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\title{Math 421-621\\Fall 2013\\Homework 5}
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\centerline{\it Due Wednesday, February 13, 2013.}


\centerline{}



\noindent 1. Consider the following commutative diagram of $R-$module homomorphisms with exact rows.
\begin{center}
$\xymatrix{A_1\ar[r]^{f_1}\ar[d]^{g_1} & A_2\ar[r]^{f_2}\ar[d]^{g_2} & A_3\ar[r]^{f_3}\ar[d]^{g_3} & A_4\ar[r]^{f_4}\ar[d]^{g_4} & A_5\ar[d]^{g_5}\\
B_1\ar[r]^{h_1} & B_2\ar[r]^{h_2} & B_3\ar[r]^{h_3} & B_4\ar[r]^{h_4} & B_5}$
\end{center}
Show the following.

\begin{itemize}
\item[a)] (5 pt) If $g_2$ and $g_4$ are onto and $g_5$ is one to one then $g_3$ is onto.
\item[b)] (5 pt) If $g_2$ and $g_4$ are one to one and $g_1$ is onto then $g_3$ is one to one.
\item[c)] (5 pt) Explain how the Short Five Lemma follows from the above results.
\end{itemize}


\centerline{}

\noindent 2. (5 pt) Consider the following short exact sequence of $R-$modules.
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$\xymatrix@1{0\ar[r]&A\ar[r]^f&B\ar[r]^g&C\ar[r]&0}$
\end{center}

Show that the following two conditions are equivalent:
\begin{itemize}
\item[i)] There is an $R-$module homomorphism $k:B\longrightarrow A$ such that $kf=1_A$.
\item[ii)] There is an $R-$module homomorphism $h:C\longrightarrow B$ such that $gh=1_C$.
\end{itemize}

\centerline{}

\noindent 3. (5 pt) Show that the $R-$module $P$ is projective if and only every short exact sequence of the form 

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$\xymatrix@1{0\ar[r]&A\ar[r]^f&B\ar[r]^g&P\ar[r]&0}$
\end{center}

\noindent is split exact.

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