\documentclass[12pt,a4paper,leqno,twoside,report]{amsart}

\textwidth=154mm
\hoffset=-14mm
\usepackage{latexsym}
\usepackage{verbatim}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{graphicx}
%\usepackage[czech]{babel}
\usepackage[utf8]{inputenc}


\theoremstyle{plain}
\newtheorem{thm}{Theorem}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{ap}[thm]{Appendix}
\newtheorem{con}[thm]{Conjecture}

\theoremstyle{definition}
\newtheorem{Def}[thm]{Definition}
\newtheorem{rem}[thm]{Remark}
\newtheorem{prob}[thm]{Problem}
\newtheorem{ex}[thm]{Exercise}

\def\N{\mathbb N}
\def\Z{\mathbb{Z}}
\def\Z2{\mathbb Z_2}
\def\R{\mathbb R}
\def\RP{\mathbb RP}

\def\C{\mathcal{C}}
\def\D{\mathcal{D}}
\def\S{\mathcal{S}}
\def\T{\mathcal{T}}
\def\L{\Lambda}
\def\d{\delta}

\def\df{\partial_\tau ^{F \ot B}}
\def\dg{\partial_\tau ^{G \ot B}}
\def\bo{\partial} %hranice

\def\bD{\bar{\Delta}}
\def\bW{\bar{W}}
\def\t{\tau}
\def\ot{\otimes}
\def\oth{\mathbin{{\otimes}_\t}}
\def\dif{\partial ^{F \ot B}}
\def\dig{\partial ^{G \ot B}}

\def\r{\rho}


\def\cb{C_* (B)}
\def\ce{C_* (E)}
\def\cg{C_* (G)}
\def\cf{C_* (F)}


\def\ep{\varepsilon}
\def\di{\partial}
\def\co{\circ}
\def\lra{\Leftrightarrow}
\def\ra{\Rightarrow}
\def\la{\Leftarrow}
\input xy
\xyoption{all}

\begin{document}


\title{Homework 2}
%\date{\today}
\maketitle

\bigskip

\begin{ex}
Finish the proof of the $5$--lemma, i.e. prove that in the following commutative diagram of Abelian groups
\[
\xymatrix{
A \ar[r] \ar[d]^{a}&B\ar[r]\ar[d]^{b}&C\ar[r]\ar[d]^{c}&D\ar[r] \ar[d]^{d}&E\ar[d]^{e}\\
A' \ar[r]&B'\ar[r]&C'\ar[r]&D'\ar[r] &E'\\
}
\]
where the rows are exact sequences and $a,b,,d,e$ are isomorphisms, the morphism $c$ is an epimorphism.
\end{ex}

\bigskip

\begin{ex}
Given the short exact sequence of Abelian groups
\[
\xymatrix{
0 \ar[r]& A \ar[r]^i &B\ar[r]^j &C\ar[r]& 0
}
\]
the following conditions are equivalent:
\begin{enumerate}
\item There exists $p \colon C \to B$ such that $j \circ p = id_C$.
\item There exists $q \colon B \to A$ such that $q \circ i = id_A$.
\item There are $p,q$ as above such that $i \circ q + p \circ j = id_B$
\end{enumerate}
We have shown that $(1) \Rightarrow (2)\text{ and }(3)$. Prove that
$(2) \Rightarrow (1)\text{ and }(3)$
\end{ex}

\bigskip

\begin{ex}
For the short exact sequence od chain complexes
$$0\to A_*\to B_*\to C_*\to 0,$$
there is a long exact sequence of homology groups
$$\dots \to H_{n+1}(C_*)\to H_n(A_*)\to H_n(B_*)\to H_n(C_*)\to H_{n-1}(A_*)\to\dots$$
Prove the exactness in $H_n(C_*)$.
\end{ex}


\end{document}