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\begin{document}

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\section*{Steenrod algebra}

\begin{quote}%
under construction


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

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{for_ordinary_cohomology}{For ordinary cohomology}\dotfill \pageref*{for_ordinary_cohomology} \linebreak
\noindent\hyperlink{for_generalized_cohomology}{For generalized cohomology}\dotfill \pageref*{for_generalized_cohomology} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{presentation_by_generators_and_relations}{Presentation by generators and relations}\dotfill \pageref*{presentation_by_generators_and_relations} \linebreak
\noindent\hyperlink{dual_steenrod_algebra}{Dual Steenrod algebra}\dotfill \pageref*{dual_steenrod_algebra} \linebreak
\noindent\hyperlink{MilnorTheorem}{Milnor's theorem}\dotfill \pageref*{MilnorTheorem} \linebreak
\noindent\hyperlink{HopfAlgebraStructure}{Hopf algebroid structure}\dotfill \pageref*{HopfAlgebraStructure} \linebreak
\noindent\hyperlink{for_ordinary_cohomology_2}{For ordinary cohomology}\dotfill \pageref*{for_ordinary_cohomology_2} \linebreak
\noindent\hyperlink{HopfAlgebroidstructureForGeneralized}{For generalized cohomology}\dotfill \pageref*{HopfAlgebroidstructureForGeneralized} \linebreak
\noindent\hyperlink{relation_to_adams_spectral_sequence}{Relation to Adams spectral sequence}\dotfill \pageref*{relation_to_adams_spectral_sequence} \linebreak
\noindent\hyperlink{for_ordinary_cohomology_3}{For ordinary cohomology}\dotfill \pageref*{for_ordinary_cohomology_3} \linebreak
\noindent\hyperlink{for_generalized_cohomology_3}{For generalized cohomology}\dotfill \pageref*{for_generalized_cohomology_3} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{ReferencesHopfAlgebroid}{$E_\bullet$-Hopf algebroid structure}\dotfill \pageref*{ReferencesHopfAlgebroid} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

For $p$ a [[prime number]], the \emph{Steenrod algebra} $\mathcal{A}$ is the [[associative algebra]] over the [[prime field]] $\mathbb{F}_p$ of [[cohomology operations]] on [[ordinary cohomology]] with [[coefficients]] in $\mathbb{F}_p$. For $p = 2$ this is the $\mathbb{F}_2$-algebra generated by the [[Steenrod square]] operations. (Often the case $p = 2$ is understood by default.)

For any prime $p$, the mod-$p$ Steenrod algebra furthermore has the structure of a [[Hopf algebra]] over $\mathbb{F}_p$, non-commutative but co-commutative. Under forming [[linear duals]] this gives the \emph{dual Steenrod algebra}, traditionally denoted $\mathcal{A}_\ast$ instead of $\mathcal{A}^\ast$. This is a [[Hopf algebra]] over $\mathbb{F}_p$ that is commutative, but non-co-commutative.

The dual $\mathbb{F}_p$ Steenrod algebra is a special case of a [[commutative Hopf algebroid]] structure canonically induced on the self [[generalized homology]] $E_\bullet(E)$ of any [[ring spectrum]] $E$ for which $E_\bullet \to E_\bullet(E)$ is a [[flat morphisms]]. Therefore in this general case one sometimes speaks of ``dual $E$-Steenrod algebras''.

The [[Ext]]-groups between [[comodules]] for these [[commutative Hopf algebroids]] $E_\bullet(E)$ prominently appear on the second page of the $E$-[[Adams spectral sequence]], see there for more.

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

\hypertarget{for_ordinary_cohomology}{}\subsubsection*{{For ordinary cohomology}}\label{for_ordinary_cohomology}

(\ldots{}) (e.g. \hyperlink{MosherTangora68}{Mosher-Tangora 68, section 6}, \hyperlink{Lurie07}{Lurie 07}) (\ldots{})

\hypertarget{for_generalized_cohomology}{}\subsubsection*{{For generalized cohomology}}\label{for_generalized_cohomology}

The Steenrod algebra and its standard properties, such as the \hyperlink{OnAdemRelations}{Adem relations}, follow abstractly from the [[Cotor]] groups of [[comodules]] over any [[commutative Hopf algebroid]].

This is due to (\hyperlink{May70}{May 70, 11.8}). A review is in (\hyperlink{Ravenel}{Ravenel, appendix 1, theorem A1.5.2}).

In particular for $E$ a suitable [[E-infinity ring]], its self-[[generalized homology]] $E_\bullet(E)$ form a (graded-)[[commutative Hopf algebroid]] over $E_\bullet$.

See at \emph{\hyperlink{HopfAlgebroidstructureForGeneralized}{Hopf algebroid structure -- For generalized cohomology} below.}

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{presentation_by_generators_and_relations}{}\subsubsection*{{Presentation by generators and relations}}\label{presentation_by_generators_and_relations}

\begin{defn}
\label{SerreCartanBasis}\hypertarget{SerreCartanBasis}{}
The \emph{Serre-Cartan basis} is the [[subset]] of elements of the Steenrod algebra on those of the form

\begin{displaymath}
Sq^{i_1} \circ Sq^{i_2} \circ \cdots \circ Sq^{i_n}
\end{displaymath}
where $i_k \in \mathbb{N}$ subject to the relation

\begin{displaymath}
i_k \geq 2 i_{k+1}.
\end{displaymath}
\end{defn}
This is indeed a [[linear basis]] for the $\mathbb{F}_2$-[[vector space]] underlying the Steenrod algebra.

\begin{prop}
\label{AdemRelations}\hypertarget{AdemRelations}{}
The [[Steenrod square]] operations satisfy the following relation, for all for all $0 \lt i \lt 2 j$:

\begin{displaymath}
Sq^i \circ Sq^j 
   = 
  \sum_{k = 0}^{[i/2]}
  \left(
    \itexarray{
      j - k - 1
      \\
      i - 2k 
    }
  \right)_{mod 2}
  Sq^{i + j -k} \circ Sq^k
\end{displaymath}
\end{prop}
These are called the \textbf{\'A{}dem relations} (\hyperlink{Adem52}{\'A{}dem 52}).

\begin{prop}
\label{}\hypertarget{}{}
The \'A{}dam relations precisely generate the [[ideal]] of relations among the Serre-Cartan basis elements, def. \ref{AdemRelations}, in the Steenrod algebra.

\end{prop}
More generally, for other [[prime numbers]]:

\begin{defn}
\label{SteenrodAlgebraForHFp}\hypertarget{SteenrodAlgebraForHFp}{}
Let $p$ be a [[prime number]]. Write $\mathbb{F}_p$ for the corresponding [[prime field]].

The \textbf{mod $p$-Steenrod algebra} $\mathcal{A}_{\mathbb{F}_p}$ is the graded co-commutative [[Hopf algebra]] over $\mathbb{F}_p$ which is

\begin{itemize}%
\item for $p = 2$ generated by elements denoted $Sq^n$ for $n \in \mathbb{N}$, $n \geq 1$;


\item for $p \gt 2$ generated by elements denoted $\beta$ and $P^n$ for $n \in \mathbb{N}$, $n \geq 1$



\end{itemize}
(called the \textbf{Serre-Cartan basis elements}) whose product is subject to the following relations (called the \textbf{\'A{}dem relations}):

\textbf{for $p = 2$}:

for $0 \lt h \lt 2k$ the

\begin{displaymath}
Sq^h Sq^k
  \;=\;
  \underoverset{i = 0}{[h/2]}{\sum}
  \left(
    \itexarray{
      k -i - 1
      \\
      h - 2i
    }    
  \right)
  Sq^{h + k -i} Sq^i
  \,,
\end{displaymath}
\textbf{for $p \gt 2$}:

for $0 \lt h \lt p k$ then

\begin{displaymath}
P^h P^k
  \;=\;
  \underoverset{i = 0}{[h/p]}{\sum}
  (-1)^{h+i}
  \left(
    \itexarray{
      (p-1)(k-i) - 1
      \\
      h - p i 
    }
  \right)
  P^{h +k - i}P^i
\end{displaymath}
and if $0 \lt h \lt p k$ then

\begin{displaymath}
\begin{aligned}
    P^h \beta P^k
    & =\;
    \underoverset{[h/p]}{i = 0}{\sum}
    (-1)^{h+i}
    \left(
      \itexarray{
        (p-1)(k-i)
        \\
        h - p i
      }
    \right)
    \beta P^{h+k-i}P^i
    \\
    & + 
    \underoverset{[(h-1)/p]}{i = 0}{\sum}
    (-1)^{h+i-1}
    \left(
      \itexarray{
        (p-1)(k-i) - 1
        \\
        h - p i - 1
      }
    \right)
    P^{h+k-i} \beta P^i
  \end{aligned}
\end{displaymath}
and whose coproduct $\Psi$ is subject to the following relations:

\textbf{for $p = 2$}:

\begin{displaymath}
\Psi(Sq^n)
  \;=\;
  \underoverset{k = 0}{n}{\sum} Sq^k \otimes Sq^{n-k}
\end{displaymath}
\textbf{for $p \gt 2$}:

\begin{displaymath}
\Psi(P^n)
  \;=\;
  \underoverset{n}{k = 0}{\sum}
  P^k \otimes P^{n-k}
\end{displaymath}
and

\begin{displaymath}
\Psi(\beta) = \beta \otimes 1 + 1 \otimes \beta
  \,.
\end{displaymath}
\end{defn}
e.g. (\hyperlink{Kochmann96}{Kochmann 96, p. 52})

\hypertarget{dual_steenrod_algebra}{}\subsubsection*{{Dual Steenrod algebra}}\label{dual_steenrod_algebra}

\begin{defn}
\label{DualSteenrodAlgebraForHPf}\hypertarget{DualSteenrodAlgebraForHPf}{}
The $\mathbb{F}_p$-[[linear dual]] of the mod $p$-Steenrod algebra (def. \ref{SteenrodAlgebraForHFp}) is itself naturally a graded [[commutative Hopf algebroid|commutative Hopf algebra]] (with coproduct the linear dual of the original product, and vice versa), called the \textbf{dual Steenrod algebra} $\mathbb{A}_{\mathbb{F}_p}^\ast$.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
There is an isomorphism

\begin{displaymath}
\mathcal{A}^\ast_{\mathbb{F}_p}
  \simeq
  H_\bullet( H \mathbb{F}_p, \mathbb{F}_p )
  = 
  \pi_\bullet( H \mathbb{F}_p \wedge H \mathbb{F}_p )
  \,.
\end{displaymath}
\end{remark}
(e.g. \hyperlink{Rognes12}{Rognes 12, remark 7.24})

\hypertarget{MilnorTheorem}{}\paragraph*{{Milnor's theorem}}\label{MilnorTheorem}

We now give the generators-and-relations description of the dual Steenrod algebra $\mathcal{A}^\ast_{\mathbb{F}_p}$ from def. \ref{DualSteenrodAlgebraForHPf}, in terms of linear duals of the generators for $\mathcal{A}_{\mathbb{F}_p}$ itself, according to def. \ref{SteenrodAlgebraForHFp}.

In the following, we use for $p = 2$ the notation

\begin{displaymath}
P^n \coloneqq Sq^{2n}
\end{displaymath}
\begin{displaymath}
\beta \coloneqq Sq^1
  \,.
\end{displaymath}
This serves to unify the expressions for $p = 2$ and for $p \gt 2$ in the following. Notice that for all $p$

\begin{itemize}%
\item $P^n$ has even degree $deg(P^n) = 2n(p-1)$;


\item $\beta$ has odd degree $deg(\beta) = 1$.



\end{itemize}
\begin{theorem}
\label{MilnorTheoremOnDualSteenrodAlgebra}\hypertarget{MilnorTheoremOnDualSteenrodAlgebra}{}
\textbf{([[Milnor's theorem]])}

The dual mod $p$-Steenrod algebra $\mathcal{A}^\ast_{\mathbb{F}_p}$ (def. \ref{DualSteenrodAlgebraForHPf}) is, as an [[associative algebra]], the free [[graded commutative algebra]]

\begin{displaymath}
\mathcal{A}^\ast_{\mathbb{F}_p}
    \simeq
  Sym_{\mathbb{F}_p}(\xi_1, \xi_2, \cdots, \;\tau_0, \tau_1, \cdots)
\end{displaymath}
on generators:

\begin{itemize}%
\item $\xi_n$ being the linear dual to $P^{p^{n-1}} P^{p^{n-2}} \cdots P^p P^1$;


\item $\tau_n$ being linear dual to $P^{p^{n-1}} P^{p^{n-2}} \cdots P^p P^1\beta$.



\end{itemize}
Moreover, the coproduct on $\mathcal{A}^\ast_{\mathbb{F}_p}$ is given by

\begin{displaymath}
\Psi(\xi_n)
  =
  \underoverset{k = 0}{n}{\sum} \xi_{n-k}^{p^k} \otimes \xi_k
\end{displaymath}
and

\begin{displaymath}
\Psi(\tau_n)
  =
  \tau_n \otimes 1
  + 
  \underoverset{k=0}{n}{\sum}
  \xi_{n-k}^{p^k} \xi_{n-k}^{p^k}\otimes \tau_k
  \,,
\end{displaymath}
where we set $\xi_0 \coloneqq 1$.

\end{theorem}
This is due to (\hyperlink{Milnor58}{Milnor 58}). See for instance (\hyperlink{Kochmann96}{Kochmann 96, theorem 2.5.1})

\hypertarget{HopfAlgebraStructure}{}\subsubsection*{{Hopf algebroid structure}}\label{HopfAlgebraStructure}

\hypertarget{for_ordinary_cohomology_2}{}\paragraph*{{For ordinary cohomology}}\label{for_ordinary_cohomology_2}

The Steenrod algebra for mod $p$ coefficients is a [[Hopf algebra]] over $\mathbb{F}_p$ which is graded commutative and non-co-commutative.

This is due to (\hyperlink{Milnor58}{Milnor 58}). A review is in \hyperlink{Ravenel}{Ravenel, ch. 3, section 1}.

\hypertarget{HopfAlgebroidstructureForGeneralized}{}\paragraph*{{For generalized cohomology}}\label{HopfAlgebroidstructureForGeneralized}

More generally:

\begin{lemma}
\label{SelfHomologyIsModuleOverCohomologyRing}\hypertarget{SelfHomologyIsModuleOverCohomologyRing}{}
Let $R$ be an [[E-∞ ring]] and let $A$ an [[E-∞ algebra]] over $R$. The self-[[generalized homology]] $A^R_\bullet(A)$ is naturally a [[module]] over the [[cohomology ring]] $A_\bullet$ via applying the [[homotopy groups]] $\infty$-functor $\pi_\bullet$ to the canonical inclusion

\begin{displaymath}
A \stackrel{\simeq}{\rightarrow}
  A \underset{R}{\wedge} R
  \stackrel{}{\rightarrow}
  A \underset{R}{\wedge} A
  \,.
\end{displaymath}
\end{lemma}
\begin{prop}
\label{}\hypertarget{}{}
Let $R$ be an [[E-∞ ring]] and let $A$ an [[E-∞ algebra]] over $R$. If the the $A_\bullet$-[[module]] $A^R_\bullet(A)$ of lemma \ref{SelfHomologyIsModuleOverCohomologyRing} is a [[flat module]], then

\begin{enumerate}%
\item $(A_\bullet, A_\bullet(A))$ is a [[commutative Hopf algebroid]] over $R_\bullet$;


\item $A^R_\bullet(X)$ is a left $A^R_\bullet(A)$-module for every $R$-[[∞-module]] $X$.



\end{enumerate}
\end{prop}
This is due to (\hyperlink{BakerLazarev01}{Baker-Lazarev 01}), further discussed in (\hyperlink{BakerJeanneret02}{Baker-Jeanneret 02}) (there expressed in terms of the presentation by [[highly structured ring spectra]]). A review is also in (\hyperlink{Ravenel}{Ravenel, chapter 2, prop. 2.2.8}).

\begin{proof}
Using the arguments of (\hyperlink{Adams74}{Adams 74}, \hyperlink{Ravenel86}{Ravenel 86}).

The [[flat module|flatness]] condition implies that there is an [[equivalence]]

\begin{displaymath}
\pi_\bullet\left(
    A \underset{R}{\wedge} A \underset{R}{\wedge} X
  \right)
  \stackrel{\simeq}{\longrightarrow}
  A_\bullet^R(A) \underset{A_\bullet}{\otimes} A_\bullet^R(X)
  \,.
\end{displaymath}
Combining this with the map in lemma \ref{SelfHomologyIsModuleOverCohomologyRing} yields the [[coaction]]

\begin{displaymath}
A_\bullet^R(X)
  \longrightarrow
  \pi_\bullet\left(
    A \underset{R}{\wedge} A \underset{R}{\wedge} X
  \right)
  \stackrel{\simeq}{\longrightarrow}
  A_\bullet^R(A) \underset{A_\bullet}{\otimes} A_\bullet^R(X)
  \,.
\end{displaymath}
\end{proof}
These (dual) $E$-Steenrod algebra Hopf algebroids have also been called ``[[brave new algebra|brave new Hopf algebroids]]'' (\hyperlink{Baker}{Baker}, \hyperlink{BakerJeanneret02}{Baker-Jeanneret 02})

\begin{example}
\label{}\hypertarget{}{}
For $E = H \mathbb{F}_2$ the [[Eilenberg-MacLane spectrum]], this reproduces the [[Hopf algebra]] structure on the dual ordinary Steenrod algebra as above.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
For $E =$ [[MU]] then by [[Quillen's theorem on MU]] $\pi_\bullet(MU) \simeq L$ is the [[Lazard ring]] and the Hopf algebroid $(\pi_\bullet(MU), MU_\bullet(MU))$ is described by the [[Landweber-Novikov theorem]].

\end{example}
\begin{example}
\label{}\hypertarget{}{}
For $E =$ [[BP]] the analog is the content of the [[Adams-Quillen theorem]]. The [[Landweber exact functor theorem]] was proven using the $B P_\bullet(B P)$-Hopf algebroid.

\end{example}
\hypertarget{relation_to_adams_spectral_sequence}{}\subsubsection*{{Relation to Adams spectral sequence}}\label{relation_to_adams_spectral_sequence}

\hypertarget{for_ordinary_cohomology_3}{}\paragraph*{{For ordinary cohomology}}\label{for_ordinary_cohomology_3}

By construction, the total [[cohomology]] $H^\bullet(X,\mathbb{Z}_2)$ of a [[topological space]] $X$, naturally is a [[module]] over the Steenrod algebra:

a cohomology element is represented by a [[cocycle]] which is a map $X \longrightarrow B^n \mathbb{Z}_2$ and the action of a [[Steenrod square]] on this is just by [[composition]].

In this form the Steenrod algebra appears in the second page of the [[Adams spectral sequence]] which computes $[\Sigma^\infty X, \Sigma^\infty Y]$ for [[topological spaces]] $X$ and $Y$: that second page is given by the [[Ext]]-groups

\begin{displaymath}
E_2 = Ext_A(H^\bullet(X, \mathbb{Z}_2), H^\bullet(Y,\mathbb{Z}_2))
\end{displaymath}
computed in the [[category of modules|category of A-modules]] for $A$ the Steenrod algebra.

\hypertarget{for_generalized_cohomology_3}{}\paragraph*{{For generalized cohomology}}\label{for_generalized_cohomology_3}

More generally, For $R$ an [[E-infinity ring]] such that its dual $R$-[[Steenrod algebra]] in the form of the self-[[homology]] $R_\bullet(R)$ is a (graded-)[[commutative Hopf algebroid]] over $R_\bullet = \pi_\bullet(R)$ (see at \href{Steenrod+algebra#HopfAlgebraStructure}{Steenrod algebra -- Hopf algebroid structure}), then the $E^2$-term of the $E$-Adams spectral sequence is an [[Ext]] of $E_\bullet(E)$-[[comodules]]

\begin{displaymath}
E^2 \simeq Ext_{R_\bullet(R)}(R_\bullet, R_\bullet(X))
  \,.
\end{displaymath}
See the \hyperlink{ReferencesHopfAlgebroid}{references below}.

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

\begin{itemize}%
\item [[cohomology operation]]


\item [[power operation]]


\item [[Pi-algebra]]


\item [[Lambda-algebra]]


\item [[Adams resolution]], [[Adams spectral sequence]], [[Adams-Novikov spectral sequence]]


\item [[Brown-Gitler spectrum]]



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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

Original articles include

\begin{itemize}%
\item [[José Ádem]], \emph{The iteration of the Steenrod squares in algebraic topology} , Proceedings of the National Academy of Sciences of the United States of America 38: 720--726 (1952)


\item [[John Milnor]], \emph{The Steenrod algebra and its dual}, Ann. of Math. 67 (1958), 150--171.


\item [[Norman Steenrod]], [[David Epstein]], \emph{Cohomology operations}, Ann. of Math. Studies, no. 50, Princeton University Press, Princeton, N. J., 1962.


\item [[John Frank Adams]], part III, section 12 of \emph{[[Stable homotopy and generalised homology]]}, University of Chicago Press (1974).



\end{itemize}
Comprehensive discussion of the ordinary Steenrod algebra, with proof is the Adem relations includes

\begin{itemize}%
\item [[Robert Mosher]], [[Martin Tangora]], \emph{Cohomology Operations and Application in Homotopy Theory}, Harper and Row (1968) (\href{www.maths.ed.ac.uk/~aar/papers/moshtang.pdf}{pdf})


\item [[Jacob Lurie]], 18.917 \emph{\href{http://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007}{Topics in Algebraic Topology: The Sullivan Conjecture, Fall 2007}}. (MIT OpenCourseWare: Massachusetts Institute of Technology), \emph{\href{http://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/lecture-notes/}{Lecture notes}}

Lecture 2 \emph{Steenrod operations} (\href{http://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/lecture-notes/lecture2.pdf}{pdf})

Lecture 3 \emph{Basic properties of Steenrod operations} (\href{http://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/lecture-notes/lecture3.pdf}{pdf})

Lecture 4 \emph{The Adem relations} (\href{http://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/lecture-notes/lecture4.pdf}{pdf})

Lecture 5 \emph{The Adem relations (cont.)} (\href{http://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/lecture-notes/lecture5.pdf}{pdf})



\end{itemize}
The general algebraic approach was laid out in

\begin{itemize}%
\item [[Peter May]], \emph{A general algebraic approach to Steenrod operations}, in \emph{The Steenrod Algebra and its Applications} (Proc. Conf. to Celebrate N. E. Steenrod's Sixtieth Birthday, Battelle Memorial Inst., Columbus, Ohio, 1970), Lecture Notes in Mathematics, Vol. 168, Springer, Berlin, 1970, pp. 153--231. (\href{http://www.math.uchicago.edu/~may/BOOKS/Algebraic.pdf}{pdf})


\item [[Robert Bruner]], [[Peter May]], J. E. McClure, M. Steinberger, \emph{$H_\infty$ ring spectra and their applications}, Lecture Notes in Math., Springer-Verlag, Berlin,



\end{itemize}
reviewed in (\hyperlink{Ravenel}{Ravenel 86, A1.5}).

Further textbook accounts include

\begin{itemize}%
\item Harvey Margolis, \emph{Spectra and the Steenrod algebra}, 1983 North-Holland


\item [[Stanley Kochmann]], section 2.5 of \emph{[[Bordism, Stable Homotopy and Adams Spectral Sequences]]}, AMS 1996


\item [[Stefan Schwede]], chapter II section 10 of \emph{[[Symmetric spectra]]}, 2012 (\href{http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf}{pdf})



\end{itemize}
Lecture notes include

\begin{itemize}%
\item [[John Rognes]], section 3 of \emph{The Adams spectral sequence}, 2012 (\href{http://folk.uio.no/rognes/papers/notes.050612.pdf}{pdf})

\end{itemize}
Reviews include

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

\end{itemize}
(appendix 1, section 5 reviews the abstract algebraic definition).

\begin{itemize}%
\item Cary Malkievich, \emph{The Steenrod Algebra} (\href{http://math.stanford.edu/~carym/steenrod.pdf}{pdf})

\end{itemize}
On the dual Steenrod algebra:

\begin{itemize}%
\item J. Palmieri, \emph{Some quotient Hopf algebras of the dual Steenrod algebra} (\href{http://www.math.washington.edu/~palmieri/Papers/quotient.pdf}{pdf})

\end{itemize}
For a commented list of furhter references see also

\begin{itemize}%
\item \emph{\href{http://www.math.uwo.ca/~sisaacso/steenrod-seminar/steenrod.xhtml}{Steenrod algebra learning seminar}}

\end{itemize}
See also

\begin{itemize}%
\item \href{http://math.berkeley.edu/~aaron/adem/}{online Adem relation calculator}

\end{itemize}
\hypertarget{ReferencesHopfAlgebroid}{}\subsubsection*{{$E_\bullet$-Hopf algebroid structure}}\label{ReferencesHopfAlgebroid}

The [[commutative Hopf algebroid]] structure on the dual $E$-Steenrod algebra $E_\bullet(E)$ and its relation to the $E^2$-term in the [[Adams spectral sequence]] is discussed in

\begin{itemize}%
\item [[Andrew Baker]], \emph{Brave new Hopf algebroids}, \href{http://www.maths.gla.ac.uk/~ajb/dvi-ps/brave-ha.pdf}{pdf}


\item [[Andrew Baker]], [[Andrey Lazarev]], \emph{On the Adams Spectral Sequence for R-modules}, Algebr. Geom. Topol. 1 (2001) 173-199 (\href{http://arxiv.org/abs/math/0105079}{arXiv:math/0105079})


\item [[Andrew Baker]] and Alain Jeanneret, \emph{Brave new Hopf algebroids and extensions of $MU$-algebras}, Homology Homotopy Appl. Volume 4, Number 1 (2002), 163-173. (\href{http://projecteuclid.org/euclid.hha/1139840059}{Euclid})


\item [[Mark Hovey]], \emph{Homotopy theory of comodules over a Hopf algebroid} (\href{http://arxiv.org/abs/math/0301229}{arXiv:math/0301229})



\end{itemize}
[[!redirects Steenrod algebras]]

[[!redirects Adem relation]] [[!redirects Ádem relation]] [[!redirects Adem relations]] [[!redirects Ádem relations]]

[[!redirects dual Steenrod algebra]] [[!redirects dual Steenrod algebras]]



\end{document}