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   \large UNIVERSITY OF  KAISERSLAUTERN \hfill  Winterterm 1997/98 \\
   Mathematical Department \hfill 01st February, 1998 \\
   Gert-Martin Greuel \hfill Thomas Keilen\\
   \vspace{0.5cm}

   \begin{center}
     \LARGE \bf Introduction to Algebraic Geometry \\
   \end{center}
   \vspace{0.5cm}
   \normalsize
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   \textbf{Due date:} 09th February, 1998
   \\[0.2cm]
   
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   \setcounter{exercise}{43}

   \begin{exercise}
     Let $K$ be algebraically closed, $X$ and $Y$ be varieties over
     $K$ with $x\in X$, $y\in Y$ such that $\ko_{X,x}\cong
     \ko_{Y,y}$. \\
     Show that there are open subvarieties $x\in U\subseteq X$ and
     $y\in V\subseteq 
     Y$ and an isomorphism $\varphi:U\rightarrow V$ such that
     $\varphi(x)=y$. 
   \end{exercise}

   \begin{exercise}
     Let $K$ be an algebraically closed field, $X$ a variety over $K$
     and $x\in X$ a regular point.\\
     Show that there is a canonical surjective map
     $$\Der_K(\ko_{X,x})\rightarrow \Der_K(\ko_{X,x},K)=T_xX$$ 
     which is as well $\ko_{X,x}$-linear as $K$-linear.\\
     Moreover, show that this induces an isomorphism
     $$\Der_K(\ko_{X,x})/\m_x\Der_K(\ko_{X,x})\cong T_xX.$$
     Is this still true if $x$ is a singular point of $X$?
   \end{exercise} 

   \begin{exercise}
     Let $K$ be an algebraically closed field, $X$ and $Y$ varieties over
     $K$, $x\in X$, $y\in Y$.
     Show that $$T_{(x,y)}(X\times Y)\cong T_xX\times T_yY.$$
   \end{exercise}

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