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\usepackage{enumerate}
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\newcommand{\ds}{\displaystyle}
\newcommand{\R}{\mathbb{R}}
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\DeclareMathOperator{\Spec}{Spec}
\DeclareMathOperator{\Tor}{Tor}

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\begin{center}\begin{large}\textbf{Math 523 - Problem Set 5\\
Due Wednesday, April 13}\end{large}\end{center}



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\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 a ring and $U \subset R$ multiplicatively closed. Define $S =U^{-1}R$. 
\begin{enumerate}
\item Show that for any prime $P \subset S$, the codimension of $P$ in $S$ is equal to the codimension of   $P \cap R$ in $R$.
\item If $Q \subset R$ is prime, how does the codimension of $Q$ compare to the codimension of $QS$ in $S$?
\end{enumerate}

\item Let $R$ be an integral domain. Recall that $R$ is \textit{\textbf{normal}} if it is its own integral closure in its field of fractions. Show that the following are equivalent.\footnote{Hint: We proved a proposition in the normalization section of the notes that will be helpful.}
\begin{enumerate}
\item[i.] $R$ is normal.
\item[ii.] $R_P$ is normal for all $P \subseteq R$ prime.
\item[iii.] $R_m$ is normal for all $m \subseteq R$ maximal.
\end{enumerate}

\item Let $R$ be Noetherian. Show that $\dim R[x,x^{-1}] = \dim R + 1$.


\end{enumerate}



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