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\title{Math 721\\Spring 2011\\Exam 1}
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\begin{center}
{\it Due Monday February 28, 2011. As is usual on exams, I am the only biological resource that you should use.}
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\centerline{}

\noindent 1. ({\it Localizations are flat}) Let $R$ be an integral domain, let $N$ be an $R-$module and $S\subseteq R$ a multiplicative set.
\begin{itemize}
\item[a)] (5 pt) Show that the set $S^{-1}N=\{\frac{n}{s}\vert n\in N, s\in S\}$ (where we declare that $\frac{n_1}{s_1}=\frac{n_2}{s_2}$ if and only if there is a $t\in S$ such that $t(s_2n_1-s_1n_2$)=0) is an $R-$module with addition given by 

\[
\frac{n_1}{s_1}+\frac{n_2}{s_2}=\frac{s_2n_1+s_1n_2}{s_1s_2}
\]

\noindent and $R-$ action

\[
r(\frac{n}{s})=\frac{rn}{s}.
\]
\item[b)] (5 pt) Show that every element of $R_S\otimes_R N$ can be written as a single tensor of the form $(\frac{1}{s})\small{\otimes} n$ with $s\in S$ and $n\in N$.
\item[c)] (5 pt) Show that there is an isomorphism $R_S\otimes_R N\cong S^{-1}N$.
\item[d)] (5 pt) Show that a tensor of the form $(\frac{1}{s})\small{\otimes} n$ is $0$ if and only if there exists $t\in S$ such that $tn=0$.
\item[e)] (5 pt) Show that $R_S$ is a flat $R-$module.
\item[f)] (5 pt) Show that $\mathbb{Q}$ is a flat $\mathbb{Z}-$module that is not projective.
\end{itemize} 

\centerline{}

\noindent 2. Let $R$ be an integral domain with quotient field $K$ and $M$ an $R-$module. Let $T(M)=\{x\in M\vert rx=0\text{ for some nonzero $r\in R$}\}$. We have shown that $T(M)$ is a submodule of $M$.
\begin{itemize}
\item[a)] (5 pt) Show that $M/T(M)$ is torsion free.
\item[b)] (5 pt) Show that $M\otimes_RK\cong M/T(M)\otimes_RK$.
\end{itemize}



\centerline{}

\noindent 3. (5 pt) We showed in class that tensor product and direct sum commute (that is, $M\otimes_R(\oplus_{i\in I}A_i)\cong \oplus_{i\in I}(M\otimes_RA_i)$). Does this result hold in general for direct products (i.e. is $M\otimes_R(\prod_{i\in I}A_i)\cong \prod_{i\in I}(M\otimes_RA_i)$)?

\centerline{}

\noindent 4. (5 pt) ({\it Adjoint Associativity}) Show that if $R$ is commutative with identity and $A,B$, and $C$ are $R-$modules, then we have an $R-$module isomorphism

\[
\text{Hom}_R(A\otimes_RB,C)\cong\text{Hom}_R(A,\text{Hom}_R(B,C)).
\]

\centerline{}

\noindent 5. Suppose that $R$ is commutative with identity, $I\subseteq R$ an ideal, and $J$ and $M$ are $R-$modules.
\begin{itemize}
\item[a)] (5 pt) Show that $R/I\otimes_RM\cong M/IM$.
\item[b)] (5 pt) Show that if $R=\mathbb{Z}$ and $J$ is injective, then $M\otimes_R J$ is injective.
\end{itemize}


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