\magnification\magstep1
\nopagenumbers

%%%%%%%%%%%Blackboardbold 
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\font\bbbf=msbm10 
\font\sbbb=msbm6 
\font\ssbbb=msbm5 
\textfont6=\bbbf
\scriptfont6=\sbbb 
\scriptscriptfont6=\ssbbb 
\def\bbb{\fam6}
\def\mP{{\bbb P}} 
\def\mA{{\bbb A}} 
\def\mB{{\bbb B}} 
\def\mR{{\bbb R}}
\def\mZ{{\bbb Z}}

%%%%Gothic
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\font\gothicf=eufm10
\font\sgothic=eufm7
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\textfont5=\gothicf
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\scriptscriptfont5=\ssgothic
\def\gothic{\fam5}


\font\Kopfont=cmbx12
\def\mapright#1{\smash{\mathop{\longrightarrow}\limits^{#1}}}
\def\mapdown#1{\Big\downarrow\rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\def\downmap#1{\downarrow\rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\def\mapup#1{\Big\uparrow\rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\def\longlongrightarrow{\relbar \joinrel \longrightarrow}
\def\cC{{\cal C}}
\def\cD{{\cal D}}
\def\gp{{\gothic p}}
\def\gq{{\gothic q}}
\def\Spec{\mathop{\rm Spec}}

\centerline{\Kopfont Schemes}

\smallskip
\centerline{Excercises 1}

\bigskip\noindent
{\bf Sheaves}

\bigskip\item{\bf 1.} Carefully prove that a map of sheaves of {\bf sets}
is an epimorphisms (in the category of sheaves of sets) if and only if the
induced maps on all the stalks are surjective.

\medskip\item{\bf 2.} Let $f : X \to Y$ be a map of topological spaces.
Prove pushforward $f_\ast$ and pullback $f^{-1}$ for sheaves of {\bf sets}
form an adjoint pair of functors.

\medskip\item{\bf 3.} Let $j : U \to X$ be an open immersion. Show
that $j^{-1}$ has a left adjoint $j_{!}$ on the category of sheaves
of sets. Characterize the stalks of $j_{!}({\cal G})$. (Hint: $j_{!}$
is called extension by zero when you do this for abelian sheaves... )

\medskip\item{\bf 4.} Let ${\cal F}$ be an abelian sheaf on $X$. Show
that ${\cal F}$ is the quotient of a (possibly very large) direct sum
of sheaves all of whose terms are of the form
$$
j_{!}(\underline{{\bbb Z}}_U)
$$
where $U \subset X$ is open and $\underline{{\bbb Z}}_U$ denotes the
constant sheaf with value ${\bbb Z}$ on $U$.

\medskip\noindent
{\bf Remark.} In the category of abelian sheaves the direct sum of
a family of sheaves $\{{\cal F}_i\}_{i\in I}$ is the sheaf associated to
the presheaf $U \mapsto \oplus {\cal F}_i(U)$. Consequently the stalk of
the direct sum at a point $x$ is the direct sum of the stalks of the 
${\cal F}_i$ at $x$.

\bye
