\documentclass[12pt]{amsart}
\usepackage{ae,amsmath, amssymb}
\usepackage[margin=0.5in]{geometry}
%\usepackage[all]{xypic}
%\newcommand{\beql}{\begin{equation}}
\newcommand{\eeq}{\end{equation}}




% old stuff 
%\setlength{\textwidth}{5.8in}
%\setlength{\textheight}{9in}
%\setlength{\oddsidemargin}{.2in}
%\setlength{\topmargin}{-0.25in}
%\setlength{\headheight}{0in}
% old stuff

%*****old*****
%\tolerance=10000
%***************
%\renewcommand{\thefootnote}{*}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\newcommand{\bA} {\mathbb{A}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\bZ} {\mathbb{Z}_p}
\newcommand{\bC}{\mathbb{C}}
\newcommand{\bQ}{\mathbb{Q}_p}
\renewcommand{\subjclassname}{%
  \textup{2010} Mathematics Subject Classification} 

\newcommand{\bs}{{\bf s}}
\newcommand{\bx}{{\bf x}}
\newcommand{\by}{{\bf y}}
\newcommand{\bz}{{\bf z}}

\newcommand{\Et} {\tau}
\newcommand{\Es}{ \sigma}
\newcommand{\Einv}{ \sigma^{-1}}
\newcommand{\tEsm}{\sigma^{m}}


\newcommand{\pt}{[q_0:\cdots :q_n]}

\newcommand{\sRS}{{\mathcal{R}(S)}}
\newcommand{\sB}{{\mathcal B}}

\newcommand{\ev}{{\rm ev}}

\newcommand{\CC}{\mathbb{C}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\AAA}{\mathbb{A}}



\tolerance=10000
\renewcommand{\thefootnote}{*}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\newtheorem{thm}{Theorem}[section]
\newtheorem{notn}[thm]{Notation}
\newtheorem{exam}[thm]{Example}
\newtheorem{quest}{Question}
\newtheorem{exercise}{Exercise}
\newtheorem{rem}[thm]{Remark}
\newtheorem{defn}[thm]{Definition}
\newtheorem{conj}[thm]{Conjecture}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{cor}[thm]{Corollary}
%\newtheorem{example}[thm]{Example
%\def\eqref#1{(\ref{#1})}
%\def\qed{~\vrule height8pt width 5pt depth -1pt\medskip}




%%%NEW 
\def\binom#1#2{{#1}\choose{#2}}

\def\slfrac#1#2{\hbox{\kern.1em %
 \raise.5ex\hbox{\the\scriptfont0 #1}\kern-.11em %
 /\kern-.15em\lower.25ex\hbox{\the\scriptfont0 #2}}}

%\font\phvr=phvr at 11pt
\newcommand\he[1]{\mbox{\phvr  #1}}
%%%%%NEW






%\begin{document}

%\begin{center}

\begin{document}
%\bibliographystyle{plain}

Assignment 3.  Due November 3 in class.  Feel free to assume the results from any earlier questions in doing the later questions.
\begin{enumerate}
\item A topological space is said to be \emph{Noetherian} if every descending chain of closed subsets terminates.    Show that every subset of a Noetherian space is compact (when given the subspace topology).  
\item Show that if $X$ is a topological space with every subset compact (with subspace topology) then $X$ is noetherian.   
\item Given an associative (but not necessarily commutative) ring $R$ with 1 we let ${\rm Spec}(R)$ denote the set of prime ideals of $R$.  (An ideal $P$ of $R$ is called prime if whenever $a,b\in R$ have the property that $arb\in P$ for all $r\in R$ we must have $a\in P$ or $b\in P$.  We put a topology on ${\rm Spec}(R)$ by declaring the closed sets to be precisely the sets of the form $C_I=\{P\in {\rm Spec}(R)\colon P\supseteq I\}$, where $I$ is a two-sided ideal of $R$.  Show that this does indeed give a topology on $R$, which we call the \emph{Zariski topology}.  (Hint: if you understand the commutative case, you probably understand this case too.)
\item Show that if $R$ is Noetherian as a ring then ${\rm Spec}(R)$ is Noetherian as a topological space.
\item Give an example of a non-Noetherian ring $R$ with ${\rm Spec}(R)$ Noetherian.  (Hint: you can do it so that ${\rm Spec}(R)$ is a point.) 
\item (Optional) A $k$-algebra $R$ is said to be $\mathbb{N}$-\emph{graded} if $R$ has a decomposition $R=\bigoplus_{n=0}^{\infty} R_n$ where each $R_i$ is a $k$-vector subspace of $R$ and $R_i R_j\subseteq R_{i+j}$ for $i,j\ge 0$.  $R$ is called connected graded if $R_0=k$.  An element $r\in R$ is called homogeneous if there is some $n$ such that $r\in R_n$ and an ideal $I$ of $R$ is called homogeneous if $I$ is generated by homogeneous elements.  Given a connected graded ring $R$, we let ${\rm Proj}(R)$ denote the set of homogeneous prime ideals that do not contain the positive part with the subspace topology (regarding Proj as a subset of Spec).  Is there an example of a connected graded commutative domain $R$ where ${\rm Proj}(R)$ is Noetherian but ${\rm Spec}(R)$ is not?
\item We'll say that a topological space is Artinian if every ascending chain of closed subsets terminates.    Show that if $X$ is an Artinian space then every $x\in X$ has a unique smallest open neighbourhood $U_x$ of $x$.  
\item Show that a Hausdorff Artinian space is discrete and finite.  
\item Let $X$ be a Noetherian topological space and let $\phi :X\to X$ be a continuous map and let $Y$ be a closed subset and let $x\in X$.  Show that if $\phi^n(x)\in Y$ for all composite numbers $n$ then $\phi^n(x)\in Y$ for  all sufficiently large natural numbers $n$. 
\item Give an example that shows the conclusion to question 9 need not hold if $X$ is not noetherian.
\item A closed subset $Y$ of $X$ is irreducible if $Y$ cannot be written as a union of two proper closed subsets.  We define the dimension of a Noetherian space $X$ to be the supremum of the lengths of descending chains of irreducible closed subsets of $X$.  Show that $X=[0,1]$ with closed subsets being the empty set, $X$, and sets of the form $[1/n,1]$ with $n\ge 1$ gives an infinite dimensional noetherian topology on $X$.
\item Let $F(x)\in \mathbb{C}[[x]]$ be an algebraic power series of degree 2; that is we have
$$\sum_{i=0}^2 p_i(x)F(x)^i =0$$ for some complex polynomials $p_i(x)$, with $p_2p_0$ nonzero.  Show that $F(x)$ satisfies a differential equation of the form
$$\sum_{i=0}^m q_j(x) F^{(j)}(x)$$ where the $q_j(x)$ are polynomials, not all zero.  Hint: consider the equation $F^2+bF+c=0$ with $b,c$ rational functions in $x$.  Differentiate to get
$$2FF' + b'F + bF' + c' = 0.$$  Now try multiplying this by $(F+b)$ and see where you can go from there.  (In fact, this fact doesn't need degree 2---it is true for any algebraic power series, but the proof is a bit more involved.)
\end{enumerate}
\end{document}

\end{document}
