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\begin{document}
\smallskip
\centerline{\sc Math 632, Lecture 3\\ January 12, 2004}

\section{More on stalks}\label{stalks}

Let $X$ be a topological space and $\calF$ a presheaf on $X$ with $x\in X$.  Recall the stalk at $x$ is
$$\calF_x=\varinjlim_{U\ni x} \calF(U).$$

\begin{enumerate}
	\item Let $G$ be ab abelian group and let $\calF=\underline{G}$. Recall we may view $\calF$
	as the sheaf of sections of the projection $G\times X\rightarrow X$ (where $G$ is given the discrete topology).
	 We have a map $G\rightarrow \calF(U)$ for each open $U\subseteq X$ given by $g\mapsto (u\mapsto g)$.
	 This induces a map $G\rightarrow \calF_x$, and this map is an isomorphism (which follows readily from the 
	 fact that elements of $\calF(U)$ are locally constant maps to $G$).
	
	\item Let $X^{\prime}\stackrel{f}{\rightarrow} X$ be a covering map and $\calF=\Gamma_{X^{\prime}/X}$ the sheaf of sections.
	We saw last time that there is a natural identification $\calF_x\simeq f^{-1}(x)$.  Observe that $\calF$ is locally constant because
	by definition of a covering space, there exists an open covering $U_{\alpha}$ of $X$ such that 
	$f^{-1}(U_{\alpha})\simeq\coprod_{i\in I_{\alpha}}U_{\alpha}$, for some discrete $I_{\alpha}$.  Therefore, 
	$\calF\big|_{U_{\alpha}}$ is the constant sheaf attached to the set $I_{\alpha}$.

	\item Let $X\stackrel{f}{\rightarrow} Y$ be a continuous map of top. spaces and $\calF$ a sheaf on $X$.
	Then $f_*\calF$ is a sheaf on $Y$ and for any $y\in Y$ we have
	$(f_*\calF)_y=\varinjlim_{U\ni y}\calF(f^{-1}(U))$.  If $y=f(x)$ then this direct limit is $\varinjlim_{f^{-1}(U)\ni x}\calF(f^{-1}(U))$
	which maps to $\varinjlim_{V\ni x}\calF(V)=\calF_x$ so we obtain a map $(f_*\calF)_f(x)\rightarrow \calF_x$.
	In general, this map is not surjective.
\end{enumerate}

\section{Ringed spaces}

Let $\calC$ be a subcategory of the category of rings containing 0 and stable under formation of direct limits.
We will be primarily interested in the categories of $A$-algebras (for a ring $A$) and rings.

\begin{definition}
	A $\calC$-ringed space $(X,\O_X)$ is a topological space $X$ equipped with a $\calC$-valued sheaf $\O_X$.
\end{definition}

\begin{definition}
	A morphism of ringed spaces is a map
	$$\varphi=(f,f^{\#}):(X,\O_X)\longrightarrow (Y,\O_Y),$$
	where $f:X\rightarrow Y$ is a continuous map of top. spaces and $f^{\#}:\O_Y\rightarrow f_*\O_X$ is a morphism of
	$\calC$-valued sheaves.
\end{definition}

\begin{warning}
	The sheaf $\O_X$ is not required to be a sheaf of functions with values in some fixed target and $f^{\#}$ is in general
	not determined by $f$, even though it interacts with $f$ via the maps $f^{\#}:\O_Y(U)\rightarrow \O_X(f^{-1}(U))$
	for each open $U\subseteq Y$.
\end{warning}

\begin{enumerate}
	\item Take $X=(M,\O_M)$ where $M$ is any $\C$-manifold and $\O_M$ the sheaf of holomorphic functions.
	Then for $Y=(N,\O_N)$ and a map $f:M\rightarrow N$ we have the induced map $f^{\#}:\O_Y\rightarrow f_*\O_X$
	given by $f^{\#}_U:\O_Y(U)\rightarrow \O_X(f^{-1}(U))$ as $\phi\mapsto \phi\circ f$, so we get a map $\varphi=(f,f^{\#})$
	of ringed spaces $X\rightarrow Y$.
	
	\item We can take $(X,\underline{\Z})$ for any topological space $X$.
	
	\item Let $X\\emptyset$ and $\O_X(X)=0$. 
	
	\item Let $X=\{\cdot\}$ be the one point topological space, and set $\O_X(X)=\C$.  Then $X$ is identified with $\Spec(\C)$.
	One might ask what $\Aut(X)$ is.    Observe
	that there is a unique map $X\rightarrow X$ as a topological space, so $\Aut(X)$ depends only on
	what category $\O_X$ takes values in. If, for example, we take the
	category of $\C$-algebras, then $\Aut(X)$ is trivial.  At the other extreme, if $\O_X$ has values in the category
	of sets then $\Aut(X)$ is uncountable. 

	\item Let $X=\Spec(\Z)$.  Then the open sets are of the form $U=X-\{p_1,\ldots,p_n\}$ for some finite list of
	primes $p_i$.  Consider the presheaf on $X$ defined by
	$$\O_X(U)=\begin{cases} 0 & U=\emptyset\\ \Z[1/(p_1\cdots p_r)] & U=X-\{p_1,\ldots,p_r\}\end{cases}.$$  
	The fact that we have an embedding $\O_X(U)\hookrightarrow \Q$ for all open $U$ implies that $\O_X$
	is, in fact, a sheaf.  Moreover, it is not hard to see that
	$\O_{X,(p)}=\Z_{(p)}$ for any prime ideal $(p)$ (in particular, $\O_{X,(0)}=\Q$).
	
	\item Let $X=\Spec(k[t]/(t^2))$.  Then as a topological space, $X$ consists of a single point, $(t)$.
	We define $\O_X$ by $\O_X(X)=k[t]/(t)^2$.
\end{enumerate}

\section{Composition of morphisms of ringed spaces}


Suppose that we have the following situation:
$$
\begin{CD}
	(X,\O_X)@>\varphi=(f,f^{\#})>>(Y,\O_Y)@>\psi=(g,g^{\#})>>(Z,\O_Z)
\end{CD}.
$$

We want to understand how to form the composition $\psi\circ\varphi:(X,\O_X)\rightarrow (Z,\O_Z)$.
On the level of topological spaces, we have the maps
$$
\begin{CD}
	X@>f>>Y@>g>>Z
\end{CD},
$$
so we obtain $g\circ f:X\rightarrow Z$.  We then want to define a map of sheaves (on $Z$) 
$$\O_Z\rightarrow (g\circ f)_*\O_X$$
and we have the sheave maps $f^{\#}:\O_Y\rightarrow f_*\O_X$ (both sides are sheaves on $Y$)
and $g^{\#}:\O_Z\rightarrow g_*\O_Y$ (now both sided are sheaves on $Z$).  
Recall that $(g\circ f)_*=g_*\circ f_*$ as maps of sheaves on $Z$.  Moreover, by the functoriality of the pushforward map $g_*$,
the map of sheaves $f^{\#}:\O_Y\rightarrow f_*\O_X$ gives rise to a morphism
$$
\begin{CD}
	g_*f^{\#}:g_*\O_Y\rightarrow g_*f_*\O_X=(gf)_*\O_X,
\end{CD}
$$
so that we have
$$
\begin{CD}
	\O_Z@>g^{\#}>> g_*\O_Y@>g_*f^{\#}>>g_*f_*\O_X,	
\end{CD}
$$
as morphisms of sheaves on $Z$.  In summary, the map $\psi\circ\varphi:(X,\O_X)\rightarrow (Z,\O_Z)$
is 
$$\psi\circ \varphi = (g\circ f, g_*f^{\#}\circ g^{\#}).$$

Explicitly, let $U\subseteq Z$ be open.  Then we have the maps of sections
$$
\begin{CD}
	\O_Z(U)@>g^{\#}_U>>\O_Y(g^{-1}U)@>f^{\#}_{g^{-1}U}>>\O_X(f^{-1}g^{-1}(U)).
\end{CD}
$$
Observe that $f^{\#}_{g^{-1}U}=g_*f^{\#}_U$ and $f^{-1}g^{-1}(U)=(f\circ g)^{-1}(U)$.

\begin{exercise}
	Check that composition thus defined is associative, and show that $\id_(X,\O_X)=(\id_X,\id_{\O_X}),$
	where $\id_{\O_X}=(\id_X)_*$, so that $\calC$-ringed spaces are thus a category.
\end{exercise}

\section{How to relate things to stalks}

If $(X,\O_X)$ and $(Y,\O_Y)$ are two $\calC$-ringed spaces and 
$$\varphi=(f,f^{\#}):(X,\O_X)\longrightarrow (Y,\O_Y)$$ a morphism,
given any $x\in X$ we have a map $f^{\#}_U:\O_Y(U)\longrightarrow \O_X(f^{-1}U)$
for all $U\subseteq Y$.  This gives a map
$$\O_{Y,f(x)}\longrightarrow (f_*\O_X)_{f(x)},$$
and composing with the map $(f_*\O_X)_{f(x)}\rightarrow \O_{X,x}$ which we defined in \S\ref{stalks},
we obtain a map 
$$\varphi_x:\O_{Y,f(x)}\longrightarrow \O_{X,x}.$$

\section{Pointed $\calC$-ringed spaces}

\begin{definition}
	A {\em pointed $\calC$-ringed space} is a pair $((X,\O_X),x)$.  A morphism of pointed $\calC$-ringed spaces
	$$\varphi=(f,f^{\#}):((X,\O_X),x)\longrightarrow ((Y,\O_Y),y)$$ is a morphism
	$$(f,f^{\#}):(X,\O_X)\longrightarrow (Y,\O_Y)$$ such that $f(x)=y$.
\end{definition}

\begin{exercise}
	Check that the association $((X,\O_X),x)\longrightarrow \O_{X,x}$ is a functor on the category of pointed ringed spaces.
\end{exercise}

\begin{enumerate}
		\item Let $M,N$ be two $C^{\infty}$ manifolds (with $\O_M$ and $\O_N$ the sheaves
		of $C^{\infty}$ functions on $M,N$ respectively, 
		and $\varphi=(f,f^{\#}:M\rightarrow N$ a $C^{\infty}$ map,
		where $f^{\#}$ is ``composition with $f$,"
		that is $f^{\#}:\O_{N,n}\rightarrow \O_{M,m}$ is given on sections over an open $U$ by $\alpha\rightarrow \alpha\circ f$.

		Suppose that $m\in M$ and $n=f(m)$.  Then we obtain a map
		$$\varphi_m:\O_{N,n}\longrightarrow \O_{M,m}$$ induced by composition with $f$.  Observe that $\O_{N,n}$ is a local
		ring with maximal ideal $\m_n$ the ideal of all functions vanishing at $n$.  If $\alpha(n)=0$ then
		$\alpha\circ f(m)=0$ so that $\alpha\circ f\in \m_m$, where $\m_m$ is the unique maximal ideal of $\O_{M,m}$
		consisting of functions vanishing at $m$.  Thus, $\varphi_m(\m_n)\subseteq \m_m$, so that $\varphi_m$
		is a local map of local rings and gives rise to a morphism of pointed ringed spaces.  Observe that all the stalks are
		local rings.
		 
		\item For an example of a ringed space where none of the stalks are local rings, consider the ringed space
		$(X,\underline{\Z}),$ where $X$ is any topological space.  The stalk at any point $\underline{\Z}_x$ is just $\Z$,
		which is not local.
\end{enumerate}





\end{document}