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\title{The Snake Lemma}
\subtitle{A Beamer Presentation}
\author{Jason Michael Starr}
\institute{Department of Mathematics \\
  Stony Brook University \\ Stony Brook, NY 11794}
%% \email{jstarr@math.sunysb.edu} 
\date{\today}

\begin{document}

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\frame{ 
\titlepage
}

%% \tableofcontents



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section: Origins of Homological Algebra
%% 
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\frame{
\frametitle{Beware of Snakes!}
\begin{center}
\includegraphics[scale=0.3]{800px-Coast_Garter_Snake.jpg}
\end{center}
}
\section{Origins of Homological Algebra}
\subsection{An Incomplete  List}

\frame{
\frametitle{Problems from Topology}
\setbeamercovered{transparent}
\begin{center}
\textsc{\textcolor{Blue}{Homological algebra originated from
    computational issues with the homology / cohomology of spaces.}}
\end{center}
\begin{enumerate}
\item<2-> The K\"unneth decomposition does not work ``on the nose'' in
  the presence of torsion, e.g., \text{\textcolor{Red}{$\mathbb{R}\textbf{P}^2 \times
  \mathbb{R}\textbf{P}^2$}}.  
\item<3-> Again in the presence of torsion, the Universal Coefficients
  Theorem for homology / cohomology with coefficients has an extra term.
\item<4-> The first computations of group
  homology / cohomology were \emph{ad hoc} (they were also topological, 
  being the homology / cohomology
  of a classifying space of the group).   
\end{enumerate}


}

\frame{
\frametitle{\emph{Ad Hoc} Constructions}
\setbeamercovered{transparent}
\begin{block}{Motto}
Homological algebra systematizes 
\emph{ad hoc} constructions from topology and algebra. 
\end{block}
\begin{itemize}
\item<2-> Group cohomology: original definitions all in terms of
  explicit cocycles.
\item<3-> Tor: MacLane's explicit description.
\item<4-> Ext: The Yoneda approach.
\item<5-> MacLane's
  \emph{Homological Algebra} develops these operations
  \textbf{without} the language of derived functors.
\item<6-> For readers who know derived functors,
  the original constructions are now difficult to read.
\end{itemize}

}

\frame{
\frametitle{Formal Development}
\setbeamercovered{transparent}
\begin{center}
\textsc{\textcolor{Blue}{The formal development of homological algebra
    began in the post-war period and continues to today.}}
\end{center}

\textbf{\underline{Highpoints.}}

\begin{enumerate}
\item<2->
\emph{Homological Algebra} by Cartan and Eilenberg,
1956.
\item<3->
Grothendieck's ``Tohoku paper'', \emph{Sur quelques points
  d'alg\`{e}bre homologique}, 1957:
\textbf{Abelian categories}, \textbf{delta functors}, \textbf{sheaf cohomology},
\textbf{Grothendieck spectral sequence}.
\item<4-> Verdier's thesis on derived categories, early 1960s.
\item<5-> Andre-Quillen cohomology, cotangent complex,  1967.
\item<6-> Giraud's non-Abelian cohomology,
Quillen's model categories, Grothendieck's topoi, operads,
  higher categories, stacks,
  etc., up to the present.
\end{enumerate}

}

\section{Exact Sequences and Diagrams} 
\subsection{Abelian Categories}

\frame{
\setbeamercovered{transparent}
\frametitle{Abelian Categories}

\begin{block}<1->{Additive Categories}
An \emph{additive category} is a category where Hom
sets are Abelian groups, composition is biadditive, there is a zero
object, and there are finite products.
\end{block}

\begin{block}<2->{Abelian Categories}
An \emph{Abelian category} is an additive category that has kernels
and cokernels which behave ``as usual'': monics are kernels of
their cokernels and epis are cokernels of their kernels.
\end{block}

}

\frame{\frametitle{}
\setbeamercovered{transparent}
In an Abelian category, most categorical facts familiar from the
category of modules hold, e.g.,
image and coimage are the
same.  Frequently Abelian categories are assumed to satisfy additional
``compatibility axioms'' with arbitrary products and coproducts.

\textbf{\underline{Examples.}}
\begin{itemize}
\item<2-> The category of (left) modules over an associative, unital
  ring.
\item<3-> The category of sheaves of Abelian groups on a space.
\end{itemize}

\begin{block}<4->{Freyd-Mitchell Theorem}
Every \emph{small} Abelian category is equivalent to a full
subcategory of the category of modules over a ring.
\end{block}


}

\subsection{Short Exact Sequences}

\frame{\frametitle{Short Exact Sequences}
\setbeamercovered{transparent}

For what follows, fix an Abelian category.  All objects and morphisms
are in this Abelian category.

\begin{block}{Short Exact Sequences}
A \emph{short exact sequence} 
$$
\xymatrix@1{
\Sigma_A: &  0 \ar[r] & A' \ar[r]^{q_A} & A \ar[r]^{p_A} & A'' \ar[r] & 0
}
$$
is a pair of morphisms
$$
\Sigma_A = (q_A:A'\to A,p_A:A \to A)
$$
such that all of the following hold: 
\begin{enumerate}
\item[(i)]
$q_A$ is a monomorphism,
\item[(ii)]
$p_A$ is an epimorphism, and
\item[(iii)]
the image of $q_A$ equals the kernel of $p_A$.
\end{enumerate}
\end{block}

\begin{itemize}
\item<2->
Briefly, $A'$ is the kernel of $p_A$ and $A''$ is the
  cokernel of $q_A$.
\end{itemize}

}

\subsection{Morphisms of Short Exact Sequences}

\frame{
\setbeamercovered{transparent}

\begin{block}{Morphisms of Short Exact Sequences}
Let $\Sigma_A=(q_A,p_A)$ and $\Sigma_B=(q_B,p_B)$ be short exact
sequences.  A 
\emph{morphism} $\Sigma_f$ from $\Sigma_A$ to $\Sigma_B$,
$$
\xymatrix{
\Sigma_A:\ar[d]_{\Sigma_f} &  0 \ar[r] & A' \ar[r]^{q_A} \ar[d]_{f'} & A \ar[r]^{p_A}
\ar[d]^{f} & A'' \ar[r] \ar[d]^{f''} & 0 \\
\Sigma_B: &  0 \ar[r] & B' \ar[r]_{q_B} & B \ar[r]_{p_B} & B'' \ar[r] & 0
}
$$
is a triple of morphisms 
$$
\Sigma_f = (f':A'\to B', f:A\to B, f'':A''\to B'')
$$
such that every square commutes, i.e., both of the following hold:
\begin{enumerate}
\item[(i)]
$q_B\circ f'$ equals $f\circ q_A$, and
\item[(ii)]
$p_B\circ f$ equals $f''\circ p_A$.
\end{enumerate}
\end{block}

}

\subsection{Short Exact Sequences Form a Category}

\frame{\frametitle{Short Exact Sequences Form a Category}
\setbeamercovered{transparent}
The short exact sequences in a fixed Abelian category form their own
category. 
\begin{itemize}
\item<2-> Composition of morphisms of short exact sequences is
  term-by-term composition for $f'$, $f$ and $f''$ separately.
\item<3-> The identity morphism of a short exact sequence $\Sigma_A$ is just
  $(\text{Id}_{A'}, \text{Id}_A, \text{Id}_{A''})$.  
\item<4-> The category of short exact sequences is additive, but it
  is \emph{not} Abelian.
\end{itemize}
}

\subsection{Kernels and Cokernels}

\frame{\frametitle{Kernels and Cokernels}
\setbeamercovered{transparent}
For a morphism of short exact sequence,
\begin{block}{The morphism $\Sigma_f$}
$$
\xymatrix{
\Sigma_A:\ar[d] &  0 \ar[r] & A' \ar[r]^{q_A} \ar[d]_{f'} & A \ar[r]^{p_A}
\ar[d]^{f} & A'' \ar[r] \ar[d]^{f''} & 0 \\
\Sigma_B: &  0 \ar[r] & B' \ar[r]_{q_B} & B \ar[r]_{p_B} & B'' \ar[r] & 0
}
$$
\end{block}
denote the kernels of $f'$, respectively $f$, $f''$ by,
$$
i':K_{\Sigma_f}' \to A', \ \text{resp.} \ \ i:K_{\Sigma_f} \to A, \
i'':K_{\Sigma_f}'' \to A''.
$$

Similarly, denote the cokernels by
$$
s':B' \to C_{\Sigma_f}', \ \text{resp.} \ \ s:B\to C_{\Sigma_f},
\ s'':B'' \to C_{\Sigma_f}''.
$$

}

\subsection{Morphisms of the Kernels}

\frame{\frametitle{Morphisms of the Kernels}
\setbeamercovered{transparent}

There are induced morphisms of the kernels.
\begin{block}{Morphisms of Kernels}
$$
\xymatrix{ 
K_{\Sigma_f}' \ar[r]^{q_K} & K_{\Sigma_f} \ar[r]^{p_K} & K_{\Sigma_f}''
}
$$
\end{block}
\begin{itemize}
\item<2-> 
Since $q_B\circ f'$ equals $f\circ q_A$, also $f\circ (q_A\circ i')$
equals $q_B(f'\circ i') = 0$.  This gives $q_K$.
\item<3->
The morphism $p_K$ is similar.
\end{itemize}
}

\subsection{Morphisms of the Cokernels}

\frame{\frametitle{Morphisms of the Cokernels}
\setbeamercovered{transparent}

There are also induced morphisms of the cokernels.
\begin{block}{Morphisms of Cokernels}
$$
\xymatrix{ 
C_{\Sigma_f}' \ar[r]^{q_C} & C_{\Sigma_f} \ar[r]^{p_C} & C_{\Sigma_f}''
}
$$
\end{block}
\begin{itemize}
\item<2-> 
The proof of the existence of the morphisms of cokernels is similar to
the proof of the existence of the morphisms of the kernels.
\end{itemize}
}

\subsection{Summary}

\frame{\frametitle{Summary}
\setbeamercovered{transparent}
To summarize, given a morphism of short exact sequences,
\begin{block}<2->{The morphism $\Sigma_f$}
$$
\xymatrix{
\Sigma_A:\ar[d] &  0 \ar[r] & A' \ar[r]^{q_A} \ar[d]_{f'} & A \ar[r]^{p_A}
\ar[d]^{f} & A'' \ar[r] \ar[d]^{f''} & 0 \\
\Sigma_B: &  0 \ar[r] & B' \ar[r]_{q_B} & B \ar[r]_{p_B} & B'' \ar[r] & 0
}
$$
\end{block}
\uncover<3->{there is an associated large commutative diagram as follows.}

}

\frame{

\begin{block}{The Associated Large Diagram}
$$
\xymatrix{
 & & K_{\Sigma_f}' \ar[r]^{q_K} \ar[d]_{i'} & K_{\Sigma_f} \ar[r]^{p_K}
 \ar[d]^{i} & K_{\Sigma_f}'' \ar[d]^{i''} \\
\Sigma_A:\ar[d]_{\Sigma_f} &  0 \ar[r] & A' \ar[r]^{q_A} \ar[d]_{f'} & A \ar[r]^{p_A}
\ar[d]^{f} & A'' \ar[r] \ar[d]^{f''} & 0 \\
\Sigma_B: &  0 \ar[r] & B' \ar[r]_{q_B} \ar[d]_{s'} & B \ar[r]_{p_B}
\ar[d]^{s} & B'' \ar[r] \ar[d]^{s''} & 0 \\ 
 & & C_{\Sigma_f}' \ar[r]_{q_C} & C_{\Sigma_f} \ar[r]_{p_C} &
 C_{\Sigma_f}'' 
}
$$
\end{block}

}
\section{The Snake Lemma}
\subsection{The Common Composition}

\frame{\frametitle{The Common Composition}
\setbeamercovered{transparent}
By hypothesis, both $f''\circ p_A$ and $p_B\circ f$ are equal.  Denote
by $t$ this common composition.
\begin{block}<2->{The common composition}
\alt<-2>{
$$
\xymatrix{
A \ar[r]^{p_A} 
\ar[d]_{f} & A'' \ar[d]^{f''} \\
B \ar[r]_{p_B} & B'' }
$$
}{
$$
\xymatrix{
A \ar[r]^{p_A} \ar[dr]^{\textbf{\textcolor{red}{t}}}
\ar[d]_{f} & A'' \ar[d]^{f''}\\
B \ar[r]_{p_B} & B''}
$$
}
\end{block}
\uncover<4->
{And denote the kernel of $t$ by}
\begin{block}<4->{The kernel of $t$} 
$$
j:K_t \to A.
$$
\end{block}

}

\subsection{The key morphisms}

\frame{\frametitle{The Key Morphisms}
\setbeamercovered{transparent}
Now $f''\circ(p_A\circ j)$ equals $t\circ j$, which is $0$.  So by the
universal property of the kernel of $f''$, there is a unique morphism
\begin{block}<2->{The first key morphism}
$$
\widetilde{p_A}:K_t \to K_{\Sigma_f}''
$$
\end{block}
\uncover<2->
{such that $i''\circ \widetilde{p_A}$ equals $p_A\circ j$.}  

\uncover<3->
{similarly,
$p_B\circ (f\circ j)$ equals $t\circ j$, which is $0$.  By the
universal property of the kernel of $p_B$, there is a unique morphism}
\begin{block}<4->{The second key morphism}
$$
\widetilde{f}:K_t \to B'
$$
\end{block}
\uncover<4->
{such that $q_B\circ \widetilde{f}$ equals $f\circ j$.}  

}

\subsection{Statement of the Snake Lemma}

\frame{\frametitle{Statement of the Snake Lemma}
\setbeamercovered{transparent}
Finally this brings us to the statement of the snake lemma for a
morphism $\Sigma_f$ of short exact sequences.  
\begin{block}<2->{The Snake Lemma}
All of the
following hold.
\begin{enumerate}
\item<3->[(i)]
The morphism $q_K$ is a monomorphism, and the morphism $p_C$ is an epimorphism.
\item<4->[(ii)]
The image of $q_K$ equals the kernel of $p_K$, and the kernel of $p_C$
equals the image of $q_C$.
\item<5->[(iii)]
There is a unique morphism $\delta_{\Sigma_f}:K_{\Sigma_f}'' \to
C_{\Sigma_f}'$ such that $\delta_{\Sigma_f} \circ \widetilde{p_A}$
equals $s'\circ\widetilde{f}$ as morphisms $K_t\to C_{\Sigma_f}'$.  
\item<6->[(iv)]
The image of $p_K$ equals the kernel of $\delta_{\Sigma_f}$, and the
kernel of $q_C$ equals the image of $\delta_{\Sigma_f}$.
\end{enumerate}
\end{block}

}

\subsection{The Snake Diagram}

\frame{\frametitle{The Snake Diagram}
\setbeamercovered{transparent}
So in the following all rows are exact, including the ``snake''.
\begin{block}<2->{The snake diagram}
$$
\xymatrix{
 & 0 \ar[r] & K_{\Sigma_f}' \ar[r]^{q_K} \ar[d]_{i'} & K_{\Sigma_f} \ar[r]^{p_K}
 \ar[d]^{i} & K_{\Sigma_f}'' \ar[d]_{i''}  \ar @{-->} 
%% '[dddll]
`dr_d[d] `[dd]^<{\delta_{\Sigma_f}} '[dd]
'[ddll] `^d[ddddll]-<30pt,0pt> `[ddddll]-<10pt,0pt> 
[ddddll]-<10pt,0pt> 
%% `r[d] `d[l] `l[d] `d[r] 
&  \\
\Sigma_A:\ar[dd]_{\Sigma_f} &  0 \ar[r] & A' \ar[r]^{q_A} \ar[dd]_<<<<<<{f'}
& A \ar[r]^{p_A} \ar[dd]^<<<<<<{f} & A'' \ar[r] \ar[dd]_<<<<<<{f''} & 0 \\
 & & & & & \\
\Sigma_B: &  0 \ar[r] & B' \ar[r]_{q_B} \ar[d]^{s'} & B \ar[r]_{p_B}
\ar[d]^{s} & B'' \ar[r] \ar[d]^{s''} & 0 \\ 
 & & C_{\Sigma_f}' \ar[r]_{q_C} & C_{\Sigma_f} \ar[r]_{p_C} &
 C_{\Sigma_f}'' \ar[r] & 0
}
$$
\end{block}

}

\end{document}


