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   UNIVERSITY OF  KAISERSLAUTERN \hfill  Summerterm 1998 \\
   Mathematical Department \hfill 09th June, 1998 \\
   Gert-Martin Greuel \hfill  Christoph Lossen\\
   \vspace{0.5cm}


   \begin{center}
     \LARGE \bf Algebraic Geometry \\[0.1cm] \normalsize Exercise set 9
   \end{center}
   \vspace{0.5cm}
   \normalsize
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   \textbf{Due date:} 22th June, 1998 \hfill \textbf{10 a.\ m.}
   \\[0.2cm]
  
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   \setcounter{exercise}{25}

   \begin{exercise}
     Let \mbox{$X=\Spec A$} be an affine scheme and let 
     $$ 0 \lra \kf' \lra \kf \lra \kf''\lra 0$$
     be an exact sequence of {\em quasi-coherent\/} $\ko_X$-modules. Show that
     the sequence 
     $$ 0 \lra \Gamma(X,\kf') \lra \Gamma(X,\kf) \lra \Gamma(X,\kf'')\lra 0$$
     is exact.
     
     \smallskip 
     Compare this with Exercise 8b.
   \end{exercise}


   \medskip
   \begin{exercise}
     Let \mbox{$f:(X,\ko_X)\to (Y,\ko_Y)$} be a morphism of ringed spaces,
     $\kf$ an $\ko_X$-module and $\ke$ a locally free $\ko_Y$-module of
     finite rank. Show that
     $$ f_\ast (\kf \otimes_{\ko_X} f^\ast \ke) \, \cong \, f_\ast (\kf)
     \otimes_{\ko_Y} \ke\,.$$
     (This is called the {\em projection formula.\/})
     \smallskip
   \end{exercise}
   
   
   \medskip
   \begin{exercise}
     Let \mbox{$f:\P^1_{K} \to \P^1_{K}$} be the morphism defined by
     \mbox{$(x:y)\mapsto (x^3:y^3)$}. Calculate $f^\ast\ko_{\P^1_K}(1)$.
     
     \medskip
     {\it Hint:\/} The result should be of the form $\ko_{\P^1_K}(d)$, \mbox{$d\in
       \Z$}.
   \end{exercise}
   


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