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   \large UNIVERSITY OF  KAISERSLAUTERN \hfill  Summerterm 1998 \\
   Mathematical Department \hfill 19th April, 1998 \\
   Gert-Martin Greuel \hfill Thomas Keilen
   \vspace{0.2cm}

   \begin{center}
     \LARGE \bf Algebraic Geometry 
   \end{center}
   \vspace{0.5cm}

   \normalsize
   \parindent0cm
   \textbf{Due date:} 27th April, 1998 \hspace{0.5cm}(10 a.~m.)\hfill
   \textbf{Room: 48-582 !!!}
%   \\[0.2cm]
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   \setcounter{exercise}{3}

   \begin{exercise}
     Let $X$ be a topological space, $\varphi:\kf\rightarrow\kg$ a
     morphism of sheaves of abelian groups on $X$.
%     Show the following:
     \begin{enumerate}
        \item The following are equivalent:
          \begin{enumerate}
             \item $\varphi$ is injective.
             \item $\ker(\varphi(U))=0$ for all $U\subseteq X$ open.
             \item $\ker(\varphi_p)=0$ for all $p\in X$.
          \end{enumerate}
        \item The following are equivalent:
          \begin{enumerate}
             \item $\varphi$ is surjective.
             \item $\coker(\varphi_p)=0$ for all $p\in X$.
          \end{enumerate}
        \item The following are equivalent:
          \begin{enumerate}
             \item $\varphi$ is an isomorphism.
             \item $\varphi_p$ is an isomorphism for all $p\in X$.
             \item $\varphi$ is injective and surjective.
          \end{enumerate}
     \end{enumerate}
     \small(Hint, the proofs of a.~and b.~are staight forward using
     proposition 2.6.)
   \end{exercise}

   \begin{exercise}
     Let $f:X\rightarrow Y$ be a continous map of topological spaces,
     $\kf$ a sheaf of abelian groups on $X$, and $\kg$ a sheaf of
     abelian groups on $Y$.\\
     Show that there are natural maps 
     \begin{center}
        $f^{-1}f_*\kf\rightarrow \kf$
        \hspace{0.5cm}and\hspace{0.5cm}
        $\kg\rightarrow f_*f^{-1}\kg,$
     \end{center}
     and use these maps to show that there is a natural bijection of
     sets 
     \[
        \Hom_X(f^{-1}\kg,\kf)\cong\Hom_Y(\kg,f_*\kf).
     \]
   \end{exercise}

   \begin{exercise}
     Let $X$ be an algebraic variety over an algebraically closed
     field $K$ and $\ko_X$ be the structure sheaf of $X$. Let
     $Y\subseteq X$ be a closed subset. Obviously the presheaf defined 
     by the the assignment
     \begin{displaymath}
       U\mapsto \ki_Y(U):=\menge{f\in\Gamma(U,\ko_X)| f(p)=0
         \;\forall p\in Y\cap U}
     \end{displaymath}
     is a sheaf. It is called the \textbf{ideal sheaf} $\ki_Y$
     of $Y\subseteq X$.\\
     Show that the structure sheaf $\ko_Y$ of the
     algebraic variety $Y$ satisfies 
     \begin{displaymath}
       i_*\ko_Y\cong\ko_X/\ki_Y,
     \end{displaymath}
     where $i:Y\hookrightarrow X$ denotes the inclusion.
   \end{exercise}



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