\documentclass{amsart}

% PACKAGES

\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb,enumerate}
\usepackage{amsthm,stmaryrd}
\usepackage[all]{xy}
\usepackage{hyperref}

\theoremstyle{definition}
\newtheorem{exer}{Exercise}

\newcommand{\bbr}{\mathbb{R}}
\newcommand{\bbc}{\mathbb{C}}
\newcommand{\bbz}{\mathbb{Z}}
\newcommand{\bbq}{\mathbb{Q}}
\newcommand{\bbn}{\mathbb{N}}
\newcommand{\be}{\mathbf{e}}
\newcommand{\ba}{\mathbf{a}}
\newcommand{\fm}{\mathfrak{m}}
\newcommand{\Hom}{\operatorname{Hom}}
\renewcommand{\ker}{\operatorname{Ker}}
\newcommand{\im}{\operatorname{Im}}

\begin{document}

\noindent MATH 720, Algebra I \\
Exercises 11\\
Due Wed 26 Nov

\

\noindent
Throughout this homework set, let $R$ be a commutative ring with identity.
Let $L$, $M$, and $N$ be unitary $R$-modules.

\

\begin{exer}
Let $f\colon M\to N$ be an $R$-module isomorphism.
Prove that the map $f^*\colon\Hom_R(N,L)\to\Hom_R(M,L)$
is an isomorphism.
\end{exer}



\begin{exer}[The Five Lemma] 
Consider the following commutative diagram of $R$-module homomorphisms with exact rows
$$\xymatrix{
A_1\ar[r]^{f_1}\ar[d]^{\alpha_1}&
A_2\ar[r]^{f_2}\ar[d]^{\alpha_2}&
A_3\ar[r]^{f_3}\ar[d]^{\alpha_3}&
A_4\ar[r]^{f_4}\ar[d]^{\alpha_4}&
A_5\ar[d]^{\alpha_5}
\\
B_1\ar[r]_{g_1}&
B_2\ar[r]_{g_2}&
B_3\ar[r]_{g_3}&
B_4\ar[r]_{g_4}&
B_5}$$
\begin{enumerate}[(a)]
\item
Prove that if $\alpha_1$ is an epimorphism and $\alpha_2$, $\alpha_4$ are monomorphisms, then $\alpha_3$ is a monomorphism.
\item
Prove that if $\alpha_5$ is a monomorphism and $\alpha_2$, $\alpha_4$ are epimorphisms, then $\alpha_3$ is an epimorphism.
\end{enumerate}
\end{exer}


\begin{exer}
Consider the following commutative diagram of $R$-module homomorphisms with exact rows
$$\xymatrix{
&
A_1\ar[r]^{f_1}\ar[d]^{\alpha_1}&
A_2\ar[r]^{f_2}\ar[d]^{\alpha_2}&
A_3\ar[r]^{f_3}\ar[d]^{\alpha_3}&
0
\\
0\ar[r]&B_1\ar[r]_{g_1}&
B_2\ar[r]_{g_2}&
B_3
}$$
\begin{enumerate}[(a)]
\item
Prove that there is an exact sequence $\ker(\alpha_1)\to\ker(\alpha_2)\to\ker(\alpha_3)$.
\item
Prove that there is an exact sequence $B_1/\im(\alpha_1)\to B_2/\im(\alpha_2)\to B_3/\im(\alpha_3)$.
\item
(Bonus) Show how this gives another proof of Proposition 3.7.9 from the notes.
\end{enumerate}
\end{exer}


\end{document}


\begin{exer}
\end{exer}















