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\begin{document}

\noindent Math 742, Spring 2016\\
Takehome exam 1\\
Due: March 28 (MONDAY) by the end of class (2:10PM)

~

\begin{itemize}
\item Do not discuss this exam with anyone else.

\item You may use your notes and textbook and use as much time as you like up until the due date. You are free to use any results in the book that we covered and any previous exercises. 

\item No late exams will be accepted. If you cannot make it to class, you can email me your solutions. 

\item This counts for 4 homework assignments and all problems will be graded. Each problem will be scored out of 10 points, so the maximum score is 60 points.
\end{itemize}

\section{Exercises}

\begin{enumerate}[(1)]

\item Let $S$ be a subring of $R$.
  \begin{enumerate}
  \item Give an example where $R$ is noetherian and $S$ is not noetherian.
  \item Assume there is an $S$-module map $\phi \colon R \to S$ which is the identity when restricted to $S$. Show that $S$ is noetherian if $R$ is noetherian.
  \end{enumerate}

\bigskip

\item Let $R$ be a ring and let $\phi \colon R^n \to R^n$ be a linear map where $n$ is finite.
  \begin{enumerate}
  \item Show that $\det \phi \in {\rm ann}(\coker \phi)$.
  \item Give an example where $\det \phi$ is a nonzerodivisor but ${\rm ann}(\coker \phi)$ is not equal to the ideal generated by $\det \phi$.
  \end{enumerate}

\bigskip

\item A ring is {\bf reduced} if its nilradical is $0$.
\begin{enumerate}
\item Show that a ring $R$ is reduced if and only if $R_\fm$ is reduced for all maximal ideals $\fm$.
\item Give an example of a field $k$ and a reduced $k$-algebra $R$ such that $R \otimes_k \ol{k}$ is not reduced where $\ol{k}$ is an algebraic closure of $k$.
\end{enumerate}


\bigskip

\item Let $k$ be a field. In each case, verify that $R$ is an integral domain and describe, as explicitly as possible, its normalization.
  \begin{enumerate}
  \item $R = k[x,y] / (x^4-y^3)$
  \item $R = k[x,y,z] / (x^2 - yz)$
  \end{enumerate}

\bigskip

\item Let $R$ be a noetherian ring and let $M$ be a finitely generated $R$-module. Recall in HW3 \#4, we defined the dual $M^\vee = \Hom_R(M,R)$ and the map $\sigma_M \colon M \to (M^\vee)^\vee$ by $\sigma_M(m)(f) = f(m)$ (where $m \in M$ and $f \in M^\vee$). 
  \begin{enumerate}
  \item Show that $\sigma_M$ is injective if and only if $M$ is isomorphic to a submodule of $R^{\oplus n}$ for some finite $n$.
  \item Show that $\sigma_M$ is bijective if and only if there exists a homomorphism $\phi \colon R^{\oplus n} \to R^{\oplus m}$ for some finite $n,m$ such that $M \cong \ker \phi$.
  \end{enumerate}

\bigskip

\item Let $S \subset R$ be an integral extension.
  \begin{enumerate}
  \item Show that $\dim S = \dim R$.
  \item Assume that $R$ is a finitely generated $S$-module. Let $\fp \subset S$ be a prime ideal. Show that there are only finitely many prime ideals $\fq \subset R$ such that $\fq \cap S = \fp$.
  \end{enumerate}

\end{enumerate}

\end{document}
