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    UNIVERSITY OF  KAISERSLAUTERN \hfill  Summerterm 1998 \\
   Mathematical Department \hfill 21st April, 1998 \\
   Gert-Martin Greuel \hfill Thomas Keilen\\
   \vspace{0.2cm}

   \begin{center}
     \LARGE \bf Algebraic Geometry \\[0.1cm] \normalsize Exercise set 3
   \end{center}
   \vspace{0.3cm}
   \normalsize
   \parindent0cm
   \textbf{Due date:} 4th May, 1998 \hspace{0.5cm}(10 a.~m.)
   \\[0.2cm]
   Exercise 7 is to be replaced by exercise 6 from the last set of exercises.
   
   \parindent0cm

   \setcounter{exercise}{6}

   \begin{exercise}
     Let $(I,\leq)$ be a directed partially ordered set, and let $\kc=(R-\mod)$ 
     be the category of $R$-modules over some commutative ring $R$
     with one.\\
     Show that the direct limit functor from the category of direct
     systems over $(I,\leq)$ in $\kc$ to the category $\kc$ is exact.
   \end{exercise}

   \begin{exercise}
     Show that the functor $\Gamma(X,.)$ from the category of sheaves
     of abelian groups
     on the topological space $X$ to the category of abelian groups is
     left exact, but not i.~g.~exact. \\
     I.~e.~show the following:
     \begin{enumerate}
        \item Let $X$ be a topological space and
          $0\rightarrow\kf'\rightarrow\kf\rightarrow\kf''$ be an exact 
          sequence of sheaves of abelian groups on $X$, then the
          sequence 
          \begin{displaymath}
            0\rightarrow\Gamma(X,\kf')\rightarrow\Gamma(X,\kf)\rightarrow\Gamma(X,\kf'')
          \end{displaymath}
          of abelian groups is exact.
        \item Let $X=\P^1_K$, $\kf=\ko_X$, $\kf'=\ki_Y$, where
          $Y\subseteq X$ is a subvariety consisting of two different
          points $p\not=q$, and $\kf''=i_*\ko_p\oplus j_*\ko_q$, where 
          $i:\menge{p}\hookrightarrow X$ and
          $j:\menge{q}\hookrightarrow X$ denote the inclusions. \\
          Then 
          \begin{displaymath}
            0\rightarrow\kf'\rightarrow\kf\rightarrow\kf''\rightarrow 0
          \end{displaymath}
          is  an exact exact sequencs of sheaves, but 
          \begin{displaymath}
            \Gamma(X,\kf)\rightarrow\Gamma(X,\kf'')
          \end{displaymath}
          is not surjective.
     \end{enumerate}
     (Note that this implies that exercise 4.b.~cannot be extended in
     the same way as exercise 4.a.)
   \end{exercise}

   \begin{exercise}
     Let $(X,\ko_X)$ be a ringed space, $\kf$ a locally free
     $\ko_X$-module of finite rank $r$, $\ke$ and $\kg$ any
     $\ko_X$-modules. We denote by
     $\ke^\vee:=\HOM{13}_{\ko_X}(\ke,\ko_X)$ the dual of the sheaf
     $\ke$. Show that there are natural isomorphisms of $\ko_X$-modules:
     \begin{enumerate}
         \item $(\kf^\vee)^\vee\cong \kf$,
         \item
           $\kf^\vee\otimes_{\ko_X}\kg\cong\HOM{13}_{\ko_X}(\kf,\kg)$, and
         \item There is a natural isomorphism of
           $\Gamma(X,\ko_X)$-modules 
           \begin{displaymath}
              \Hom(\ke\otimes_{\ko_X}\kf,\kg)\cong\Hom(\ke,\HOM{13}_{\ko_X}(\kf,\kg)).
           \end{displaymath}
     \end{enumerate}
     \small (Hint, find the natural morphisms and use the following
     lemma to show that they induce isomorphisms on the stalks.)
   \end{exercise}

   \begin{exercise-lemma}
     Let $(X,\ko_X)$ be a ringed space, $\kf, \kg$ be
     $\ko_X$-modules. Moreover, let $x\in U\subseteq X$ open such that 
     we have an exact sequence
     \begin{equation}\label{blubber}
       \ko_{X|_U}^r\stackrel{\delta}{\longrightarrow}\ko_{X|_U}^s
       \stackrel{\gamma}{\longrightarrow}\kf_{|U}\rightarrow 0.  
     \end{equation}\\
     The natural $\ko_X$-module morphism
     $$\HOM{13}_{\ko_X}(\kf,\kg)_x\stackrel{\alpha_x}{\longrightarrow}
     \Hom_{\ko_{X,x}}(\kf_x,\kg_x)$$ 
     is then an isomorphism. \\
     Note: if every point $x$ of $X$ has a neighbourhood $U$
     such that $\kf_{|U}$ satisfies an equation of type (\ref{blubber}),
     the sheaf $\kf$ is called a 
     $\ko_X$-module \textbf{of finite presentation}, or
     \textbf{coherent} if $(X,\ko_X)$ is a sheme.
   \end{exercise-lemma}
   \begin{proof}
     The same proof as in exercise 3 gives a natural morphism of
     $\ko_{X,x}$-modules (instead of abelian groups)
     $\alpha_x:\HOM{13}_{\ko_X}(\kf,\kg)_x\rightarrow
     \Hom_{\ko_{X,x}}(\kf_x,\kg_x)$. It remains to show, that $\alpha_x$ 
     under the above condition is bijective. W.~l.~o.~g.~$U=X$.
     \begin{enumerate}
        \item \underline{Injectivity:}\footnote{For this we only need the
          surjection $\xymatrix{\gamma:\ko_X^s\ar@{->>}[r]&\kf}$.} Let
          $\varphi\in\HOM{13}_{\ko_X}(\kf,\kg)_x$ be such that
          $\alpha_x(\varphi)\equiv 0$.\\
          Idea: lift $\varphi$ on some neigbourhood $W$ of $x$ via
          $\gamma$ to a free sheaf and
          use the fact that $\gamma$ is an epimorphism (in the
          categorical sense). \\
          Choose a representative $(\tilde{\varphi}, V)$ of $\varphi$
          with
          $\tilde{\varphi}\in\HOM{13}_{\ko_{X|V}}(\kf_{|V},\kg_{|V})$.
          Remind yourself that $\alpha_x([(\tilde{\varphi},V)])$ is
          just the germ $\tilde{\varphi}_x$ of $\tilde{\varphi}$ at
          $x$. Since $\tilde{\varphi}_x=0$, also
          $(\gamma\circ\tilde{\varphi})_x =
          \gamma_x\circ\tilde{\varphi}_x=0$.
          This is equivalent to the fact that the standard basis
          $e_{1,x},\ldots,e_{s,x}$ of $\ko_{X,x}^s$ gets mapped to zero, that is there
          exist open neighbourhoods $V_i(x)$ such that
          $(\gamma\circ\tilde{\varphi})(V_i)(e_{i|V_i})=0$. Setting
          $W:=V_1\cap\ldots\cap V_s$, we get that
          $(\gamma\circ\tilde{\varphi})_{|W}=0$,
          since $\ko_{X|W}^s$ is free with free generators
          $e_{1|\tilde{W}},\ldots,e_{s|\tilde{W}}$ for
          $\tilde{W}\subseteq W$ open. \\
          Thus we have the following commutative diagram:
          \begin{displaymath}
            \xymatrix{\ko_{X|W}\ar[r]^{\gamma_{|W}} &
              \kf_{|W}\ar@<1ex>[r]^{\tilde{\varphi}_{|W}}
              \ar@<-1ex>[r]_0 &\kg_{|W}}.
          \end{displaymath}
          The fact that $\gamma$ is an epimorphism\footnote{Since the
            category (Shv/X) is abelian, the exactness of
            $\ko_X^s\stackrel{\gamma}{\rightarrow} \kf\rightarrow 0$
            implies that $\gamma$ is an epimorphism. If you don't
            want to use that (Shv/X) is abelian, replace the above
            diagram by the cokernel 
            diagram of $\gamma$ and use the fact that the cokernel is
            $0$. (See also the following footnote \ref{follow}.)}  implies thus  
          $\tilde{\varphi}_{|W}=0$. 
        \item \underline{Surjectivity:} Let
          $\psi\in\Hom_{\ko_{X,x}}(\kf_x,\kg_x)$.\\
          This gives the following ``cokernel diagram'' of abelian
          groups:
          \begin{displaymath}
            \xymatrix{
              &\ko_{X,x}^r\ar[r]^{\delta_x}\ar@/_/[drr]^{0} &
              \ko_{X,x}^s\ar[r]^{\gamma_x}\ar[dr]^{\Psi_x} & \kf_x
              \ar[r]\ar[d]^\psi & 0\\ 
              & & & \kg_x & }.       
          \end{displaymath}
          Idea: extend $\Psi_x$ to some neighbourhood $V$ of $x$ and
          use the universal property of the sheaf cokernel to get a
          unique extension of $\psi$ making the cokernal diagram
          commute. \\
          Let $e_{1,x},\ldots,e_{s,x}$ be the standard basis of
          $\ko_{X,x}^s$ and set $f_{i,x}:=\Psi_x(e_{i,x})$. Choosing
          representatives $(f_i,V_i)$ for the $f_{i,x}$ and setting
          $W:=V_1\cap\ldots\cap V_s$, we can define a unique
          $\ko_{X|V}$-module morphism $\Psi$ by linear extension of
          $\Psi(\tilde{W}): e_{i|\tilde{W}}\mapsto f_{i|\tilde{W}}$
            (for $\tilde{W}\subseteq W$), since $\ko_{X|W}^s$ is free.
          Moreover, the germ of $\Psi$ at $x$ is indeed
          $\Psi_x$. By (\ref{blubber}) $\kf$ is the cokernel of      
          $\delta:\ko_X^r\rightarrow\ko_X^s$, and using the 
          universal property of the cokernel\footnote{\label{follow}On the level of
            presheaves the universal property of the presheaf cokernel is
            obvious, and it is induced on the sheaf cokernel via the
            universal property of the sheafification.} we get the
          following commutative diagram:
          \begin{displaymath}
            \xymatrix{
              &\ko_{X|W}^r\ar[r]^{\delta_{|W}}\ar@/_/[drr]^{0} &
              \ko_{X|W}^s\ar[r]^{\gamma_{|W}}\ar[dr]^{\Psi_{|W}} & \kf_{|W}
              \ar[r]\ar@{-->}[d]^{\exists_!\tilde{\psi}} & 0\\ 
              & & & \kg_{|W}& }.       
          \end{displaymath}
          Taking limits we see that
          $\alpha_x[(\tilde{\psi},W)]=\tilde{\psi}_x=\psi$. 
     \end{enumerate}
   \end{proof}
\end{document} 
