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   UNIVERSITY OF  KAISERSLAUTERN \hfill  Summerterm 1998 \\
   Mathematical Department \hfill 08th May, 1998 \\
   Gert-Martin Greuel \hfill Thomas Keilen\\
   \vspace{0.5cm}

   \begin{center}
     \LARGE \bf Algebraic Geometry \\[0.1cm] \normalsize Exercise set 5
   \end{center}
   \vspace{0.5cm}
   \normalsize
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   \textbf{Due date:} 18th May, 1998\hfill 10 a.~m. 
   \\[0.2cm]
   
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   \setcounter{exercise}{12}

   \begin{exercise} 
     Show the following:
     \begin{enumerate}
        \item $\Spec(\Z)$ is the final object in the category of
          schemes, i.\ e.\ each scheme admits a unique morphism into
          $\Spec(\Z)$.
        \item To give a morphism from a scheme $(X,\ko_X)$
          into $\Spec(K)$ for a field $K$ is the same as to equip
          $\ko_X$ with the structure of a sheaf of $K$-algebras.
        \item To give a morphism from $\Spec(K)$ into a scheme
          $(X,\ko_X)$ is the same as to choose a point $x\in X$ and to 
          give a homomorphism 
          $K(x)=\ko_{X,x}/\m_{X,x}\rightarrow K$.
     \end{enumerate}
   \end{exercise}

   \begin{exercise}
     Let $\varphi:A\rightarrow B$ be a morphism of rings, and
     $(f,f^\#):\Spec(B)\rightarrow \Spec(A)$ the induced morphism of
     the affine schemes. Which of the following statements is correct?
     Prove or give a counter example (counter examples without
     details).
     \begin{enumerate}
        \item If $\varphi$ is injective, then $f$ is injective.
        \item If $\varphi$ is injective, then $f$ is dominant.
        \item If $\varphi$ is injective, then $f$ is surjective.
        \item If $\varphi$ is injective, then $f^\#$ is injective.
        \item If $f^\#$ is injective, then $\varphi$ is injective.
        \item If $\varphi$ is surjective, then $\im(f)\subseteq\Spec(A)$ is closed.
        \item If $\varphi$ is surjective, then $f$ is surjective.
        \item If $\varphi$ is surjective, then $f$ is injective.
        \item If $\varphi$ is surjective, then $f$ is a homeomorphism
          onto $\im(f)$.
        \item If $\varphi$ is surjective, then $f^\#$ is surjective.
        \item If $f^\#$ is surjective, then $\varphi$ is surjective.
        \item If $\im(f)\subseteq\Spec(A)$ is closed, $f$ is a homeomorphism onto
          $\im(f)$ and $f^\#$ is surjective, then $\varphi$ is surjective.
     \end{enumerate}
%     Show the following:
%     \begin{enumerate}
%        \item The following are equvalent:
%          \begin{enumerate}
%             \item $\varphi$ is injective.
%             \item $f^\#$ is injective.
%          \end{enumerate}
%          In that case $f$ is dominant, i.~e.~$f(\Spec(B))$ is dense
%          in $\Spec(A)$.
%        \item The following are equivalent:
%          \begin{enumerate}
%             \item $\varphi$ is surjective.
%             \item $f^\#$ is surjective and $f$ is a homeomorphism onto 
%               a closed subset of $\Spec(A)$, i.~e.~$(f,f^\#)$ is a
%               closed embedding.
%          \end{enumerate}
%     \end{enumerate}
   \end{exercise}

   \begin{exercise}
     Let $K$ be an algebraically closed field, and let
     $\textswab{t}:\textswab{Var}_{\textswab{aff}}(K)\rightarrow
     \textswab{Sch}(K)$ be the functor associating to an 
     affine variety $X$ the spectrum of its coordinate ring, where
     $\textswab{Var}_{\textswab{aff}}(K)$ denotes the category of affine varieties 
     over $K$ and $\textswab{Sch}(K)$ the category of
     schemes over $K$ respectively.\\
     Show the following:
     \begin{enumerate}
        \item $\textswab{t}$ is a fully faithful functor, i.~e.~$\textswab{t}$
          induces a natural bijection 
          \[ 
             \Hom_{\textswab{\scriptsize Var}_{\textswab{\scriptsize aff}}(K)}(X,Y)
             \longrightarrow\Hom_{\textswab{\scriptsize
                 Sch}(K)}(\Spec(K[X]),\Spec(K[Y])).
          \]
        \item ``The functor induces a homeomorphism from an affine
          variety to the set of closed points of the corresponding
          scheme'',
          i.~e.~$$\beta:X\rightarrow\mSpec(K[X]):P\mapsto\m_P$$ is a 
          homeomorphism, where $\mSpec(K[X])\subseteq\Spec(K[X])$
          carries the subspace topology.
        \item ``The restriction of the structure sheaf 
          of $\Spec(K[X])$ via this homeomorphism is the sheaf of
          regular functions of the affine variety $X$'', i.~e.~the map 
          $$\beta^\#:\ko_{\Spec(K[X])|\mSpec(K[X])}\rightarrow\beta_*\ko_X$$ 
          is an isomorphism of sheaves. 
     \end{enumerate}
   \end{exercise}

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