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   UNIVERSITY OF  KAISERSLAUTERN \hfill  Summerterm 1998 \\
   Mathematical Department \hfill 05th June, 1998 \\
   Gert-Martin Greuel \hfill Christoph Lossen\\
   \vspace{0.5cm}

   \begin{center}
     \LARGE \bf Algebraic Geometry \\[0.1cm] \normalsize Exercise set 8
   \end{center}
   \vspace{0.5cm}
   \normalsize
   \parindent0cm
   \textbf{Due date:} 15th June, 1998 \hfill \textbf{10 a.\ m.}
   \\[0.2cm]
   
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   \setcounter{exercise}{22}

%   \begin{exercise}
%     Let $(X,\ko_X)$ be a ringed space and $\kf$ a sheaf of
%     $\ko_X$-modules of finite type. Given $x\in U\subseteq X$ open and
%     $s_1,\ldots,s_r\in\kf(U)$ such that they generate $\kf_x$, then 
%     there is an open neighbourhood $V$ of $x$ such that they generate 
%     $\kf_y$ for each $y\in V$.
%   \end{exercise}
%   \begin{proof}
%     C.~f.~Jean-Pierre Serre, Faisceaux alg\'ebrique coh\'erent,
%     Annals of Mathematics, 61,2 (1955) 197-278, I.2 prop.~1.
%   \end{proof}

   \begin{exercise}
     Let $(X,\ko_X)$ be a ringed space and $0\rightarrow
     \kf'\stackrel{\alpha}{\rightarrow}\kf\stackrel{\beta}{\rightarrow }\kf''\rightarrow 0$ an exact
     sequence of $\ko_X$-modules.  Show, if $\kf$ and $\kf''$ are
     coherent, then so is $\kf'$.\\
     \small (Hint, if $\gamma:\ko_X^n\rightarrow\kf$ is a surjection,
     then show that there is an induced surjection from
     $\ker(\beta\circ\gamma)$ to $\kf'$.)
   \end{exercise}
   \begin{proof}
     C.~f.~Jean-Pierre Serre, Faisceaux alg\'ebrique coh\'erent,
     Annals of Mathematics, 61,2 (1955) 197-278, I.2 thm.~1.
   \end{proof}


   \begin{exercise}
     Let $(X,\ko_X)$ be a ringed space with $\ko_X$ coherent, and let
     $\kf$ be a coherent
     $\ko_X$-module. Define the annihilator sheaf of $\kf$ (in a
     sensible way) and show
     that it is a coherent sheaf of ideals of $\ko_X$.
   \end{exercise}
   \begin{proof}
     Define the annihilator $\ann(\kf)$ of $\kf$ to be the kernel of the
     morphism of sheafs
     \begin{displaymath}
       \varphi:\ko_X\rightarrow \HOM{13}_{\ko_X}(\kf,\kf),
     \end{displaymath}
     given by the multiplication. Then for each open set $U$
     we have that $\ann(\kf)(U)=\ann(\kf(U))$, and moreover
     $\ann(\kf)_x=\ann(\kf_x)$ for each $x\in X$. The latter follows
     from the fact that $\ker(\varphi)_x\cong\ker(\varphi_x)$ and - since
     $\kf$ is of finite presentation -
     $\HOM{13}_{\ko_X}(\kf,\kf)_x\cong\Hom_{\ko_{X,x}}(\kf_x,\kf_x)$.\\
     Thus $\ann(\kf)$ is coherent, since $\ko_X$ and
     $\HOM{13}_{\ko_X}(\kf,\kf)$ are - the latter since $\kf$ is so.
   \end{proof}


   \begin{exercise}
     Let \mbox{$X=\Spec(A)$}, \mbox{$Y=\Spec (B)$} be noetherian affine
     schemes and \mbox{$f:X\to Y$} a morphism.
     \begin{enumerate}
        \item Show by example that if $\kf$ is a coherent $\ko_X$-module and
          $f$ is {\em quasi-finite\/}, then $f_\ast \kf$ need not be a
          coherent $\ko_Y$-module. 
        \item Let $f$ be a {\em finite\/} morphism and $\kf $ a coherent
          $\ko_X$-module. Prove that $f_\ast \kf$ is a
          coherent $\ko_Y$-module.
     \end{enumerate}
   \end{exercise}

%   \begin{exercise}
%     Let $X=\Spec(A)$ be a noetherian affine scheme.
%     \begin{enumerate}
%        \item Every quasi-coherent sheaf $\kf$ of $\ko_X$-modules is the
%          union of its coherent subsheaves, i.~e.~for each $U\subseteq 
%          X$ open this is true for $\kf(U)$.
%        \item If $U\subseteq X$ is open and $\kf$ a coherent sheaf of
%          $\ko_U$-modules, 
%          then there is coherent sheaf $\kf'$ of $\ko_X$-modules, such 
%          that $\kf'_{|U}\cong\kf$.\\
%          \small(Hint, if $i: U\hookrightarrow X$ denotes the
%          inclusion, then apply part a.~to the quasi-coherent sheaf
%          $i_*\kf$.) 
%     \end{enumerate}
%     Remark: the statements holt also for an arbitrary noetherian
%     scheme $X$.
%   \end{exercise}
%   \begin{proof}
%     Kein Beweis. Die Aufgabe stammt aus Hartshorne, II.5 ex.~18.
%   \end{proof}


   \begin{exercise-remark}
     Replace the word ``coherent'' in lemma 1 on the exercise sheets
     three and four and in exercise 10 by the expression ``of finite presentation''.
   \end{exercise-remark}

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