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\begin{document}
\smallskip
\centerline{\sc Math 632, Lecture 2\\ January 9, 2004}

\section{Sheaves and Ringed Spaces}

\subsection{More examples of presheaves}
\begin{enumerate} 
	\item Let $k$ be an algebraically closed field and $X$ an algebraic set.  Then	
	$$\O_X: U\mapsto \O_X(U)=\{\text{regular functions on } U\}$$
	is a presheaf of $k$-algebras.
	
	\item Let $X$ be a top. space and $G$ an abelian group.  The {\em constant presheaf} is
	$$U\mapsto \begin{cases}G & U\neq \emptyset \\ 0 & U=\emptyset\end{cases}$$
	together with the obvious restriction maps.
	
	\item The constant sheaf for $G$ on $X$, $\underline{G}$ is
	$$U\mapsto \{\text{continuous $G$-valued functions on $U$}\},$$ where $G$ is given the
	discreet topology.  Observe that if $X$ is any locally connected space (like a manifold)
	then 
	$$\underline{G}(U)=\prod_{\text{conn. cpts. of } U} G.$$
	
	\item If $X=\Z_p$ or $X=\Spec(\prod_{\Z} k)$ with $k$ a field, then $\underline{G}$ is hard to make 
	as explicit as above.
	
	\item $X$ any $\C$-manifold and $\O_X$ the presheaf of holomorphic functions on $X$.  
	Then let
	$$\O_X^{\times}:U\mapsto \{f\in \O_X(U): f\neq 0\ \text{on}\ U\}.$$
\end{enumerate}

\subsection{Examples of morphisms of (pre)sheaves}

If $\phi:\calF\rightarrow \calG$ is a morphism of presheaves on $X$, we define the presheaf $\ker \phi$ on $X$
by $$\ker\phi (U)=\ker\left(\phi(U):\calF(U)\rightarrow \calG(U)\right).$$

\begin{enumerate}
	\item Let $X$ be a $\C$-manifold and let $\Z(1)$ be the abelian group $2\pi\i\Z$.  Consider the
	exponential morphism $\exp: \O_X\rightarrow \O_X^{\times}$ given on the sections over $U$
	as $f\mapsto e^f$.  Then one has $\ker \exp = \underline{\Z(1)}$, the constant sheaf on $X$ associated
	to $\Z(1)$. 
\end{enumerate}

\subsection{Sheaves}

\begin{definition}
	Suppose $\mathcal{C}$ is a subcategory of the category of sets.  Then a $\mathcal{C}$-valued presheaf $\calF$
	on $X$ is a {\em sheaf} if for any open cover $\{U_{\alpha}\}$ of $U$ and any collection
	of $s_{\alpha}\in \calF(U_{\alpha})$ that satisfy $s_{\alpha}\big|_{U_{\alpha}\cap U_{\beta}}=s_{\beta}\big|_{U_{\alpha}\cap U_{\beta}}$
	for all $\alpha,\beta$
	(as elements of $\calF(U_{\alpha}\cap U_{\beta})$), there is a {\em unique} $s\in \calF$ with $s\big|_{U_{\alpha}}=s_{\alpha}$
	for all $\alpha$.
\end{definition}

Roughly speaking, a sheaf is a presheaf with the additional property that local, compatible data uniquely determines
global data.  Observe that the compatibility criterion is vacuously satisfied if $U_{\alpha}\cap U_{\beta}=\emptyset$.
We make the set of sheaves on $X$ into a category by stipulating that morphisms of sheaves are just morphisms
of presheaves as already defined.


\begin{definition}
	If $\calF$ is a sheaf, then a {\em subsheaf} of $\calF$ is a subpresheaf $\calG\subseteq \calF$ that is a sheaf.
	By the uniqueness of glueing, glueing in $\calG$ is the same as glueing in $\calF$.
\end{definition}

\begin{enumerate}
	\item The constant presheaf is {\em not} a subsheaf of the constant sheaf.
	
	\item Let $X$ be a $\C^{\infty}$ manifold and $\Omega_X^k$ the presheaf of $\C^{\infty}$ $k$-forms (a sheaf).
	Let $\wedge_{\O_X}^k$ be the presheaf $U\mapsto \wedge^k_{\O_X(U)} (\Omega^1_{X}(U))$
	(that is, the $k$ th exterior power of the $\C$-vector space $\Omega^1_X(U)$).
	Then $\wedge_{\O_X}^k$ is {\em not} in general a sheaf.  However, we have a map $\wedge_{\O_X}^k\rightarrow \Omega_X^k$
	given on sections over $U$ as $\omega_1\wedge\ldots\wedge\omega_n\mapsto \omega_1\wedge\ldots\wedge\omega_n$.
	This map is usually neither injective nor surjective.	
\end{enumerate}

\begin{definition}
	Let $X$ be a top. space, $U\subseteq X$ an open set, and suppose that $\calF$ is a presheaf on $X$.  Then we define
	the presheaf $\calF\big|_{U}$ on $U$ via $ U\supseteq V\mapsto \calF(V)$.  Since $U$ is open in $X$ and $V$ is open in
	$U$, we see that $V$ is open in $X$ so that this definition makes sense.
\end{definition}

It is not hard to see that $\calF\big|_{U}$ is a sheaf if $\calF$ is.

\begin{enumerate}
	\item Let $X$ be a $\C^{\infty}$ manifold and $\calF=\O_X$.  Then $\calF\big|_{U}=\O_U$.
	
	\item Let $X$ be a top. space and $G$ an abelian group.  Let $\calF=(\underline{G},X)$ be the sheaf
	$\underline{G}$ on $X$.  Then $\calF\big|_{U}=(\underline{G},U)$ is the sheaf $\underline{G}$
	of locally constant $G$-valued functions on $U$.
	
	\item Let $f:X^{\prime}\rightarrow X$ be a continuous map of top. spaces and 
	$\Gamma_{X^{\prime}/X}$ the sheaf of sections $$U\mapsto \{s:U\rightarrow f^{-1}(U) : f\circ s=\id_U\}.$$
	If $W\subseteq X$ is open then $\Gamma_{X^{\prime}/X}\big|_{W}=\Gamma_{f^{-1}(W)/W}$. 
\end{enumerate}

\begin{definition}
	Let $f:X\rightarrow Y$ be a continuous map of topological spaces and $\calF$ a presheaf on $X$.  The {\em pushforward},
	$f_*\calF$ is the presheaf on $Y$ defined by
	$$(f_*\calF)(U)=\calF(f^{-1}(U)),$$ with the restriction maps $\rho_{f^{-1}(U),f^{-1}(V)}:(f_*\calF)(U)\rightarrow (f_*\calF)(V)$
	inherited from $\calF$.  
\end{definition}

Observe that pushforward if functorial.  Indeed, if $\varphi:\calF\rightarrow\calG$ is a morphism of presheaves on $X$ and $f:X\rightarrow Y$
as above, we obtain a morphism   $f_*\varphi:f_*\calF\rightarrow f_*\calG$ of presheaves on $Y$, where
 $(f_*\varphi)(U):(f_*\calF) (U)\rightarrow (f_*\calG)(U)$ is just the map $\varphi(f^{-1}(U)):\calF(f^{-1}(U))\rightarrow \calG(f^{-1}(U))$.

Observe that if $\calF$ is a sheaf on $X$ and $f:X\rightarrow Y$ a continuous map of top. spaces then $f_*\calF$
is a sheaf on $Y$.  
This is more or less a tautology. 
In complete detail, let $U\subset Y$ be open and $\{U_{\alpha}\}$ an open cover of $U$.  Let 
$s_{\alpha}\in (f_*\calF)(U_{\alpha})$ be a collection of sections with 
$s_{\alpha}\big|_{U_{\alpha}\cap U_{\beta}}=s_{\beta}\big|_{U_{\alpha}\cap U_{\beta}}$ as elements of $(f_*\calF)(U_{\alpha}\cap U_{\beta})$.  
By the definition of $f_*$, we have $s_{\alpha}\in \calF(f^{-1}(U_{\alpha}))$ with 
$s_{\alpha}\big|_{f^{-1}(U_{\alpha}\cap U_{\beta})}=s_{\beta}\big|_{f^{-1}(U_{\alpha}\cap U_{\beta})}$
as elements of $\calF(f^{-1}(U_{\alpha}\cap U_{\beta}))$.
Since $f^{-1}(U_{\alpha}\cap U_{\beta})=f^{-1}(U_{\alpha})\cap f^{-1}(U_{\beta})$, we can rename (for psychological purposes really)
$f^{-1}(U_{\alpha})=V_{\alpha}\subseteq X$ and $f^{-1}(U)=V\subset X$ (so that $V_{\alpha}$ is a covering of $V$)
 and we see that the sheaf property of unique glueing for $\calF$ gives an element $s\in \calF(V)=(f_*\calF)(U)$
 with $s\big|_{U_{\alpha}}=U_{\alpha}$.
 
 \begin{enumerate}
 	\item Let $\varphi:X\rightarrow Y$ be a $\C^{\infty}$ map of $\C^{\infty}$ manifolds, and let $\O_X,\O_Y$
	be the sheaves of $\C^{\infty}$ functions on $X$ and $Y$ respectively.
	Now $f_*\O_X$ is a sheaf on $\O_Y$ and we have a map
	$$\varphi^{\#}:\O_Y\rightarrow f_*\O_X$$
	given by $g\mapsto g\circ \varphi$ as a map $\O_Y(U)\rightarrow \O_X(\varphi^{-1}(U))$ over any $U\subseteq Y$.
	Observe that this map makes sence as $\varphi$ is $\C^{\infty}$ co that the composition $g\circ\varphi$ is $\C^{\infty}$
	(and hence an element of $\O_X(\varphi^{-1}(U))$).
	
	\item Suppose that $X\stackrel{f}{\rightarrow}Y\stackrel{g}{\rightarrow}Z$.  Then $(g\circ f)_*=g_*\circ f_*$
	in the sense that for any presheaf $\calF$ on $X$ we have a natural isomorphism of sheaves
	$(g\circ f)_*\calF \simeq g_*(f_*\calF)$.  This is really just the statement
	$$\calF((g\circ f)^{-1}(U))=\calF(f^{-1}\circ g^{-1}(U))=(f_*\calF)(g^{-1}U).$$  Moreover, this equality 
	is transitive, i.e. $(h\circ g)_*(f_*\calF)=h_* ((g\circ f)_*\calF)$ as is easily checked.
 \end{enumerate}  
 
 \subsection{Stalks}
 
\begin{definition}
	Let $\calF$ be a presheaf on $X$ and $x\in X$.  Define the {\em stalk} of $\calF$ at $x$ as
	$$\calF_x = \varinjlim_{U\ni x}\calF(U),$$ where the direct limit is formed using the restriction maps.
\end{definition}

\begin{enumerate}
	\item If $X$ is a $\C$-manifold and $\calF=\O_X$ then $\O_{X,x}$ is the set of germs of functions at $x$.
	
	\item If $f:X^{\prime}\rightarrow X$ is a covering space and $\calF=\Gamma_{X^{\prime}/X}$ then
	$\calF_x=f^{-1}(x)$.  This follows from the fact that $f$ is a local homeomorphism.  
	Since $f$ is a covering map, for small enough $U\ni x$ we have $f^{-1}(U)=\coprod_{s\in f^{-1}(x)} V_s$
	for open sets $V_s\subset X^{\prime}$ which are homeomorphic to $U$ via $f$.
	Then
	an element $s\in f^{-1}(x)$ is just the map $s: U\rightarrow V_s$ given by $f^{-1}$.  Evidently
	$f\circ s(x)=x$ and this condition uniquely determines any map $s:U\rightarrow f^{-1}(U)$ when
	$U$ is small enough (which is all we care about since we are taking a direct limit).
\end{enumerate}


\end{document}