization of $X$

\begin{displaymath}
X \longrightarrow 
  L_E X
  \longrightarrow
  \underset{\leftarrow}{\lim} E^{\wedge_S^{\bullet+1}} \wedge_S X
  \,.
\end{displaymath}
Under suitable condition the second map here is indeed an [[equivalence]], in which case the [[totalization]] of the dual [[Cech nerve]] exhibits the $E$-localization. This happens for instance in the discussion of the [[Adams spectral sequence]], see the examples given there.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
For $p \in \mathbb{N}$ a [[prime number]], let

\begin{displaymath}
E \coloneqq H \mathbb{Z}/p\mathbb{Z}
\end{displaymath}
be the corresponding [[Eilenberg-MacLane spectrum]]. Then a spectrum which corresponds to a [[chain complex]] under the \href{module+spectrum#StableDoldKanCorrespondence}{stable Dold-Kan corespondence} is $E$-local, def. \ref{AcyclicAndLocal}, if that chain complex has [[chain homology]] groups being $\mathbb{Z}[p^{-1}]$-modules.

The $E$-localization of a spectrum in this case is called \emph{[[p-localization]]}.

\end{example}
(e.g. \hyperlink{LurieLecture}{Lurie, Lecture 20, example 8})

\hypertarget{chromatic_layers}{}\subsubsection*{{Chromatic layers}}\label{chromatic_layers}

Let

\begin{itemize}%
\item $k$ be a [[perfect field]] of [[characteristic]] $p$;


\item $f$ be a [[formal group]] of [[height of a formal group|height]] $n$ over $k$.



\end{itemize}
\begin{defn}
\label{}\hypertarget{}{}
Write $W(k)$ for the [[ring of Witt vectors]]. Write

\begin{displaymath}
R \coloneqq W(k)[ [ v_1, \cdots, v_{n-1} ] ]
\end{displaymath}
for the ring of [[formal power series]] over this ring, in $n-1$ [[variables]]; called the \emph{[[Lubin-Tate ring]]}.

\end{defn}
\begin{theorem}
\label{}\hypertarget{}{}
The [[Lubin-Tate formal group]] $\overline{f}$ is the [[universal property|universal]] deformation of $f$ in that for every [[infinitesimal object|infinitesimal thickening]] $A$ of $k$, $\overline{f}$ induces a [[bijection]]

\begin{displaymath}
Hom_{/k}(R,A) \stackrel{\simeq}{\longrightarrow} Def(A)
\end{displaymath}
between the $k$-[[associative algebra|algebra]]-[[homomorphisms]] from $R$ into $A$ and the [[deformations]] of $A$.

\end{theorem}
(e.g. \hyperlink{LurieLecture}{Lurie 10, lect 21, theorem 5})

By the discussion there, this is [[Landweber exactness|Landweber exact]], hence defines a [[cohomology theory]]. Therefore by the [[Landweber exact functor theorem]] there is an [[even periodic cohomology theory]] $E(n)^\bullet$ [[Brown representability theorem|represented by]] a [[spectrum]] $E(n)$ with the property that its [[homotopy groups]] are

\begin{displaymath}
\pi_\bullet(E(n))
  \simeq
  W(k)[ [v_1, \cdots, v_{n-1} ] ] [ \beta^{\pm 1} ]
\end{displaymath}
for $\beta$ of degree 2. This is called alternatively $n$th \emph{[[Morava E-theory]]}, or \emph{[[Lubin-Tate theory]]} or \emph{[[Johnson-Wilson theory]]}.

(e.g. \hyperlink{LurieLecture}{Lurie, lect 22})

For each [[prime]] $p \in \mathbb{N}$ and for each [[natural number]] $n \in \mathbb{N}$ there is a [[Bousfield localization of spectra]]

\begin{displaymath}
L_n \coloneqq L_{E(n)}
  \,,
\end{displaymath}
where $E(n)$ is the $n$th [[Morava E-theory]] (for the given [[prime]] $p$), called the $n$th \emph{[[chromatic localization]]}. These arrange into the \emph{[[chromatic tower]]} which for each spectrum $X$ is of the form

\begin{displaymath}
X \to \cdots \to L_n X \to L_{n-1} X \to \cdots \to L_0 X
  \,.
\end{displaymath}
The [[homotopy fibers]] of each stage of the tower

\begin{displaymath}
M_n(X) \coloneqq fib(L_{E(n)}X \longrightarrow L_{E(n-1)}(X))
\end{displaymath}
is called the $n$th \emph{[[monochromatic layer]]} of $X$.

[[!include chromatic tower examples - table]]

\hypertarget{iv_adams_spectral_sequence_for_local_homotopy_groups}{}\subsection*{{\textbf{IV)} Adams spectral sequence for $E$-local homotopy groups}}\label{iv_adams_spectral_sequence_for_local_homotopy_groups}

Summing up the above, we need a means to compute [[homotopy groups]] of $E$-[[Bousfield localization of spectra|localized]] [[spectra]]. In (\hyperlink{LurieHigherAlgebra}{Lurie, Higher Algebra, section 1.2.2}) is given a general [[spectral sequence of a filtered stable homotopy type]] which computes homotopy groups of [[spectra]], and in (\hyperlink{LurieLecture}{Lurie 10, lectures 8 and 9}) is discussed that the [[totalization]] of the [[coskeleton]] filtration on the dual [[Cech nerve]] of an [[E-∞ algebra]] yields the $E$-localization. Taken together this is just what we need\ldots{} and this is the general \emph{$E$-[[Adams spectral sequence]]}. We follow the nice exposition in (\hyperlink{Wilson13}{Wilson 13}).

First we recall

\begin{itemize}%
\item \hyperlink{SpectralSequencesForHomotopyGroupsOfFilteredObjects}{Spectral sequences computing homotopy groups of filtered objects}

\end{itemize}
for the general case of filtered objects in suitable [[stable (∞,1)-categories]]. Then we consider the specialization of that to the

\begin{itemize}%
\item \hyperlink{CanonicalCosimplicialResolutionOfEInfinityAlgebras}{Canonical cosimplicial resolutions of E-∞ algebras}.

\end{itemize}
In conclusion this yields for each suitable [[E-∞ algebra]] $E$ over $S$ and $S$-[[∞-module]] $X$ a [[spectral sequence]] converging to the [[homotopy groups]] of the $E$-[[Bousfield localization of spectra|localization]] of $X$, and this is

\begin{itemize}%
\item \hyperlink{TheEAdamsSpectralSequence}{The E-Adams-Novikov spectral sequence}.

\end{itemize}
\hypertarget{SpectralSequencesForHomotopyGroupsOfFilteredObjects}{}\subsubsection*{{Spectral sequences for homotopy groups of filtered spectra}}\label{SpectralSequencesForHomotopyGroupsOfFilteredObjects}

We discuss the [[spectral sequence of a filtered stable homotopy type]].

Let throughout $\mathcal{C}$ be a [[stable (∞,1)-category]], $\mathcal{A}$ an [[abelian category]], and $\pi \;\colon\; \mathcal{C}\longrightarrow \mathcal{A}$ a [[homological functor]] on $\mathcal{C}$, i.e., a functor that transforms every [[cofiber sequence]]

\begin{displaymath}
X\to Y\to Z\to \Sigma X
\end{displaymath}
in $\mathcal{C}$ into a long exact sequence

\begin{displaymath}
\dots \to \pi(X)\to \pi(Y)\to \pi(Z)\to \pi(\Sigma X) \to \dots
\end{displaymath}
in $\mathcal{A}$. We write $\pi_n=\pi\circ \Sigma^{-n}$.

\begin{example}
\label{}\hypertarget{}{}
\begin{itemize}%
\item $\mathcal{C}$ is arbitrary, $\mathcal{A}$ is the category of [[abelian groups]] and $\pi$ is taking the 0th [[homotopy group]] $\pi_0 \mathcal{C}(S,-)$ of the [[mapping spectrum]] out of some [[object]] $S\in\mathcal{C}$


\item $\mathcal{C}$ is equipped with a [[t-structure]], $\mathcal{A}$ is the [[heart of a stable (∞,1)-category|heart]] of the t-structure, and $\pi$ is the canonical functor.


\item $\mathcal{C} = D(\mathcal{A})$ is the [[derived category]] of the abelian category $\mathcal{A}$ and $\pi=H_0$ is the degree-0 [[chain homology]] functor.


\item Any of the above with $\mathcal{C}$ and $\mathcal{A}$ replaced by their [[opposite categories]].



\end{itemize}
\end{example}
\begin{defn}
\label{GeneralizedFilteredObject}\hypertarget{GeneralizedFilteredObject}{}
A \emph{[[filtered object in an (∞,1)-category]]} in $\mathcal{C}$ is simply a [[sequential diagram]] $X \colon (\mathbb{Z}, \lt) \to \mathcal{C}$

\begin{displaymath}
\cdots
  X_{n-1} \to X_n \to X_{n+1} \to \cdots
  \,.
\end{displaymath}
\end{defn}
This appears as ([[Higher Algebra|Higher Algebra, def. 1.2.2.9]]).

We will take the view that the object being filtered is the [[homotopy limit]]

\begin{displaymath}
X \coloneqq \underset{\leftarrow}{\lim}_n X_n.
\end{displaymath}
We could also consider the sequential diagram as a filtering of its [[homotopy colimit]], but this is really an equivalent point of view since we can replace $\mathcal{C}$ by $\mathcal{C}^{op}$.

\begin{defn}
\label{ChainComplexInStableInfinityCategory}\hypertarget{ChainComplexInStableInfinityCategory}{}
Let $I$ be a [[linearly ordered set]]. An $I$-chain complex in a [[stable (∞,1)-category]] $\mathcal{C}$ is an [[(∞,1)-functor]]

\begin{displaymath}
F \;\colon\; I^{\Delta[1]} \longrightarrow \mathcal{C}
\end{displaymath}
such that

\begin{enumerate}%
\item for each $n \in I$, $F(n,n) \simeq 0$ is the [[zero object]];


\item for all $i \leq j \leq k$ the induced [[diagram]]

\begin{displaymath}
\itexarray{
     F(i,j) &\longrightarrow& F(i,k)
     \\
     \downarrow && \downarrow
     \\
     F(j,j) &\longrightarrow& F(j,k)
  }
\end{displaymath}
is a [[homotopy pullback]] square.



\end{enumerate}
\end{defn}
This is [[Higher Algebra|Higher Algebra, def. 1.2.2.2]].

\begin{remark}
\label{}\hypertarget{}{}
Given a $\mathbb{Z}$-chain complex $F$ in $\mathcal{C}$ as in def. \ref{ChainComplexInStableInfinityCategory}, setting

\begin{displaymath}
C_n \coloneqq \Sigma^{-n} F(n,n+1)
\end{displaymath}
and defining a [[differential]] induced from the [[connecting homomorphisms]] of the defining [[homotopy fiber sequences]]

\begin{displaymath}
F(n-1,n) \to F(n-1, n+1) \to F(n,n+1)
\end{displaymath}
yields an ordinary [[chain complex]] $C_\bullet$ in the [[homotopy category of an (∞,1)-category|homotopy category]].

\end{remark}
([[Higher Algebra|Higher Algebra, remark 1.2.2.3]])

\begin{prop}
\label{ChainComplexesFromFilteredObjects}\hypertarget{ChainComplexesFromFilteredObjects}{}
Consider the inclusion of [[posets]]

\begin{displaymath}
(\mathbb{Z}, \leq)
  \to 
  (\mathbb{Z}\cup \{\infty\}, \leq)^{\Delta[1]}
\end{displaymath}
given by

\begin{displaymath}
n \mapsto (n,\infty)
  \,.
\end{displaymath}
The induced [[(∞,1)-functor]]

\begin{displaymath}
Func((\mathbb{Z}\cup \{\infty\}, \leq)^{\Delta[1]} , \mathcal{C})
  \longrightarrow
  Func((\mathbb{Z}, \leq), \mathcal{C})
\end{displaymath}
restricts to an [[equivalence of (∞,1)-categories|equivalence]] between the (∞,1)-category of $\mathbb{Z}\cup \{\infty\}$-chain complexes in $\mathcal{C}$ (def. \ref{ChainComplexInStableInfinityCategory}) and that of generalized filtered objects in $\mathcal{C}$ (def. \ref{GeneralizedFilteredObject}).

\end{prop}
This is [[Higher Algebra|Higher Algebra, lemma 1.2.2.4]]. The inverse functor can be described informally as follows: given a filtered object $X_\bullet$, the associated chain complex $X(\bullet,\bullet)$ is given by

\begin{displaymath}
X(n, n+r) = \operatorname{fib}(X_n\to X_{n+r}).
\end{displaymath}
\begin{defn}
\label{}\hypertarget{}{}
Let $X_\bullet$ be a filtered object in the sense of def. \ref{GeneralizedFilteredObject}. Write $X(\bullet,\bullet)$ for the corresponding chain complex, according to prop. \ref{ChainComplexesFromFilteredObjects}.

Then for all $i \leq j \leq k$ there is a [[long exact sequence of homotopy groups]] in $\mathcal{A}$ of the form

\begin{displaymath}
\cdots \to 
  \pi_n X(i,j) 
  \to 
  \pi_n X(i,k)
  \to 
  \pi_n X(j,k)
  \to
  \pi_{n-1}X(i,j)
  \to \cdots
  \,.
\end{displaymath}
Define then for $p,q \in \mathbb{Z}$ and $r \geq 1$ the object $E^r_{p,q}$ by the canonical \href{abelian+category#FactorizationOfMorphisms}{epi-mono factorization}

\begin{displaymath}
\pi_{p} X(q-r+1,q+1)
    \twoheadrightarrow
    E^r_{p,q}
    \hookrightarrow
    \pi_{p} X(q, q+r)
\end{displaymath}
in the abelian category $\mathcal{A}$, and define the [[differential]]

\begin{displaymath}
d^r \;\colon\; E_{p,q}^r \to E_{p-1, q-r}^r
\end{displaymath}
to be the restriction of the [[connecting homomorphism]]

\begin{displaymath}
\pi_{p} X(q,q+r) \to \pi_{p-1} X(q-r, q)
\end{displaymath}
from the above long exact sequence (with $i=q-r$, $j=q$, and $k=q+r$).

\end{defn}
([[Higher Algebra|Higher Algebra, construction 1.2.2.6]])

\begin{prop}
\label{}\hypertarget{}{}
$d^r\circ d^r = 0$ and there are natural (in $X_\bullet$) isomorphisms

\begin{displaymath}
E^{r+1}\cong \operatorname{ker}(d^r)/\operatorname{im}(d^r).
\end{displaymath}
Thus, $\{E^r_{*,*}\}_{r\geq 1}$ is a bigraded [[spectral sequence]] in the [[abelian category]] $\mathcal{A}$, functorial in the filtered object $X_\bullet$, with

\begin{displaymath}
E^1_{p,q} = \pi_p \operatorname{fib}(X_q\to X_{q+1}), \qquad d^r: E^r_{p,q}\to E^r_{p-1,q-r}.
\end{displaymath}
\end{prop}
([[Higher Algebra|Higher Algebra, prop. 1.2.2.7]])

If [[sequential limits]] and [[sequential colimits]] exist in $\mathcal{A}$, we can form the limiting term $E^\infty_{*,*}$ of this spectral sequence.

On the other hand, the [[graded object]] $\pi_\bullet (X)$ admits a [[filtered object|filtration]] by

\begin{displaymath}
F_q \pi_p (X) = \operatorname{ker}(\pi_p (X)\to \pi_p(X_q))
\end{displaymath}
and we would like to compare $E^\infty_{*,*}$ with the [[associated graded]] of this filtration. We say that

\begin{defn}
\label{WeakAndStrongConvergence}\hypertarget{WeakAndStrongConvergence}{}
The spectral sequence \textbf{converges weakly} if there is a canonical isomorphism

\begin{displaymath}
E^\infty_{p,q} \cong F_q\pi_p(X)/ F_{q-1}\pi_p(X)
\end{displaymath}
for every $p,q\in\mathbb{Z}$.

We say that the spectral sequence \textbf{converges strongly} if it converges weakly and if, in addition, the filtration $F_\bullet\pi_p(X)$ is complete on both sides.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
The meaning of the word \emph{canonical} in def. \ref{WeakAndStrongConvergence} is somewhat subtle since, in general, there is no map from one side to the other. However, there always exists a canonical \emph{[[relation]]} between the two, and we ask that this relation be an isomorphism (see \hyperlink{HiltonStammbach}{Hilton-Stammbach, VIII.7}).

\end{remark}
\begin{prop}
\label{FiltrationSpectralSequence}\hypertarget{FiltrationSpectralSequence}{}
Let $\mathcal{C}$ be a [[stable (∞,1)-category]] and let $\pi:\mathcal{C}\to\mathcal{A}$ be a homological functor where $\mathcal{A}$ is an [[abelian category]] which admits [[sequential limits]]. Let $X_\bullet$ be a filtered object in $\mathcal{C}$ such that $\underset{\leftarrow}{\lim} X_\bullet$ exists. Suppose further that:

\begin{enumerate}%
\item For every $n$, the diagram $r\mapsto \operatorname{fib}(X_{n-r}\to X_n)$ has a limit in $\mathcal{C}$ and that limit is preserved by $\pi$.
\item For every $n$, $\pi_n(X_r)=0$ for $r\gg 0$.

\end{enumerate}
Then the [[spectral sequence]] $\{E^r_{*,*}\}_{r\geq 1}$ in $\mathcal{A}$ converges strongly (def. \ref{WeakAndStrongConvergence}). We write:

\begin{displaymath}
E_{p,q}^1
  =
  \pi_{p} \operatorname{fib}(X_q\to X_{q+1})
  \Rightarrow
  \pi_{p} (\underset{\leftarrow}{\lim} X_\bullet)
\end{displaymath}
\end{prop}
There is also a dual statement in which limits are replaced by colimits, but it is in fact a special case of the proposition with $\pi$ replaced by $\pi^{op}$. A proof of this proposition (in dual form) is given in ([[Higher Algebra|Higher Algebra, prop. 1.2.2.14]]). Review is in (\hyperlink{Wilson13}{Wilson 13, theorem 1.2.1}).

For the traditional statement in the [[category of chain complexes]] see at \emph{[[spectral sequence of a filtered complex]]}.

Plenty of types of [[spectral sequences]] turn out to be special cases of this general construction.

[[!include Lurie spectral sequences -- table]]

\hypertarget{CanonicalCosimplicialResolutionOfEInfinityAlgebras}{}\subsubsection*{{Canonical cosimplicial resolution of $E_\infty$-algebras}}\label{CanonicalCosimplicialResolutionOfEInfinityAlgebras}

We discuss now the special case of coskeletally filtered totalizations coming from the canonical cosimplicial objects induced from [[E-∞ algebras]] (dual [[Cech nerves]]/[[Sweedler corings]]/[[Amitsur complexes]]).

In this form this appears as (\hyperlink{LurieLecture}{Lurie 10, theorem 2}). A review is in (\hyperlink{Wilson13}{Wilson 13, 1.3}). For the analog of this in the traditional formulation see (\hyperlink{Ravenel}{Ravenel, ch. 3, prop. 3.1.2}).

\begin{defn}
\label{FiltrationOfTotalizationByTotalizationOfCoskeleta}\hypertarget{FiltrationOfTotalizationByTotalizationOfCoskeleta}{}
Given an [[cosimplicial object in an (∞,1)-category]] with [[(∞,1)-colimits]]

\begin{displaymath}
Y \;\colon\; \Delta  \longrightarrow \mathcal{C}
\end{displaymath}
its [[totalization]] $Tot Y \simeq \underset{\leftarrow}{\lim}_n Y_n$ is [[filtered object in an (infinity,1)-category|filtered]], def. \ref{GeneralizedFilteredObject}, by the totalizations of its [[coskeleta]]

\begin{displaymath}
Tot Y 
  \to 
  \cdots 
  \to 
  Tot (cosk_2 Y) 
  \to
  Tot (cosk_1 Y) 
  \to
  Tot (cosk_0 Y) 
  \to 0
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{SpectralSequenceOfSimplicialStableHomotopyType}\hypertarget{SpectralSequenceOfSimplicialStableHomotopyType}{}
The [[spectral sequence of a filtered stable homotopy type|filtration spectral sequence]], prop. \ref{FiltrationSpectralSequence}, applied to the filtration of a [[totalization]] by [[coskeleton|coskeleta]] as in def. \ref{FiltrationOfTotalizationByTotalizationOfCoskeleta}, we call the \emph{[[spectral sequence of a simplicial stable homotopy type]]}.

\end{defn}
([[Higher Algebra|Higher Algebra, prop. 1.2.4.5]])

\begin{prop}
\label{E2PageByMooreComplex}\hypertarget{E2PageByMooreComplex}{}
The [[spectral sequence of a simplicial stable homotopy type]] has as first page/$E_1$-term the [[cohomology groups]] of the [[Moore complex]] associated with the [[cosimplicial objects]] of [[homotopy groups]]

\begin{displaymath}
E_2^{p,q}
  = 
  H^p(\pi_q(Tot (cosk_\bullet(Y))))
  \Rightarrow
  \pi_{p-q} Tot(Y)
  \,.
\end{displaymath}
\end{prop}
By the discussion at \emph{[[∞-Dold-Kan correspondence]]} and \emph{[[spectral sequence of a filtered stable homotopy type]]}. This appears as ([[Higher Algebra|Higher Algebra, remark 1.2.4.4]]). Review is around (\hyperlink{Wilson13}{Wilson 13, theorem 1.2.4}).

\begin{defn}
\label{}\hypertarget{}{}
Let $S$ be an [[E-∞ ring]] and let $E$ be an [[E-∞ algebra]] over $S$, hence an [[E-∞ ring]] equipped with a [[homomorphism]]

\begin{displaymath}
S \longrightarrow E
  \,.
\end{displaymath}
The \emph{canonical [[cosimplicial object]]} associated to this (the $\infty$-[[Cech nerve]]/[[Sweedler coring]]/[[Amitsur complex]]) is that given by the iterated [[smash product]]/[[tensor product]] over $S$:

\begin{displaymath}
E^{\wedge^{\bullet+1}_S} \;\colon\; \Delta \to \mathcal{C}
  \,.
\end{displaymath}
More generally, for $X$ an $S$-[[∞-module]], the canonical [[cosimplicial object]] is

\begin{displaymath}
E^{\wedge^{\bullet+1}_S}\wedge_S X \;\colon\; \Delta \to \mathcal{C}
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{FlatnessCondition}\hypertarget{FlatnessCondition}{}
If $E$ is such that the self-[[generalized homology]] $E_\bullet(E) \coloneqq \pi_\bullet(E \wedge_S E)$ (the dual $E$-[[Steenrod operations]]) is such that as a [[module]] over $E_\bullet \coloneqq \pi_\bullet(E)$ it is a [[flat module]], then there is a [[natural equivalence]]

\begin{displaymath}
\pi_\bullet
  \left(
    E^{\wedge^{n+1}_S}
    \wedge_S 
    X
  \right)
  \simeq
  E_\bullet(E^{\wedge^n_S})
  \otimes_{E_\bullet}
  E_\bullet(X)
  \,.
\end{displaymath}
\end{prop}
Reviewed for instance as (\hyperlink{Wilson13}{Wilson 13, prop. 1.3.1}).

\begin{remark}
\label{}\hypertarget{}{}
This makes $(E_\bullet, E_\bullet(E))$ be the [[commutative Hopf algebroid]] formed by the $E$-[[Steenrod algebra]]. See there for more on this.

\end{remark}
\begin{example}
\label{}\hypertarget{}{}
The condition in prop. \ref{FlatnessCondition} is satisfied for

\begin{itemize}%
\item $E = H \mathbb{F}_p$ an [[Eilenberg-MacLane spectrum]] with $mod\;p$ [[coefficients]];


\item $E = B P$ the [[Brown-Peterson spectrum]];


\item $E = MU$ the [[complex cobordism cohomology theory|complec cobordism spectrum]].



\end{itemize}
It is NOT satisfied for

\begin{itemize}%
\item $E = H \mathbb{Z}$ the [[Eilenberg-MacLane spectrum]] for [[integer|integers]] [[coefficients]];


\item $E = M S U$.



\end{itemize}
\end{example}
\begin{remark}
\label{ExtGroupsByMooreComplex}\hypertarget{ExtGroupsByMooreComplex}{}
Under good conditions (\ldots{}), $\pi_\bullet$ of the canonical [[cosimplicial object]] provides a [[resolution]] of [[comodule]] [[tensor product]] and hence computes the [[Ext]]-groups over the [[commutative Hopf algebroid]]:

\begin{displaymath}
H^p(\pi_q(Tot(cosk_\bullet(E^{\wedge^{\bullet+1}_S } \wedge_S X))))
  \simeq
  Ext^p_{E_\bullet(E)}(\Sigma^q E_\bullet, E_\bullet(X))
  \,.
\end{displaymath}
(\ldots{})

\end{remark}
\begin{remark}
\label{CanonicalMapFromELocalizationToTotalization}\hypertarget{CanonicalMapFromELocalizationToTotalization}{}
There is a canonical map

\begin{displaymath}
L_E X \stackrel{}{\longrightarrow} \underset{\leftarrow}{\lim}_n (E^{\wedge^{n+1}_S}\wedge_S X)
\end{displaymath}
from the $E$-[[Bousfield localization of spectra]] of $X$ into the [[totalization]].

\end{remark}
(\href{LurieLecture}{Lurie 10, lecture 30, prop. 1})

We consider now conditions for this morphism to be an [[equivalence]].

\begin{defn}
\label{CoreOfARing}\hypertarget{CoreOfARing}{}
For $R$ a [[ring]], its \emph{core} $c R$ is the [[equalizer]] in

\begin{displaymath}
c R 
    \longrightarrow 
  R 
    \stackrel{\longrightarrow}{\longrightarrow}
  R \otimes R
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{SufficientConditionsForTotalizationToBeELocalization}\hypertarget{SufficientConditionsForTotalizationToBeELocalization}{}
Let $E$ be a [[connective spectrum|connective]] [[E-∞ ring]] such that the core or $\pi_0(E)$, def. \ref{CoreOfARing} is either of

\begin{itemize}%
\item the [[localization of a ring|localization]] of the [[integers]] at a set $J$ of [[primes]], $c \pi_0(E) \simeq \matbb{Z}[J^{-1}]$;


\item $\mathbb{Z}_n$ for $n \geq 2$.



\end{itemize}
Then the map in remark \ref{CanonicalMapFromELocalizationToTotalization} is an equivalence

\begin{displaymath}
L_E X \stackrel{\simeq}{\longrightarrow} 
  \underset{\leftarrow}{\lim}_n (E^{\wedge^{n+1}_S}\wedge_S X)
  \,.
\end{displaymath}
\end{prop}
(\href{Bousfield+localization+of+spectra#Bousfield79}{Bousfield 79}, \href{LurieLecture}{Lurie 10, lecture 30, prop. 3}, \href{LurieLecture}{Lurie 10, lecture 31,}).

\hypertarget{TheEAdamsSpectralSequence}{}\subsubsection*{{The $E$-Adams-Novikov spectral sequence}}\label{TheEAdamsSpectralSequence}

Summing this up yields the general $E$-Adams(-Novikov) spectral sequence

\begin{cor}
\label{}\hypertarget{}{}
Let $E$ a [[connective spectrum|connective]] [[E-∞ ring]] that satisfies the conditions of prop. \ref{SufficientConditionsForTotalizationToBeELocalization}. Then by prop. \ref{FiltrationSpectralSequence} and prop. \ref{SufficientConditionsForTotalizationToBeELocalization} there is a strongly convergent multiplicative [[spectral sequence]]

\begin{displaymath}
E^{p,q}_\bullet \Rightarrow \pi_{q-p} L_{c \pi_0 E} X
\end{displaymath}
converging to the [[homotopy groups]] of the $c \pi_0(E)$-[[Bousfield localization of spectra|localization]] of $X$. If moreover the dual $E$-[[Steenrod algebra]] $E_\bullet(E)$ is [[flat module|flat]] as a [[module]] over $E_\bullet$, then, by prop. \ref{E2PageByMooreComplex} and remark \ref{ExtGroupsByMooreComplex}, the $E_1$-term of this spectral sequence is given by the [[Ext]]-groups over the $E$-[[Steenrod algebra|Steenrod]] [[commutative Hopf algebroid|Hopf algebroid]].

\begin{displaymath}
E^{p,q}_\bullet
  =
  Ext^p_{E_\bullet(E)}(\Sigma^q E_\bullet, E_\bullet X)
  \,.
\end{displaymath}
\end{cor}
\hypertarget{references}{}\subsection*{{References}}\label{references}

An introduction to the chromatic perspective on the homotopy groups of spheres and the image of $J$ is in:

\begin{itemize}%
\item [[Mark Mahowald]], [[Doug Ravenel]], \emph{Towards a Global Understanding of the Homotopy Groups of Spheres}, 1998 (\href{http://www.math.rochester.edu/people/faculty/doug/mypapers/global.pdf}{pdf})

\end{itemize}
The bulk of the basic constructions is in

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Chromatic Homotopy Theory]]}, lecture notes (2010)

\end{itemize}
\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Higher Algebra]]}

\end{itemize}
Recent surveys of the modern picture are in

\begin{itemize}%
\item \href{http://math.mit.edu/conferences/talbot/}{Talbot 2013: Chromatic Homotopy Theory}, \emph{\href{http://math.northwestern.edu/~htanaka/pretalbot2013/index.php?pageID=links}{2013 Pre-Talbot Seminar Chromatic homotopy theoy}}

\end{itemize}
and of relevance for the above discussion are particularly the following contributions there

\begin{itemize}%
\item [[Mark Behrens]], section 1 of Introduction talk at \emph{\href{http://math.mit.edu/conferences/talbot/index.php?year=2013&sub=talks}{Talbot 2013: Chromatic Homotopy Theory}} (\href{http://math.mit.edu/conferences/talbot/2013/1-Behrens-intro.pdf}{pdf}, \href{http://math.mit.edu/conferences/talbot/2013/Behrens-Introduction-chromatic-homotopy-thy-4-22-13.pdf}{pdf})

\end{itemize}
\begin{itemize}%
\item [[Dylan Wilson]] \emph{Spectral Sequences from Sequences of Spectra: Towards the Spectrum of the Category of Spectra} lecture at \emph{\href{http://math.northwestern.edu/~htanaka/pretalbot2013/index.php}{2013 Pre-Talbot Seminar}} (\href{http://math.northwestern.edu/~htanaka/pretalbot2013/notes/2013-03-21-Dylan-Wilson-ANSS.pdf}{pdf})

\end{itemize}
\begin{itemize}%
\item Ben Knudsen, \emph{First chromatic layer of the sphere spectrum = homotopy of the $K(1)$-local sphere}, talk at \emph{\href{http://math.northwestern.edu/~htanaka/pretalbot2013/index.php}{2013 Pre-Talbot Seminar}} (\href{http://math.northwestern.edu/~htanaka/pretalbot2013/notes/2013-04-04-Rob-Legg-K1-local-sphere.pdf}{pdf})

\end{itemize}
\begin{itemize}%
\item [[Vitaly Lorman]], \emph{Chromatic homotopy theory at height 1 and the image of $J$}, talk at \emph{\href{http://math.mit.edu/conferences/talbot/index.php?year=2013&sub=talks}{Talbot 2013: Chromatic Homotopy Theory}} (\href{http://math.mit.edu/conferences/talbot/2013/Image%20of%20J-1.pdf}{pdf})

\end{itemize}
Loads of details for computations in the Adams spectral sequence are in

\begin{itemize}%
\item [[Doug Ravenel]], \emph{[[Complex cobordism and stable homotopy groups of spheres]]}, 1986

\end{itemize}


\end{document}