\documentclass{amsart}
\title{Algebra 2. Problem Set 8}
\author{}

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\begin{document}
\maketitle
\thispagestyle{empty}

\begin{center}
{\em In all problems below, $R$ is a commutative ring (with $1\neq 0$).}\\
\end{center}

\vskip 0.2in

\hrule\hrule

\vskip 0.2in

\problem{R} Let $S\subset R$, $1\in S$, $0\not\in S$ be a multiplicatively closed
set. Let $M,N$ be two $R$--modules. Prove that we have the following isomorphisms
of $S^{-1}R$--modules.
\[
S^{-1}M\otimes_{S^{-1}R} S^{-1}N \cong S^{-1}M\otimes_R N
\cong S^{-1}(M\otimes_R N)
\]

Formulate and prove the similar assertion for $\Hom$.\\


\vskip 0.2in

\hrule\hrule

\vskip 0.2in

\problem{1} Let $K$ be a field and let $R$ be the localization of the polynomial ring $K[x,y]$ at
the ideal $(x,y)\subset K[x,y]$. Let $\mathfrak{m}$ denote the unique
maximal ideal of $R$. Take $M = K(t)$ (rational functions in one variable $t$)
with $R$--action:
\[
\lp \frac{f(x,y)}{g(x,y)}\rp \cdot \frac{p(t)}{q(t)} = \frac{f(t,0)}{g(t,0)}\frac{p(t)}{q(t)}
\]
where $f(x,y),g(x,y)\in K[x,y]$ are such that $g(0,0)\neq 0$, and
$p(t),q(t)\in K[t]$ are such that $q(t)\neq 0$.\\

\noindent Prove that $\Tor^R_1(R/\mathfrak{m},M)=0$ and that $M$ is not flat.\\

\problem{2} Let $R$ be a local principal ideal domain. Let $M$ be a finitely--generated $R$--module.
Prove that $M\cong M_{\tor}\oplus (M/M_{\tor})$.\\

\problem{3} Prove or disprove: every torsion--free module over an integral domain
is flat.\\

\problem{4} Let $M,N$ be two flat $R$--modules. Prove that so is $M\otimes_R N$.\\

\problem{5} Let $R$ be an integral domain and $K$ be its field of fractions. For
any $R$--module, $N$, prove that $\Tor^R_1(K/R,N) = N_{\tor}$.\\

\problem{6} Let $k$ be a field and consider $R = k[x,y]/(y^2-x^3)$. Let $R' = k[t]$
with a structure of an $R$--module via the ring homomorphism $x \mapsto t^2$ and $y\mapsto t^3$.
Prove or disprove: $R'$ is a flat $R$--module.
\end{document}
