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   UNIVERSITY OF  KAISERSLAUTERN \hfill  Summerterm 1998 \\
   Mathematical Department \hfill 20th June, 1998 \\
   Gert-Martin Greuel \hfill  Christoph Lossen\\
   \vspace{0.5cm}

   \begin{center}
     \LARGE \bf Algebraic Geometry \\[0.1cm] \normalsize Exercise set 10
   \end{center}
   \vspace{0.5cm}
   \normalsize
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   \textbf{Due date:} 29th June, 1998 \hfill \textbf{10 a.\ m.}
  
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\bigskip
   \setcounter{exercise}{28}


\begin{exercise}
Let $X$ be a noetherian scheme, and let $\kf$ be a coherent $\ko_X$-module. 
Prove the
following: 
\begin{enumerate}
\item[(a)] If the stalk $\kf_x$ is a free $\ko_{X,x}$-module for some point
    \mbox{$x\in 
    X$}, then there is a neighbourhood $U$ of $x$ such that $\kf \big|_U$ is
  free.
\item[(b)] $\kf$ is locally free if and only if its stalks $\kf_x$ are free
  $\ko_{X,x}$-modules for all \mbox{$x\in X$}.
\item[(c)] $\kf$ is invertible (i.e.\ locally free of rank 1) if and only if
  there is a coherent sheaf $\kg$ such that \mbox{$\kf \otimes \kg \cong
  \ko_X$}. 
\end{enumerate}

(Note that the statement in (c) justifies the terminology ``invertible'': it
means that $\kf$ is an invertible element of the monoid of coherent sheaves
under the operation $\otimes$.)
\end{exercise}

\medskip

\begin{exercise}
Let $X$ be a noetherian scheme, and $\kf$ a coherent $\ko_X$-module. We shall
consider the function
$$ \varphi(x) \,:=\, \dim_{k(x)} \kf_x \otimes_{\ko_{X,x}} k(x)\,,$$
where \mbox{$k(x)=\ko_{X,x}/\mathfrak{m}_{X,x}$} is the residue field at the
point $x$. Use Nakayama's lemma to prove the following results:
\begin{enumerate}
\item[(a)] The function $\varphi$ is {\it upper semi-continuous,} i.e., for any
    \mbox{$n\in \Z$}, the subset 
$$\bigl\{\!\: x\in X\:\big|\: \varphi(x)\geq n\!\:\bigr\}\,\subset \,X$$
is closed.
\item[(b)] If $\kf$ is locally free, and $X$ is connected, then $\varphi$ is a
  constant function. 
\item[(c)] Conversely, if $X$ is reduced, and $\varphi$ is constant, then $\kf$
  is locally free. 
\end{enumerate}
\end{exercise}

\medskip

\begin{exercise}
Let $X$ be a scheme and let $\kl,\km$ be two very ample
invertible sheaves on $X$. Show that \mbox{$\kl \otimes_{\ko_X} \!\!\:\km$} 
is also very ample. 

\smallskip
{\it Hint:\/} Use a Segre embedding.
\end{exercise}




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