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   \large UNIVERSITY OF  KAISERSLAUTERN \hfill  Summerterm 1998 \\
   Mathematical Department \hfill 15th April, 1998 \\
   Gert-Martin Greuel \hfill Thomas Keilen\\
   \vspace{0.5cm}

   \begin{center}
     \LARGE \bf Algebraic Geometry \\
   \end{center}
   \vspace{0.5cm}
   \normalsize
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   \textbf{Due date:} 20th April, 1998 
   \\[0.2cm]
   
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   \setcounter{exercise}{0}
   
   \begin{exercise}
     Let $X$ be a topological space, $\cal F$ a presheaf of abelian groups on $X$,
     $U\subseteq X$ open, and $s\in\kf(U)$.\\
     Show that the support $\supp(s)=\menge{x\in U|s_x\not=0}$ of $s$
     is a closed subset of $U$, and find an example which shows that
     the support $\supp(\cal F)=\menge{x\in X|\kf_x\not=0}$ of $\cal
     F$ in $X$ need not be closed. 
   \end{exercise}

   \begin{exercise}
     Let $X$ be a topological space, $\kf$ and $\kg$ be sheaves of
     abelian groups on
     $X$. \\
     Show that the presheaf of abelian groups $\kf\oplus\kg$ defined
     by $U\mapsto 
     \Gamma(U,\kf)\oplus\Gamma(U,\kg)$ for $U\subseteq X \mbox{ open}$ with
     the natural restriction maps is indeed a sheaf, and prove that
     for all $x\in X$
     there is a canonical isomorphism $$\left(\kf\oplus\kg\right)_x
     \cong \kf_x\oplus\kg_x.$$
   \end{exercise}

   \begin{exercise}
     Let $X$ be a topological space, $U\subseteq X$ open, $\kf$ and
     $\kg$ sheaves of abelian groups on $X$. We know already that the $\kf_{|U}$ with
     $\Gamma(V,\kf_U)=\Gamma(V,\kf)$ for $V\subseteq U$ open and with
     the natural restriction maps gives again a sheaf of abelian groups. \\
     Show the following:
     \begin{enumerate} 
        \item The set $\Hom(\kf_{|U},\kg_{|U})$ of sheaf morphisms
           carries a natural structure of an abelian group.
        \item The assignment $V\mapsto \Hom(\kf_{|V},\kg_{|V})$ for
          $V\subseteq X$ open gives rise to a sheaf of abelian groups, 
          which we denote by $\HOM{13}(\kf,\kg)$.
        \item For any $x\in X$ we have a natural homomorphism
          of abelian groups
          $$\alpha:\HOM{13}(\kf,\kg)_x\rightarrow\Hom(\kf_x,\kg_x).$$ 
          Note: $\alpha$ is not i.~g.~injective nor surjective.
     \end{enumerate}
   \end{exercise}

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