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   \Blatt{4}
   \Abgabe{13/05/2002}



   \Nr
   Let $\big(f,f^\#\big):(X,\ka_X)\rightarrow(Y,\ka_Y)$ be a morphism
   of ringed spaces. Show that for any $x\in X$ we have
   $f^\#_x=\widehat{f}_x$.

   \Nr
   Let $\big(f,f^\#\big):(X,\ka_X)\rightarrow(Y,\ka_Y)$ be a morphism
   of ringed spaces. Show that $\big(f,f^\#\big)$ is an isomorphism if
   and only if $\big(f,f^\#\big)$ has a two-sided inverse.

   \Nr
   Let $K$ be an algebraically closed field and 
   $f:\A^1_K\rightarrow \A^2_K:t\mapsto\big(t^2,t^3)$. We know that
   $X:=\im(f)=V\big(x^3-y^2\big)$ and that $f$ induces a morphism of
   locally ringed spaces
   $\big(f,f^\#\big):\big(\A^1_K,\ko_{\A^1_K}\big)\rightarrow(X,\ko_X)$
   via  $f^\#(U):\ko_X(U)\rightarrow\ko_{\A^1_K}\big(f^{-1}(U)\big):
   \varphi\mapsto\varphi\circ f$. 
   
   Show that for any $t_0\in\A^1_K$ the morphism
   \begin{displaymath}
     f_{t_0}^\# : \ko_{X,f(t_0)} \rightarrow \ko_{\A^1_K,t_0}
   \end{displaymath}
   is a monomorphism, and that $f_{t_0}^\#$ is surjective if and only
   if $t_0\not=0$.
   

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