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\begin{document}

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\section*{3x3 lemma}

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

\hypertarget{diagram_chasing_lemmas}{}\paragraph*{{Diagram chasing lemmas}}\label{diagram_chasing_lemmas}

[[!include diagram chasing lemmas - contents]]

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

[[!include homological algebra - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{statement}{Statement}\dotfill \pageref*{statement} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{in_abelian_categories}{In abelian categories}\dotfill \pageref*{in_abelian_categories} \linebreak
\noindent\hyperlink{in_nonabelian_categories}{In non-abelian categories}\dotfill \pageref*{in_nonabelian_categories} \linebreak


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

The \emph{$3 \times 3$-lemma} or \emph{nine lemma} is one of the basic [[diagram chasing lemmas]] in [[homological algebra]].

\hypertarget{statement}{}\subsection*{{Statement}}\label{statement}

\begin{lemma}
\label{}\hypertarget{}{}
Let

\begin{displaymath}
0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0
\end{displaymath}
be a [[short exact sequence]] of [[chain complexes]]. Then if two of the three complexes $A_\bullet, B_\bullet, C_\bullet$ are [[long exact sequence|exact]], so is the remaining third.

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

\begin{displaymath}
\itexarray{   
     && 0 && 0 && 0 &&
     \\
     && \downarrow && \downarrow && \downarrow && 
     \\
     0 &\to& A' &\to& B' &\to& C' &\to& 0
     \\
     && \downarrow && \downarrow && \downarrow && 
     \\
     0 &\to& A &\to& B &\to& C &\to& 0
     \\
     && \downarrow && \downarrow && \downarrow && 
     \\
     0 &\to& A'' &\to& B'' &\to& C'' &\to& 0
     \\
     && \downarrow && \downarrow && \downarrow && 
     \\
     && 0 && 0 && 0 &&
  }
\end{displaymath}
be a [[commuting diagram]] in some [[abelian category]] such that each of the three columns is an [[exact sequence]]. Then

\begin{enumerate}%
\item If the two bottom rows are exact, then so is the top.


\item If the top two rows are exact, then so is the bottom.


\item If the top and bottom rows are exact \emph{and} $A \to C$ is the [[zero morphism]], then also the middle row is exact.



\end{enumerate}
\end{lemma}
A proof by way of the [[salamander lemma]] is spelled out in detail at \emph{\href{http://ncatlab.org/nlab/show/salamander+lemma#3x3Lemmas}{Salamander lemma - Implications - 3x3 lemma}}.

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

\begin{itemize}%
\item [[salamander lemma]]


\item [[snake lemma]], [[5-lemma]]


\item [[horseshoe lemma]]



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

\hypertarget{in_abelian_categories}{}\subsubsection*{{In abelian categories}}\label{in_abelian_categories}

An early appearance of the $3 \times 3$-lemma is as lemma (5.5) in

\begin{itemize}%
\item D. A. Buchsbaum, \emph{Exact categories and duality}, Transactions of the American Mathematical Society Vol. 80, No. 1 (1955), pp. 1-34 (\href{http://www.jstor.org/stable/1993003}{JSTOR})

\end{itemize}
In

\begin{itemize}%
\item [[Charles Weibel]], \emph{[[An introduction to homological algebra]]}

\end{itemize}
it appears as exercise 1.3.2.

The sharp $3 \times 3$-lemma appears as lemma 2 in

\begin{itemize}%
\item Temple Fay, [[Keith Hardie]], [[Peter Hilton]], \emph{The two-square lemma}, Publicacions Matem\`a{}tiques, Vol 33 (1989) (\href{http://dmle.cindoc.csic.es/pdf/PUBLICACIONSMATEMATIQUES_1989_33_01_10.pdf}{pdf})

\end{itemize}
Also lemma 3.2-3.4 of

\begin{itemize}%
\item [[Saunders MacLane]], \emph{Homology}, Grundlehren der math. Wissenshaften vol 114, Springer (1995)

\end{itemize}
\hypertarget{in_nonabelian_categories}{}\subsubsection*{{In non-abelian categories}}\label{in_nonabelian_categories}

Discussion of generalization to non-abelian categories is in

\begin{itemize}%
\item Marino Gran, Diana Rodelo, \emph{Goursat categories and the $3 \times 3$-lemma}, Applied Categorical Structures, Vol. 20, No 3, 2012, 229-238. (\href{http://www.google.com/url?q=http%3A%2F%2Flink.springer.com%2Farticle%2F10.1007%252Fs10485-010-9236-x&sa=D&sntz=1&usg=AFQjCNG1kEggBxeeDfQlkWOa4ShUo8vwLA}{journal}, \href{http://ct2010.disi.unige.it/slides/Rodelo_CT2010.pdf}{pdf slides})


\item Marino Gran, Zurab Janelidze and Diana Rodelo, \emph{$3 \times 3$ lemma for star-exact sequences}, Homology, Homotopy and Applications, Vol. 14 (2012), No. 2, pp.1-22. (\href{http://www.intlpress.com/HHA/v14/n2/a1/}{journal})


\item [[Dominique Bourn]], \emph{$3 \times 3$-lemma and protomodularity}, Journal of Algebra, Volume 236, Number 2, 15 February 2001 , pp. 778-795(18)


\item [[Dominique Bourn]], \emph{The denormalized $3 \times 3$ lemma}, Journal of Pure and Applied Algebra, Volume 177, Issue 2, 24 January 2003, Pages 113-129, doi:\href{https://doi.org/10.1016/S0022-4049%2802%2900143-3}{10.1016/S0022-4049(02)00143-3}



\end{itemize}
[[!redirects nine lemma]] [[!redirects 9 lemma]] [[!redirects 3x3-lemma]]

[[!redirects sharp 3x3 lemma]]



\end{document}