\documentclass[12pt]{article} 

%\usepackage{epsfig}
%\usepackage{latexsym}
%\usepackage{amssymb} 
%\usepackage{theorem}
%\usepackage{euler}
%\usepackage{epic}
%\usepackage{eepic}
%\usepackage{color}
%\usepackage{curves}

\input msc.sty

%%%   \documentclass[12pt]{article}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 5.3.1996
% Privater Stil fuer A4-Papier
% (Bernd Kreussler)
%
% Nach oben rechts und links 
% sind es jeweils 30 mm bis 
% zum Papierrand. 
% Unten 60pt = ca. 21mm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  Eine Alternative waere:
%   \usepackage{a4}
%%%%%%%%%%%%%%%%%%%
%  Standartwerte:
%   \headheight12pt
%   \headsep25pt
%   \topskip10pt   %11pt %12pt  je nach Schriftgroesse
%   \footskip30pt

%\pagestyle{myheadings}  % Seitennummer oben
%\markright{Drucktest}

%%%% \parindent0pt

\setlength{\topmargin}{30mm}     % Grundlinie der 1. Zeile 30mm von Oberkante
\addtolength{\topmargin}{-1in}
\addtolength{\topmargin}{-\headsep}
\addtolength{\topmargin}{-\headheight}
\addtolength{\topmargin}{-\topskip}

\setlength{\textheight}{267mm} % Unterkante der Fussbox 30pt-voffset ueber Rand
\addtolength{\textheight}{\topskip}
\addtolength{\textheight}{-\footskip}
\addtolength{\textheight}{-30pt}

\setlength{\oddsidemargin}{-1in} % Linker Rand 30mm vom Papierrand
\addtolength{\oddsidemargin}{30mm}
\setlength{\evensidemargin}{\oddsidemargin}

\setlength{\textwidth}{150mm} % Rechter Rand 30mm vom Papierrand


 

\begin{document} 
 
   \pagestyle{empty} 
   \parindent0cm
   \large UNIVERSITY OF  KAISERSLAUTERN \hfill  Winterterm 1997/98 \\
   Mathematical Department \hfill October 25, 1997 \\
   Gert-Martin Greuel \hfill Thomas Keilen\\
   \vspace{0.5cm}

   \begin{center}
     \LARGE \bf Introduction to Algebraic Geometry \\
   \end{center}
   \vspace{0.5cm}
   \normalsize
   \parindent0cm
   \textbf{Due date:} Mo, November 10, 1997, 12:00
   \\[0.2cm]
   \setcounter{exercise}{5}
   \parindent0cm

   \begin{exercise}
     Let $X$ be any topological space.\\
     Show that if $X$ is noetherian, then $X$ is quasi compact, i.\ e. every open cover has a
          finite subcover.
     How about the converse?
   \end{exercise}

   \begin{exercise}
     Let $K$ be any field, $I\unlhd K[X_1,\ldots,X_n]=K[\underline{X}]$ an ideal. \\
     Proof the following statements, or find counterexamples.
     \begin{enumerate}
        \item If $V(I)=\A^n(K)$, then $I=(0)$.
        \item If $V(I)=\emptyset$, then $I=K[\underline{X}]$.
        \item If $I$ is a prime ideal, then $V(I)$ is irreducible.
     \end{enumerate}
     Is it possible to weaken respectively to strengthen the hypotheses such
     that the results remain respectively become true?
   \end{exercise}

   \begin{exercise}
     Let $K$ be algebraically closed, and $I\unlhd K[\underline{X}]$
     be a radical ideal. \\
     Show that $I=\bigcap\menge{\m | I\subseteq
       \m\in\mSpec(K[\underline{X}])}$, where
     $\mSpec(K[\underline{X}])$ is the set of maximal ideals of $K[\underline{X}]$.
     Interprete this geometrically. 
   \end{exercise}

   \begin{exercise}
     Let $K$ be any field, and let $\Spec(K[X,Y])$ be the set of prime 
     ideals of $K[X,Y]$.\\
     Show that $\Spec(K[X,Y])$ is the union of the following sets:
     \begin{itemize}
        \item \menge{(0)},
        \item \menge{(f)|f\in K[X,Y] \mbox{ irreducible}},
        \item $\mSpec(K[X,Y])$.
     \end{itemize}
     \small
     Use the following theorem without proof:\\
     \textit{A noetherian integral domain is a unique
       factorization domain if and only if every prime ideal of height 
       1 is principal.  (cf. R. Hartshorne, I.1.12.A.)}
%       \textit{$V$ is an irreducible $n-1$-dimensional affine algebraic subset of $\A_K^n$
%       if and only if $I(V)=(f)$ for some irreducible polynomial $f\in 
%       K[\underline{X}]$. (cf. R. Hartshorne, I.1.13.)}
   \end{exercise}

   \begin{exercise}
     Find in the examples of exercise 2  ascending
     chains of irreducible algebraic subsets of maximal length, and
     find the corresponding prime ideals.
   \end{exercise}
   



\end{document} 
