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\begin{center}\begin{large}\textbf{Math 221 - Problem Set 3\\
Due Wednesday, Oct 23}\end{large}\end{center}



\pagestyle{empty}
All rings are commutative.

\begin{enumerate} 
\item Let $R$ be a ring, $S$ an $R$-algebra, $M$ an $S$-module. Then $M$ is naturally an $R$-module. If $S$ is finite over $R$, and $M$ is a finitely-generated $S$-module, show that $M$ is a finitely-generated $R$-module. (We've been assuming this in class.)

\item Let $k$ be a field and $R = k[t]/(t^2)$. Set $$ p(x) = tx^3+tx^2-x^2-x \in R[x].$$ 

\begin{enumerate}
\item Show that $S = R[x]/(p)$ is a free $R$ module of rank 2.
\item Recall a theorem that we proved in class (Prop 4.1 in the book) that says $S$ is a free-module if and only if $(p)$ is generated by a monic polynomial. Clearly $p$ is not monic. How do you reconcile these two facts?

\end{enumerate}

\item Let $R$ be a local ring, with maximal ideal $P$. Let $I \subset R$ be an ideal, and suppose $x \in P$ is a nonzerodivisor on $R/I$. 

\begin{enumerate}
\item Show that if $I$ has a minimal set of generators, then it maps to a minimal generating set for the image of $I$ in $R/(x)$. 
\item Give an example to show that (a) can fail if $x$ is a zerodivisor on $R/I$.
\end{enumerate}



\item For each $n \in \Z$, find the integral closure $R$  of $\Z[\sqrt{n}]$ as follows:
\begin{enumerate}
\item Show that you can reduce to the case where $n$ is square-free.
\item Show that $R$ is the integral closure of $\Z$ in the field $\Q[\sqrt{n}]$, and the minimal polynomial of $\alpha = a + b\sqrt{n}$ (with $a, b \in \Q$) is $$m(x) = x^2 -(2a)x + (a^2 -b^2n).$$ Conclude $\alpha \in R$ if and only if the coefficients of $m$ are integers.
\item Show that if $\alpha \in R$, then $a \in {\frac{1}{2}}\Z$. If $a = 0$, show $\alpha \in R$ if and only if $b \in \Z$. If $a = 1/2 $ and $\alpha \in R$, show that $b \in {\frac{1}{2}}\Z$. Thus, subtracting a multiple of $\sqrt{n}$, we can assume $b = 0$ or $1/2$.
\item Conclude that $R  = \Z[1/2 + 1/2 \sqrt{n}]$ if $n \equiv 1 \pmod 4$, and $\Z[\sqrt{n}]$ otherwise.
\end{enumerate}

\item Let $R$ be an integral domain. Show $R$ is normal if and only if $R[x]$ is normal.

\item Let $R = \C[x,y]/(y^2 - x^2(x+1))$, which corresponds to the nodal curve we saw in class. Set $t = y/x$.
\begin{enumerate}
\item Show that $R[t] = \C[t]$. Conclude that $R[t]$ is the normalization of $R$.
\item Consider the corresponding normalization map $\phi: \Spec(\C[t] ) \to \Spec(R)$. Show that the point $(x,y) \in \Spec(R)$ (corresponding to the node at the origin) has exactly two points in its fiber. (In fact, it's an isomorphism away from that point!)
\end{enumerate}


\end{enumerate}
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