\documentclass[12pt]{amsart}

\usepackage[margin=1in]{geometry}
\usepackage{enumerate}
\usepackage{eucal, hyperref}

%%defines mathcal, mathbb, and mathfrak capital letters
\def\makeCal#1{%
\expandafter\newcommand\csname c#1\endcsname{\mathcal{#1}}}
\def\makeBf#1{%
\expandafter\newcommand\csname b#1\endcsname{\mathbf{#1}}}
\def\makeFrak#1{%
\expandafter\newcommand\csname f#1\endcsname{\mathfrak{#1}}}

\makeatletter
\count@=0
\loop
\advance\count@ 1
\edef\y{\@Alph\count@}%
\expandafter\makeCal\y
\expandafter\makeBf\y
\expandafter\makeFrak\y
\ifnum\count@<26
\repeat
\makeatother

\newcommand{\Set}{\mathsf{Set}}
\DeclareMathOperator{\Ann}{Ann}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\Spec}{Spec}
\DeclareMathOperator{\Supp}{Supp}
\newcommand{\op}{\mathrm{op}}

\usepackage{xypic}
\renewcommand{\cD}{\mathcal{D}}
\newcommand{\fm}{\mathfrak{m}}
\newcommand{\ol}{\overline}

\begin{document}

\noindent Math 742, Spring 2016\\
Homework 5\\
Due: February 26

\section{Exercises}

\begin{enumerate}[(1)]

\item Let $k$ be a field and let $\fm = (x-1)$ be a maximal ideal in $k[x]$. Let $Q = \fm \cap k[x^2-1]$ where $k[x^2-1]$ is the subring generated by $x^2-1$. Show that $1/(x+1) \in k[x]_\fm$ is not integral over $k[x^2-1]_Q$.

This shows that even if $S \subset R$ is integral, $R_\fm$ need not be integral over $S_{S \cap \fm}$.

\bigskip

\item Let $K$ be a field extension of the field $k$. A subset $S \subset K$ is {\bf algebraically independent over $k$} if, given any nonzero polynomial $p(x_1,\dots,x_r)$ with coefficients in $k$ and distinct $s_1,\dots,s_r \in S$, we have $p(s_1,\dots,s_r) \ne 0$. Order the set of algebraically independent subsets by inclusion. If $S$ is a maximal algebraically independent subset, then $S$ is a {\bf transcendence basis}.
  \begin{enumerate}
  \item Use Zorn's lemma to show that transcendence bases exist.
  \item If $S$ is a finite algebraically independent subset and $T$ is another algebraically independent subset with $|T| > |S|$, show that there exists $x \in T$ such that $S \cup \{x\}$ is also algebraically independent.
  \item Now assume that $K$ is finitely generated over $k$ (i.e., $K$ is the fraction field of a finitely generated $k$-algebra). Show that $K$ has a finite transcendence basis over $k$. Conclude from (b) that all transcendence bases have the same size; this number $d$ is the {\bf transcendence degree} of $K$ over $k$. 

In particular, if $K$ is algebraic over a subfield generated by $d$ elements, then these elements must be algebraically independent.
  \end{enumerate}


\bigskip

\item Let $k$ be a field and let $R = k[x_1,\dots,x_n]$. In class, we used the fact that $\dim R_\fm = n$ for all maximal ideals $\fm$ of $R$ without proving it. The goal of this exercise is to show that $\dim R_\fm$ is independent of $\fm$ without using the full version of (15.1), so don't quote any results including and after (15.9) from the text for this problem (also don't use (15.1)).
  \begin{enumerate}
  \item Show that $\dim R_\fm$ is independent of $\fm$ when $k$ is algebraically closed.
  \item Show that $\dim R_\fm$ is independent of $\fm$ in general by considering the integral extension $R \subset \ol{k}[x_1,\dots,x_n]$ where $\ol{k}$ is an algebraic closure of $k$.
  \end{enumerate}

\bigskip

\item Give an example of a non-noetherian ring such that every ascending chain of prime ideals $P_1 \subseteq P_2 \subseteq \cdots$ stabilizes, i.e., $P_j = P_{j+1} = \cdots$ for some $j$. This shows that the obvious variation of Theorem 16.10 is not true.

\end{enumerate}

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

From Altman--Kleiman:
\begin{itemize}
\item Chapter 14: 4, 13, 14, 17
\item Chapter 15: 11, 17, 19
\end{itemize}

\end{document}
