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   \Blatt{2}
   \Abgabe{09/05/2002}



   \Nr
   Let $X$ be a topological space.
   If $\kf$ is a presheaf of sets on $X$ we define
   $E:=\Gamma\kf:=\coprod_{x\in X}\kf_x$ and
   $\pr:\Gamma\kf\rightarrow X$ the canonical projection. 

   For $s\in\kf(U)$ with $U\subseteq X$ open, we define
   $\tilde{s}:U\rightarrow \Gamma\kf:x\rightarrow s_x$, and we use
   these maps to define a topology on $\Gamma\kf$. Namely, 
   $V\subseteq \Gamma\kf$ is open in $\Gamma\kf$ if and only if for any $U\subseteq
   X$ open and for any $s\in\kf(U)$ the preimage
   $\tilde{s}^{-1}(V)\subseteq U$ is open in $U$.

   Show that $\pr:\Gamma\kf\rightarrow X$ is a covering space.

   \Nr
   Let $K$ be an algebraically closed field, $A=K[x_1,\ldots,x_n]/I$,
   $X=V(I)\subseteq\A_K^n$, $M$ an $A$-module and $\widetilde{M}$ the
   sheafification of $M$. Show that:
   \begin{enumerate}
   \item $\big(\widetilde{M}\big)_{\underline{z}}\cong
     M_{\m_{\underline{z}}}$ for all $\underline{z}=(z_1,\ldots,z_n)\in
     X$, 
     where $\m_{\underline{z}}=(x_1-z_1,\ldots,x_n-z_n)$.
   \item $\Gamma\big(D(f),\widetilde{M}\big)\cong M_f$ for all $f\in
     A$.
   \item $\Gamma\big(X,\widetilde{M}\big)\cong M$.
   \end{enumerate}
   {\small Hint, recall the proof of $K[X]\cong\ko(X)$ from the
     lecture Algebraic Geometry I.}

   \Nr 
     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 morphisms of sheaves 
     \begin{center}
        $\alpha^\kf:f^{-1}f_*\kf\rightarrow \kf$
        \hspace{0.5cm}and\hspace{0.5cm}
        $\beta^\kg:\kg\rightarrow f_*f^{-1}\kg$
     \end{center}
     inducing a natural bijection of sets of sheaf morphisms
     \[
        \Hom_X(f^{-1}\kg,\kf)\cong\Hom_Y(\kg,f_*\kf)
     \]
     via $\varphi\mapsto (f_*\varphi)\circ\beta^\kf$ with inverse
     $\psi\mapsto\alpha^\kg\circ \big(f^{-1}\psi\big)$.
   
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