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\begin{document}

\noindent Math 742, Spring 2016\\
Homework 3\\
Due: February 12

\section{Exercises}

\begin{enumerate}[(1)]

\item Let $I, J$ be ideals of a ring $R$ and let $M$ be an $R$-module.
  \begin{enumerate}
  \item Show that $R/I \otimes_R M \cong M/IM$.
  \item Show that $R/I \otimes_R R/J \cong R/(I+J)$.
  \item In particular, show that $\bZ/n \otimes_\bZ \bZ/m = 0$ if $n,m$ are relatively prime integers.
  \end{enumerate}

\bigskip

\item Let $R$ be a local ring with maximal ideal $\fm$. Let $M,N$ be finitely generated $R$-modules. 
  \begin{enumerate}
  \item Show that if $M \otimes_R (R/\fm) = 0$, then $M = 0$.
  \item Show that if $M \otimes_R N = 0$, then either $M=0$ or $N=0$.
  \end{enumerate}

\bigskip

  \item Show that the tensor product of two free $R$-modules is again free. More specifically, given sets $\Lambda, \Lambda'$, construct an isomorphism\footnote{It's instructive to think about how to prove this only using universal mapping properties.}
\[
R^{\oplus \Lambda} \otimes_R R^{\oplus \Lambda'} \xrightarrow{\cong} R^{\oplus (\Lambda \times \Lambda')}.
\]
Conclude that the tensor product of two projective $R$-modules is again projective.

\bigskip

\item Given any module $M$, its {\bf dual} is the module $M^\vee = \Hom_R(M,R)$. Define a map $\sigma_M \colon M \to (M^\vee)^\vee$ by $\sigma_M(m)(f) = f(m)$ (where $m \in M$ and $f \in M^\vee$). Show that $\sigma_M$ is an isomorphism if $M$ is a finitely generated projective module. 

\bigskip

\item Let $P,P'$ be finitely generated projective modules and let $M,M'$ be arbitrary modules. Construct an isomorphism
\[
\Hom_R(P,M) \otimes_R \Hom_R(P',M') \xrightarrow{\cong} \Hom_R(P \otimes_R P', M \otimes_R M').
\]
Deduce the following isomorphisms:
\begin{align*}
P^\vee \otimes_R (P')^\vee &\xrightarrow{\cong} (P \otimes_R P')^\vee,\\
P^\vee \otimes_R M' &\xrightarrow{\cong} \Hom_R(P, M').
\end{align*}

\bigskip

\item Let $R$ be a ring and let $M$ be an $R$-module. Show that if $M$ is flat, then $M$ is torsion-free\footnote{In general, torsion-free means that if $m \in M$ is nonzero and $rm = 0$ for $r \in R$, then $r$ is a zerodivisor.}. Show that the converse is true if $R$ is a principal ideal domain.

Conclude that if $M$ is finitely generated and $R$ is a PID, then $M$ is flat if and only if $M$ is free.

\bigskip

\item Let $R$ be an integral domain and let $\alpha \colon R^n \to R^n$ be an endomorphism. Show that if $\alpha$ is nilpotent, i.e., $\alpha^r=0$ for some $r>0$, then in fact, $\alpha^n = 0$. Give an example to show that this statement can fail if $R$ is not required to be an integral domain.

\end{enumerate}

\section{Suggested exercises (don't submit)}

From Altman--Kleiman:
\begin{itemize}
\item Chapter 8: 12, 19, 24, 26
\item Chapter 9: 4, 8, 10, 12, 13, 25
\item Chapter 10: 8, 9, 14, 16
\end{itemize}

\section{Further reading}

The Tor functor (and more generally, derived functors) are not treated in the text. This is a useful tool for studying tensor products when the modules aren't flat, but takes more time to develop than I think we'll have. Some discussion can be found in the last page of \S XVI.3 of Lang, but it can also be found in any textbook which treats homological algebra.

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