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\newcommand{\Z}{\mathbb{Z}}
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\begin{document}


\begin{center}\begin{large}\textbf{Math 221 - Problem Set 5\\
Due Wednesday, Nov 20}\end{large}\end{center}



\pagestyle{empty}
All rings are commutative.

\begin{enumerate} 

\item Let $R$ be a ring and $M$ an $R$-modules. Show that the following three statements are equivalent.
\begin{enumerate}[i]
\item $M$ is flat.
\item $\Tor_i^R(M,N) = 0$ for all $R$-modules $N$ and all $i > 0$.
\item $\Tor_1^R(M,N) = 0$ for all $R$-modules $N$.
\end{enumerate}

\item Let $0 \to M_1 \to M_2 \to M_3 \to 0$ be a short exact sequence of $R$-modules. If $\{i,j,k\} = \{1,2,3\}$, for which values of $i$ and $j$ is the following statement true:  If $M_i$ and $M_j$ are flat, then $M_k$ is flat?

\item Calculate $\Tor_i^R (M,N)$ for all $i >0$ in the following cases.
\begin{enumerate}
\item $R = \C[x,y]$, $M = R/(x)$, $N = R/(xy)$.
\item $R = \Z$, $M = \Z/(m)$, $N = \Z/(n)$.
\item $R = \C[x,y,z]$, $M = R/(x,y)$, $N = R/(x,y,z)$.
\end{enumerate}

\item Let $M$ and $N$ be $R$-modules, and $S$ a flat $R$-algebra. Show that $$S \otimes \Tor_i^R(M,N) \cong \Tor_i^S(S \otimes M, S \otimes N).$$ In particular, $\Tor$ commutes with localization.

\item Let $R$ be Noetherian and $M$ a finitely generated $R$ module. Show $M$ is flat if and only if $\Tor_1^R(M,R/P) = 0$ for every prime ideal $P$ of $R$. (In fact, this is true for infinitely generated modules as well, by a reduction like the one we mentioned in class.)





\end{enumerate}
\end{document}