\documentclass[12pt]{amsart}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage[all]{xy}
\voffset=-1in
\hoffset=-1in
\textheight=10in
\textwidth=6in
\begin{document}
\author{}
\title{Math 726\\Summer 2005\\Homework 5}
\date{}
\maketitle
\thispagestyle{empty}
\pagestyle{empty}



\centerline{\it Due Friday August 5, 2005.}

\centerline{}


\noindent 1. (5pt) Let $A$ and $B$ be abelian groups. Show that $\text{Ext}^n_{\mathbb{Z}}(A,B)=0$ for all $n\geq 2$.

\centerline{}

\noindent 2. (5pt) Compute $\text{Ext}^1_{\mathbb{Z}}(A,B)$ where $A$ is a finitely generated abelian group.

\centerline{}

\noindent 3. (5 pt) Let $A$ and $B$ be abelian groups. Show that $\text{Tor}_n^{\mathbb{Z}}(A,B)=0$ for all $n\geq 2$.

\centerline{}

\noindent 4. (5pt) Let $R$ be an integral domain. Show that if the $R-$module $B$ is a torsion module, then $\text{Tor}_n^R(A,B)$ is torsion for all $n\geq 0$.

\centerline{}

\noindent 5. (5pt) Let $R$ be an integral domain. Show that $\text{Tor}_n^R(A,B)$ is torsion for all $R-$modules $A$ and $B$ and $n\geq 1$. (Hint: You may assume that any torsion free $R-$module may be embedded in a vector space over $K$, the quotient field of $R$).







\end{document}