\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{Adams resolution}

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

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

[[!include stable homotopy theory - contents]]

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{ClassicalDefinition}{Classical definition}\dotfill \pageref*{ClassicalDefinition} \linebreak
\noindent\hyperlink{ViaInjectiveResolutions}{Via injective resolutions}\dotfill \pageref*{ViaInjectiveResolutions} \linebreak
\noindent\hyperlink{injective_spectra}{$E$-Injective spectra}\dotfill \pageref*{injective_spectra} \linebreak
\noindent\hyperlink{adams_resolutions}{$E$-Adams resolutions}\dotfill \pageref*{adams_resolutions} \linebreak
\noindent\hyperlink{adams_towers}{$E$-Adams towers}\dotfill \pageref*{adams_towers} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{itemize}%
\item \hyperlink{ClassicalDefinition}{Classical definition}


\item \hyperlink{ViaInjectiveResolutions}{Via injective resolutions}



\end{itemize}
\hypertarget{ClassicalDefinition}{}\subsubsection*{{Classical definition}}\label{ClassicalDefinition}

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

\begin{displaymath}
\itexarray{
    \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
  }
\end{displaymath}
where

\begin{itemize}%
\item each $K_i$ is a wedge of [[suspensions]] of $E$;


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


\item each $f_n$ is a surjection on [[cohomology]].



\end{itemize}
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

\begin{displaymath}
\itexarray{
    \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)
  }
  \,,
\end{displaymath}
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]] $A_p$.

\begin{displaymath}
\itexarray{
    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  
  }
\end{displaymath}
The computaton of the [[cohomology]] of $X$ by means of this resolution is given by the [[Adams spectral sequence]].

\hypertarget{ViaInjectiveResolutions}{}\subsubsection*{{Via injective resolutions}}\label{ViaInjectiveResolutions}

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

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

\begin{displaymath}
[-,-] \;\colon\; HoSpectra^{op} \times HoSpectra \longrightarrow Ab
\end{displaymath}
for the [[hom-functor]] with values in [[abelian groups]].

\begin{defn}
\label{HomotopyFunctor}\hypertarget{HomotopyFunctor}{}
For $S \in HoSpectra$, the \emph{homotopy functor it represents} is the [[representable functor]]

\begin{displaymath}
[S,-] \;\colon\; HoSpectra \longrightarrow Ab
\end{displaymath}
(as opposed to the other, contravariant, functor).

\end{defn}
\begin{example}
\label{}\hypertarget{}{}
For $S = \Sigma^\infty S^n \simeq \Sigma^n \mathbb{N}$ then

\begin{displaymath}
[\Sigma^\infty S^n ,- ]\simeq \pi_n
\end{displaymath}
is the $n$th [[homotopy group]]-functor.

\end{example}
Throughout, let $E$ be a [[ring spectrum]].

\hypertarget{injective_spectra}{}\paragraph*{{$E$-Injective spectra}}\label{injective_spectra}

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

\begin{defn}
\label{ExactSequences}\hypertarget{ExactSequences}{}
Say that

\begin{enumerate}%
\item a sequence of spectra

\begin{displaymath}
A_1 \longrightarrow A_2 \longrightarrow \cdots \longrightarrow A_n
\end{displaymath}
is

\begin{enumerate}%
\item a (long) \emph{exact sequence} if the induced sequence of homotopy functors, def. \ref{HomotopyFunctor}, is a [[long exact sequence]] in $[HoSpectra,Ab]$;


\item (for $n = 2$) a \emph{short exact sequence} if

\begin{displaymath}
0 \longrightarrow A_1 \longrightarrow A_2 \longrightarrow A_3 \longrightarrow 0
\end{displaymath}
is (long) exact;



\end{enumerate}

\item a morphism $A \longrightarrow B$ is

\begin{enumerate}%
\item a \emph{monomorphism} if $0 \longrightarrow A \longrightarrow B$ is an exact sequence;


\item an \emph{epimorphism} if $A \longrightarrow B \longrightarrow 0$ is an exact sequence.



\end{enumerate}


\end{enumerate}
For $E$ a [[ring spectrum]], then a sequence of spectra is (long/short) \emph{$E$-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 $E$.

\end{defn}
\begin{example}
\label{}\hypertarget{}{}
Every [[homotopy cofiber sequence]] of spectra is exact in the sense of def. \ref{ExactSequences}.

\end{example}
\begin{remark}
\label{}\hypertarget{}{}
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}.

\end{remark}
\begin{lemma}
\label{}\hypertarget{}{}
\begin{enumerate}%
\item 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]]

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

\item 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]]

\begin{displaymath}
s \vee f \colon B\vee C \stackrel{\simeq}{\longrightarrow} A
  \,.
\end{displaymath}


\end{enumerate}
\end{lemma}
\begin{defn}
\label{EInjective}\hypertarget{EInjective}{}
For $E$ a [[ring spectrum]], say that a spectrum $S$ is \emph{$E$-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

\begin{displaymath}
\itexarray{
    A &\longrightarrow & S 
    \\
    \downarrow & \nearrow_{\mathrlap{\exists}}
    \\
    B
  }
  \,.
\end{displaymath}
\end{defn}
\begin{lemma}
\label{}\hypertarget{}{}
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 $E \wedge X$.

\end{lemma}
\hypertarget{adams_resolutions}{}\paragraph*{{$E$-Adams resolutions}}\label{adams_resolutions}

\begin{defn}
\label{EAdamsResolution}\hypertarget{EAdamsResolution}{}
For $E$ a [[ring spectrum]], then an \emph{$E$-Adams resolution} of an spectrum $S$ is a long exact sequence, in the sense of def. \ref{ExactSequences}, of the form

\begin{displaymath}
0 \longrightarrow S \longrightarrow I_0 \longrightarrow I_1 \longrightarrow I_2 \longrightarrow \cdots
\end{displaymath}
such that each $I_j$ is $E$-injective, def. \ref{EInjective}.

\end{defn}
\begin{lemma}
\label{}\hypertarget{}{}
Any two consecutive maps in an $E$-Adams resolution compose to the [[zero morphism]].

\end{lemma}
\begin{lemma}
\label{}\hypertarget{}{}
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]]

\begin{displaymath}
\itexarray{
     X &\longrightarrow& I_\bullet
     \\
     \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{g_\bullet}}
     \\
     Y &\longrightarrow& J_\bullet
  }
  \,.
\end{displaymath}
\end{lemma}
\begin{example}
\label{}\hypertarget{}{}
\textbf{(standard resolution)}

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

\begin{displaymath}
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
  \,.
\end{displaymath}
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}.

\end{example}
\hypertarget{adams_towers}{}\paragraph*{{$E$-Adams towers}}\label{adams_towers}

\begin{defn}
\label{EAdamsTower}\hypertarget{EAdamsTower}{}
An \emph{$E$-Adams tower} of a spectrum $X$ is a [[commuting diagram]] in [[HoSpectra]] of the form

\begin{displaymath}
\itexarray{
    && \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
  }
\end{displaymath}
such that

\begin{enumerate}%
\item each hook is a [[homotopy fiber sequence]] (hence it is a [[tower of homotopy fibers]]);


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

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

\begin{displaymath}
0 \to X \stackrel{i_0}{\to} I_0 \stackrel{i_2}{\to} I_2 \stackrel{}{\to} \cdots
  \,.
\end{displaymath}


\end{enumerate}
Call this the \emph{associated $E$-Adams resolution} of the $E$-Adams tower.

The \emph{associated inverse sequence} is

\begin{displaymath}
X = X_0 \stackrel{\gamma_0}{\longleftarrow} \Omega C_1 \stackrel{\gamma_1}{\longleftarrow} C_2 \longleftarrow \cdots
\end{displaymath}
where $C_{k+1} \coloneqq hocofib(i_k)$.

\end{defn}
(In (\hyperlink{Ravenel}{Ravenel}) it is is the associated inverse sequence that is called the associated resolution.)

=--

\begin{example}
\label{}\hypertarget{}{}
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.

\end{example}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Whitehead tower]], [[Postnikov tower]]


\item [[E-nilpotent completion]]


\item [[Adams spectral sequence]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Reviews include

\begin{itemize}%
\item [[Doug Ravenel]], around Chapter 2, def. 2.1.3 of \emph{[[Complex cobordism and stable homotopy groups of spheres]]}


\item [[Stanley Kochmann]], section 3.6 of \emph{[[Bordism, Stable Homotopy and Adams Spectral Sequences]]}, AMS 1996



\end{itemize}
A streamlined presentation close in spirit to constructions in [[homological algebra]] was given in

\begin{itemize}%
\item [[Haynes Miller]], \emph{On relations between Adams spectral sequences, with an application to the stable homotopy of a Moore space}, J. Pure Appl. Algebra 20 (1981) (\href{http://math.mit.edu/~hrm/papers/miller-relations-between-adams-spectral-sequences.pdf}{pdf})

\end{itemize}
and is reproduced and expanded on in

\begin{itemize}%
\item [[Mike Hopkins]], section 5 of \emph{Complex oriented cohomology theories and the language of stacks}, course notes 1999 (\href{http://www.math.rochester.edu/u/faculty/doug/otherpapers/coctalos.pdf}{pdf})


\item [[Nersés Aramian]], \emph{The Adams spectral sequence} ([[AramianANSS.pdf:file]])



\end{itemize}
[[!redirects Adams resolutions]]

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



\end{document}