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\begin{document}

\noindent Math 742, Spring 2016\\
Homework 6\\
Due: March 4

\section{Exercises}

\begin{enumerate}[(1)]

\item A ring $R$ is {\bf coherent} if every finitely generated ideal $I$ is a finitely presented module.\footnote{It follows easily from what we've seen that every noetherian ring is coherent.}
\begin{enumerate}
\item Suppose that $R$ is coherent. Let $f \colon M \to N$ be a homomorphism of finitely presented modules. Show that $\ker f, \coker f, \image f$ are all finitely presented.
\item Let $R$ be a noetherian ring. Show that the polynomial ring in infinitely many variables $R[x_1,x_2,\dots]$ is coherent.
\end{enumerate}

\bigskip

\item Show that if $R$ is noetherian, then so is the power series ring $R[\![x_1,\dots,x_n]\!]$.\footnote{For a definition, see Altman--Kleiman, Example 3.10.}

\bigskip

\item Let $R$ be a noetherian ring. Show that the following are equivalent:
  \begin{enumerate}
  \item $R$ is artinian,
  \item $\Spec(R)$ is discrete (i.e., every subset is open) and finite,
  \item $\Spec(R)$ is discrete.
  \end{enumerate}
Give an example of a noetherian ring $R$ such that $\Spec(R)$ is finite, but $R$ is not artinian.

\bigskip

\item Let $R$ be a principal ideal domain and $M$ a finitely generated $R$-module. Explain how to calculate the associated primes of $M$ in terms of the structure theorem from HW2\#4.

\bigskip

\item Let $k$ be a field and $R = k[x_1,\dots,x_n]$ a polynomial ring. A {\bf monomial} is an element of the form $x_1^{d_1} \cdots x_n^{d_n}$ for some $d_i \ge 0$. A {\bf monomial ideal} is an ideal generated by monomials. 
  \begin{enumerate}
  \item Describe which monomial ideals are prime.
  \item Describe which monomial ideals are radical.
  \item Describe which monomial ideals are primary.
  \end{enumerate}

\end{enumerate}

\section{Suggested exercises (don't submit)}

From Altman--Kleiman:
\begin{itemize}
\item Chapter 16: 18, 20, 28, 29, 30
\item Chapter 17: 6, 8, 11, 22
\item Chapter 18: 6, 7, 8
\end{itemize}

\end{document}
