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\begin{document}

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\section*{multiplicative spectral sequence}

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

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

[[!include homological algebra - contents]]

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

[[!include higher algebra - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{constructions}{Constructions}\dotfill \pageref*{constructions} \linebreak
\noindent\hyperlink{from_spectral_products_on_cartaneilenberg_systems}{From spectral products on Cartan-Eilenberg systems}\dotfill \pageref*{from_spectral_products_on_cartaneilenberg_systems} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{AHSSForMultiplicativeCohomology}{AHSS for multiplicative cohomology}\dotfill \pageref*{AHSSForMultiplicativeCohomology} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A [[spectral sequence]] is called \emph{multiplicative} or a \emph{spectral ring} if there is a bi-[[graded algebra]] structure on each page such that the differentials act as graded [[derivations]] of total degree 1.

For example the [[Serre spectral sequence|Serre]]-[[Atiyah-Hirzebruch spectral sequence]] with [[coefficients]] in a [[ring spectrum]].

\hypertarget{constructions}{}\subsection*{{Constructions}}\label{constructions}

\hypertarget{from_spectral_products_on_cartaneilenberg_systems}{}\subsubsection*{{From spectral products on Cartan-Eilenberg systems}}\label{from_spectral_products_on_cartaneilenberg_systems}

The following gives sufficient conditions for a [[Cartan-Eilenberg spectral sequence]] to be multiplicative. This is due to (\hyperlink{Douady58}{Douady 58}). The following is taken from (\hyperlink{Goette15}{Goette 15a}).

\begin{defn}
\label{SpectralProduct}\hypertarget{SpectralProduct}{}
Let $(H,\eta,\partial)$, $(H',\eta',\partial')$ und $(H'',\eta'',\partial'')$ be [[Cartan-Eilenberg systems]]. A \textbf{spectral product} $\mu\colon(H',\partial')\times(H'',\partial'')\to(H,\partial)$ is a sequence of [[homomorphisms]]

\begin{displaymath}
\mu_r\colon H'(m,m+r)\otimes H''(n,n+r)\to H(m+n,m+n+r)
\end{displaymath}
such that for all $m$, $n$, $r\ge 1$, the following two [[commuting diagram|diagrams commute]]:

\begin{displaymath}
\itexarray{
    H'(m, m+r) \otimes H''(n, n+r)
      &\stackrel{\mu_r}{\longrightarrow}&
    H(m+n, m+n+r)
    \\
    \downarrow^{\mathrlap{\eta' \oplus \eta''}} && \downarrow^{\mathrlap{\eta}}
    \\
    H'(m, m+1) \otimes H''(n, n+1)
     &\stackrel{\mu_1}{\longrightarrow}&
    H(m+n, m+n+1)
  }
\end{displaymath}
and

\begin{displaymath}
\itexarray{
    H'(m, m+r) \otimes H''(n, n+r) &\stackrel{\mu_r}{\longrightarrow}& H(m+n,m+n+r)
    \\
    \downarrow^{\mathrlap{\partial' \otimes \eta'' \oplus \eta' \otimes \partial''}}
    && 
    \downarrow^{\mathrlap{\partial}}
    \\
    H'(m+r, m+r+1) \otimes H''(n,n+1) 
    \\
    \oplus     &\stackrel{\mu_1 + \mu_1}{\longrightarrow}& 
    H_{p+q-1}(m+n+r, m+n+r+1)
    \\
    H'(m,m+1) \otimes H''(n+r, n+r+1)
  }
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
The first diagram in def. \ref{SpectralProduct} is weaker than in (\hyperlink{Douady58}{Douady 58}). The second may be read as a [[Leibniz rule]].

\end{remark}
Write $E$ for the [[Cartan-Eilenberg spectral sequence]] induced from the Cartan-Eilenberg system $H$.

\begin{prop}
\label{MultiplicativeStructureFromSpectralProduct}\hypertarget{MultiplicativeStructureFromSpectralProduct}{}
A spectral product $\mu\colon(H',\partial')\times(H'',\partial'')\to(H,\partial)$ as in def. \ref{SpectralProduct} induces products

\begin{displaymath}
\mu^r\colon E^{\prime r}_m\otimes E^{\prime\prime r}_n\to E^r_{m+n}\;,
\end{displaymath}
such that

\begin{enumerate}%
\item $\mu^1=\mu_1$


\item $d^r_{m+n}\circ\mu^r=\mu^r\circ(d^{\prime r}_m\otimes\mathrm{id})\pm\mu^r\circ(\mathrm{id}\circ d^{\prime\prime r}_n)$,


\item $\mu^{r+1}$ is induced by $\mu^r$.



\end{enumerate}
\end{prop}
(\hyperlink{Goette15}{Goette 15a}, following \hyperlink{Douady58}{Douady 58, theorem II}).

\begin{proof}
Assume by [[induction]] that $\mu^r$ is induced by $\mu_1$. In particular,

\begin{displaymath}
Z^{\prime r}_m\otimes Z^{\prime\prime r}_n\stackrel{\mu_1}\to Z^r_{m+n}\;,
\end{displaymath}
\begin{displaymath}
B^{\prime r}_m\otimes Z^{\prime\prime r}_n\stackrel{\mu_1}\to B^r_{m+n}\;,
\end{displaymath}
\begin{displaymath}
Z^{\prime r}_m\otimes B^{\prime\prime r}_n\stackrel{\mu_1}\to B^r_{m+n}\;.
\end{displaymath}
This is clear for $r=1$ if we put $\mu^1=\mu_1$ because $E^1_p=Z^1_p=H(p,p+1)$ and $B^1_p=0$.

Let $[a]\in Z^{\prime r}_m$, $[b]\in Z^{\prime\prime r}_n$ be represented by $a=\eta'(a_0)\in H'(m,m+1)$, $b=\eta''(b_0)\in H''(n,n+1)$ with $a_0\in H'(m,m+r)$, $b_0\in H''(n,n+r)$. Using the first diagram and the construction of $d^r_{m+n}$, we conclude that

\begin{displaymath}
(d^r_{m+n}\circ\mu^r)([a]\otimes[b])=d^r_{m+n}[\mu_1(a\otimes b)]=d^r_{m+n}[\eta(\mu_r(a_0\otimes b_0))]=(\partial\circ\mu_r)(a_0\otimes b_0)
 \;.
\end{displaymath}
From the second diagram, we get

\begin{displaymath}
(\partial\circ\mu_r)(a_0\otimes b_0)=\mu_1(\partial'a_0\otimes\eta''b_0)\pm\mu_1(\eta'a_0\otimes\partial''b_0)=\mu^r(d^{\prime r}_m[a]\otimes[b])\pm\mu^r([a]\otimes d^{\prime\prime r}_n[b])
\;.
\end{displaymath}
This proves the Leibniz rule (2).

From the Leibniz rule and the facts that $\ker(d^r_p)=Z^{r+1}_p/B^r_p$ and $\mathrm{im}(d^r_p)=B^{r+1}_p/B^r_p$, we conclude that $\mu^r$ induces a product on $E^{r+1}_p\cong\ker(d^r_p)/\mathrm{im}(d^r_p)$, which proves (3). Because $\mu^r$ is induced by $\mu_1$, so is $\mu^{r+1}$, and we can continue the induction.

\end{proof}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{AHSSForMultiplicativeCohomology}{}\subsubsection*{{AHSS for multiplicative cohomology}}\label{AHSSForMultiplicativeCohomology}

We discuss that the multiplicative structure on the cohomology [[Serre spectral sequence|Serre]]-[[Atiyah-Hirzebruch spectral sequence]] for [[multiplicative cohomology theory|multiplicative]] [[generalized cohomology]]. This is taken from (\hyperlink{Goette15}{Goette 15b}).

\begin{defn}
\label{CESystemForMultiplicativeGeneralizedCohomology}\hypertarget{CESystemForMultiplicativeGeneralizedCohomology}{}
For $\pi\colon X\to B$ a [[Serre fibration]] over a [[CW-complex]] $B$. And for $(\tilde h^\bullet,\delta,\wedge)$ a [[multiplicative cohomology theory|multiplicative]] [[reduced cohomology|reduced]] [[generalized (Eilenberg-Steenrod) cohomology]] theory, define a [[Cartan-Eilenberg system]] $(H,\eta,\partial)$ by

\begin{displaymath}
H(p,q)=\tilde h^\bullet(X^{q-1}/X^{p-1})
\end{displaymath}
(where $X^k=\pi^{-1}(B^k)$) for $p\le q$ with the obvious maps $\eta\colon H(p',q')\to H(p,q)$ for $p\le p'$, $q\le q'$.

The [[Cartan-Eilenberg spectral sequence]] of this Cartan-Eilenberg system is the [[Serre spectral sequence|Serre]]-[[Atiyah-Hirzebruch spectral sequence]].

\end{defn}
\begin{defn}
\label{SpectralProductForAHSS}\hypertarget{SpectralProductForAHSS}{}
The spectral product $\mu\colon(H,\eta,\partial)\times(H,\eta,\partial)\to(H,\eta,\partial)$, def. \ref{SpectralProduct}, on the Cartan-Eilenberg system of def. \ref{CESystemForMultiplicativeGeneralizedCohomology} is that given by the following morphism

\begin{displaymath}
\begin{aligned}
  F_{m,n,r}
    & \colon
    (X\wedge X)^{m+n+r-1}/(X\wedge X)^{m+n-1}
    \cong
    \bigcup_{a+b=m+n+r-1}(X^a\wedge X^b) / \bigcup_{c+d=m+n-1}(X^c\wedge X^d)
    \\
    & 
    \twoheadrightarrow
    \bigcup_{a+b=m+n+r-1}
     (X^a\wedge X^b)/(\bigcup_{a=0}^m(X^{a-1}\wedge X^{m+n+r-a})
    \cup\bigcup_{b=0}^n(X^{m+n+r-b}\wedge X^{b-1})
    \\
    & 
    \cong \bigcup_{a=m+1}^{m+r}(X^{a-1}\wedge X^{m+n+r-a})
    /
    \bigl(X^{m+r-1}\wedge X^{n-1}\cup X^{m-1}\wedge X^{n+r-1}\bigr)
    \\
    & \hookrightarrow  
    X^{m+r-1}\wedge X^{n+r-1}/(X^{m+r-1}\wedge X^{n-1}\cup X^{m-1}\wedge X^{n+r-1})
    \\
    & \cong (X^{m+r-1}/X^{m-1})\wedge(X^{n+r-1}/X^{n-1})
  \;.
  \end{aligned}
\end{displaymath}
Together with the [[diagonal]] map $\Delta$, for $r\ge 1$, we define

\begin{displaymath}
\begin{aligned}
    \mu_r 
     & \colon 
    H(m,m+r)\otimes H(n,n+r)
    \\
    & \cong\tilde h(X^{m+r-1}/X^{m-1})\otimes\tilde h(X^{n+r-1}/X^{n-1})
    \\
    &\stackrel\wedge\longrightarrow\tilde h\bigl((X^{m+r-1}/X^{m-1})\wedge(X^{n+r-1}/X^{n-1})\bigr)
    \\
    &\stackrel{F_{m,n,r}^*}\longrightarrow\tilde h\bigl((X\wedge X)^{m+n+r-1}/(X\wedge X)^{m+n-1}\bigr)
    \\
    &\stackrel{\Delta_X^*}\longrightarrow\tilde h(X^{m+n+r-1}/X^{m+n-1})=H(m+n,m+n+r)\;.
\end{aligned}
\end{displaymath}
\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
With def. \ref{SpectralProductForAHSS}, then for all $m$, $n$, $r\ge 1$, the following [[commuting diagram|diagram commutes]]

\begin{displaymath}
\itexarray{
    H(m,m+1) \otimes H(n,n+1) &\stackrel{\mu_1}{\longrightarrow}& H(m+n, m+n+1)
    \\
    \uparrow^{\mathrlap{\eta \oplus \eta}}
    &&
    \uparrow^{\mathrlap{\eta}}
    \\
    H(m,m+r) \otimes H(n,n+r) &\stackrel{\mu_r}{\longrightarrow}& H(m+n,m+n+r)
    \\
    \downarrow^{\mathrlap{\partial \otimes \eta \oplus \eta \otimes \partial}}
    && 
    \downarrow^{\mathrlap{\partial}}
    \\
    H(m+r, m+r+1) \otimes H(n,n+1)
    \\
    \oplus &\stackrel{\mu_1 \pm \mu_1}{\longrightarrow}& 
       H(m+n+r, m+n+r+1)
    \\
    H(m,m+1) \otimes H(n+r, n+r+1)
  }
  \,.
\end{displaymath}
Hence by prop. \ref{MultiplicativeStructureFromSpectralProduct} the spectral product of def. \ref{SpectralProductForAHSS} defines a mutliplicative structure on the Serre-WhiteheadAtiyah-Hirzebruch spectral sequence for multiplicative generalizted cohomology.

\end{prop}
\begin{proof}
The upper square commutes because the maps $F_{m,n,r}$ are [[natural transformations]]. For the lower square, we consider the boundary morphism $\delta$ of the triple

\begin{displaymath}
\begin{aligned}
    (&
     X^{m+r}\wedge X^{n+r-1}\cup X^{m+r-1}\wedge X^{n+r},
    \\
    &
    X^{m+r}\wedge X^{n-1}\cup X^{m+r-1}\wedge X^{n+r-1}\cup X^{m-1}\wedge X^{n+r},
    \\
    & X^{m+r}\wedge X^{n-1}\cup X^{m-1}\wedge X^{n+r})
  \end{aligned}
  \;.
\end{displaymath}
The following diagram commutes:

\begin{displaymath}
\itexarray{
    \tilde h^{-p}(X^{m+r-1}/ X^{m-1})
    \otimes
    \tilde h^{-q}(X^{n+r-1}/X^{n-1})
    &\stackrel{\wedge}{\longrightarrow}&
    \tilde h^{-p-q}((X^{m+r-1}/X^{m-1}) \wedge (X^{n+r-1}/X^{n-1})
    \\
    \downarrow^{\mathrlap{\delta \wedge id \oplus id \wedge \delta}}
    &&
    \downarrow^{\mathrlap{\delta}}
    \\
    \tilde h(1-p)(X^{m+r}/X^{m+r-1}) \otimes \tilde h^{-q}(X^{n+r-1}/X^{n-1})
    &&
    \tilde h^{1-p-q}((X^{m+r}/X^{m+r-1}) \wedge (X^{n+r-1}/X^{n-1}))
    \\
    \oplus
    &\stackrel{\wedge \oplus \wedge}{\longrightarrow}&
    \\
    \tilde h^{-p}(X^{m+r-1}/X^{m-1}) \otimes \tilde h^{1-q}(X^{n+r}/X^{n+r-1})
    &&
    \tilde h^{1-p-q}( (X^{m+r-1}/ X^{m-1}) \wedge (X^{n+r}/ X^{n+r-1}) )
  }
  \,.
\end{displaymath}
By extend this diagram to the right using the maps $F_{m,n,r}$ once concludes that the lower square above also commutes.

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

\begin{itemize}%
\item [[multiplicative cohomology theory]]

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

\begin{itemize}%
\item [[Adrien Douady]], \emph{La suite spectrale d'Adams : structure multiplicative} S\'e{}minaire Henri Cartan, 11 no. 2 (1958-1959), Exp. No. 19, 13 p (\href{http://www.numdam.org/item?id=SHC_1958-1959__11_2_A10_0}{Numdam})


\item Brayton Gray, \emph{Products in the Atiyah-Hirzebruch spectral sequence and the calculation of $M SO_\ast$, Trans. Amer. Math. Soc. 260 (1980), 475-483 (\href{http://www.ams.org/journals/tran/1980-260-02/S0002-9947-1980-0574793-9/}{web})}


\item [[Stanley Kochmann]], prop. 4.2.9 of \emph{[[Bordism, Stable Homotopy and Adams Spectral Sequences]]}, AMS 1996


\item [[John McCleary]], section 2.3 in \emph{A User's Guide to Spectral Sequences}, Cambridge University Press (2000)


\item [[Daniel Dugger]], \emph{Multiplicative structures on homotopy spectral sequences I} (\href{http://de.arxiv.org/abs/math/0305173}{arXiv:math/0305173})


\item [[Daniel Dugger]], \emph{Multiplicative structures on homotopy spectral sequences II} (\href{http://de.arxiv.org/abs/math/0305187}{arXiv:math/0305187})


\item [[Sebastian Goette]], \href{http://mathoverflow.net/a/231235/381}{MO comment a}, \href{http://mathoverflow.net/a/231236/381}{MO comment b} Feb 15, 2015



\end{itemize}
[[!redirects multiplicative spectral sequences]]



\end{document}