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   \Blatt{12}
   \Abgabe{04/02/2002}

   Throughout this example sheet $K$ will be an algebraically closed field.

%   \Nr
%   Let $X\subseteq \P_K^n$ be a projective variety  with homogenous
%   coordinate ring $K[X]$ and let $p\in
%   X$.  Show that
%   \begin{displaymath}
%     \ko_{X,p}\cong K[X]_{(\m_p)}:=
%     \left\{\tfrac{f}{g}\in \Quot\big(K[X]\big)
%       \;\Big|\; f,g\in K[X] \mbox{ homog.}, \deg(f)=\deg(g),
%       g(p)\not=0 \right\}.
%   \end{displaymath}

%   {\scriptsize Note that an element in $K[X]=K[\underline{x}]/I(X)$ is homogenous if and
%     only if it has a homogenous representative.}


   \Nr\\
   Show that $\left\{\frac{\partial\phantom{x}}{\partial x_1},\ldots,
     \frac{\partial\phantom{x}}{\partial x_n}\right\}$ is a basis of 
   $\Der_K\big(K[x_1,\ldots,x_n]\big)$ as 
   $ K[x_1,\ldots,x_n]$-module. 

   \Nr
   Let $X$ and $Y$ be quasi-projective varieties and $(p,q)\in X\times
   Y$. Show that as $K$-vectorspaces
   \begin{displaymath}
     T_{(p,q)}(X\times Y)\cong T_p(X)\oplus T_q(Y).
   \end{displaymath}

   \Nr
   Let $X$ and $Y$ be quasi-projective varieties and let $p\in X$ and
   $q\in Y$. Suppose that $\varphi:\ko_{Y,q}\rightarrow \ko_{X,p}$ is
   a local $K$-algebra isomorphism.
   Show that there are 
   open neighbourhoods $U(p)\subseteq X$ of $p$ and $U(q)\subseteq Y$
   of $q$ and an isomorphism $f:U(p)\rightarrow U(q)$ with $f(p)=q$.
   
   \bigskip

   {\scriptsize Note, a local $K$-algebra is a $K$-algebra which is
     local as a ring, and a morphism of two local $K$-algebras is said
     to be local, if the image of the unique maximal ideal is
     contained in the unique maximal ideal.

     Hint: By Exercise 31a you may assume that $X$ and $Y$ are closed
     subsets of $\A^n_K$ and that $p=q=0$. Consider then the
     $\varphi(x_i)$.}

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