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\begin{document}
\smallskip
\centerline{\sc Math 632, Lecture 7\\ January 23, 2004}

\section{More sheaf constructions}

\begin{definition}
	If $\calF\stackrel{\iota}{\hookrightarrow} \calG$ is a subsheaf, we define the sheaf $\calG/\calF$ to be the sheaf $\coker\iota$.
\end{definition}

As an example, for any complex manifold $X$ the exact sequence
\begin{diagram}
	0 & \rTo & \underline{\Z(1)} & \rTo & \O_X & \rTo^{\exp} & \O_X^{\times}  
\end{diagram}
induces $\O_X/\underline{\Z(1)}\simeq \O_X^{\times}$.

In general, for any (local) surjection $\varphi:\calG\rightarrow\calH$ of sheaves we obtain $\calG/\ker\varphi\simeq \calH$,
where the existence of a map $\calG/\ker\varphi\rightarrow \calH$ follows from the universal property of
a quotient sheaf as a cokernel, and one checks this is an isomorphism on stalks.

\begin{definition}
	A short exact sequence of sheaves is a sequence
	\begin{diagram}
		0 & \rTo & \calF' & \rTo^{\iota} & \calF & \rTo^{\varphi} & \calF'' & \rTo & 0
	\end{diagram}
	in which 
	\begin{enumerate}
		\item $\varphi\circ\iota = 0$.
		\item $\varphi$ is a (local) surjection.
		\item $\iota$ is injective.
		\item $\varphi$ induces an isomorphism $\calF/\calF'\simeq \calF''$.
	\end{enumerate}
	Observe from (1), (2), and (3) that (4) is equivalent to $\iota$ inducing $\calF'\simeq\ker\varphi$.
\end{definition}

Surjectivity is a slightly delicate notion: indeed, a map $X'\rightarrow X$ of varieties over a field $k$
can be surjective on $\overline{k}$-points without being surjective on $k$-points.  Consider the self-map
of $\A_k^1-\{0\}$ given by $t\mapsto t^2$ over a field $k$ in which $k\neq k^2$.

\section{Pullbacks}

Let $f:X\rightarrow Y$ be a continuous map of topological spaces and $\calF$ a sheaf on $Y$.

\begin{definition}
	Let $``f^{-1}\calF"$ be the presheaf on $X$ given by $U\mapsto \varinjlim_{V\supseteq f(U)}\calF(V)$,
	where the direct limit is over all open $V$,  One checks that this is a presheaf with the obvious restriction
	maps.  We define the sheaf $f^{-1}\calF$ to be the sheafification of $``f^{-1}\calF"$.
\end{definition}

The pullback construction generalizes some already familiar notions: let $y\in Y$ be a point and $X=\{y\}$
with the induced topology and $\iota:X\rightarrow Y $ the inclusion map.  If $\calF$ is any sheaf on $Y$
then $\iota^{-1}\calF(X)=\varinjlim_{V\supseteq \{y\}}\calF(V)=\calF_y$ is the stalk at $y$.

\begin{remark}
	Pullback and pushforward of sheaves are analogous to restriction and extension of scalars in the category
	of modules over a ring.  To see this, let $A,B$ be a rings, and $A$-module and $f:A\rightarrow B$
	a map of rings.  For any $A$-module $M$ we obtain the $B$-module $M\otimes_A B$, and for any $B$-
	module $N$, we get an $A$-module ${}_A N$ by restriction of scalars.  
	The $A$-module ${}_A N$ is very much like the pushforward construction, while the $B$-module $M\otimes_A B$
	is analogous to the pullback.  We will see this more clearly when we investigate sheaves of modules; but for now,
	observe that we have the adjoint property
	$$\Hom_B (B\otimes_A M,N)=\Hom_A(M,{}_A N).$$
\end{remark}

\begin{theorem}
	Let $\calF$ be a presheaf of sets on the topological space $Y$ and let 
	$$\begin{diagram}X' & \rTo^g & X & \rTo & Y\end{diagram}$$ be continuous maps of topological 
	spaces.
	There exists a natural isomorphism
\begin{align}
		(f^{-1}\calF)_x \stackrel{\sim}{\longrightarrow}  \calF_{f(x)} \label{iso1}
	\end{align}
	Moreover, we have
	\begin{align}
		f^{-1}\theta:f^{-1}\calF & \stackrel{\sim}{\longrightarrow}  f^{-1}(\calF^+)\\
		\alpha_{g,f}:g^{-1}f^{-1}\calF & \stackrel{\sim}{\longrightarrow} (f\circ g)^{-1}\calF  \label{compo}
	\end{align}
	and (\ref{compo}) induces, via (\ref{iso1})
	\begin{align*}
		\id:\calF_{f(g(x'))} \stackrel{\sim}{\longrightarrow} (f^{-1}\calF)_{g(x')} \stackrel{\sim}{\longrightarrow} \calF_{f\circ g(x')}.
	\end{align*}
	Finally, (\ref{iso1}) is functorial in $\calF$, so that if $\varphi:\calF\rightarrow \calG$ is a map of presheaves, 
	we have
	\begin{diagram}
		(f^{-1}\calF)_x & \rTo^{\sim} & \calF_{f(x)} \\
		\dTo^{(f^{-1}\varphi)_x} &  & \dTo>\varphi_{f(x)}\\
		(f^{-1}\calG)_x & \rTo^{\sim} &\calG_{f(x)}.
	\end{diagram}
\end{theorem}

\begin{proof}
	We will construct (\ref{iso1}).  First observe that $(f^{-1}\calF)_x=``f^{-1}\calF"_x$, so that
	$$(f^{-1}\calF)_x=\varinjlim_{U\ni x}\varinjlim_{V\supseteq f(U)}\calF(V).$$
	Now any $V$ containing $f(U)$ with $U$ containing $x$ must contain $f(x)$.  This gives a map
	$$(f^{-1}\calF)_x=\varinjlim_{U\ni x}\varinjlim_{V\supseteq f(U)}\calF(V)\longrightarrow \varinjlim_{W\ni f(x)}\calF(W).$$
	Conversely, $W\supseteq f(f^{-1}(W))$ and since $f$ is continuous, $f^{-1}(W)$ is open for any open $W$ and
	contains $x$ when $f(x)\in W$, so we
	obtain a map in the other direction.  It is not difficult to see that these maps are inverse to eachother.
\end{proof}


\subsection{Adjointness}

Let $f:X\rightarrow Y$ be a continuous map of topological spaces and $\calF,\calG$ sheaves on $X,Y$
respectively.  Then we have 
$$\Hom_X(f^{-1}\calG,\calF)=\Hom_Y(\calG,f_*\calF).$$
To see this, it suffices to produce natural maps $\calG\longrightarrow f_*(f^{-1}\calG)$ and $f^{-1}(f_*\calF)\rightarrow\calF$,
for given such maps the functoriality of $f_*$ and $f^{-1}$ show that any map $f^{-1}\calG\longrightarrow\calF$
gives rise to a map  $$\calG\longrightarrow f_*f^{-1}\calG\longrightarrow f_*\calF$$ and any map
$f\calG\longrightarrow f_*\calF$
gives rise to a map  $$f^{-1}\calG\longrightarrow f^{-1}f_*\calF\longrightarrow \calF.$$

The commutative algebra analog is instructive: Let $A,B$ be rings, $f:A\rightarrow B$ a ring-map
and $N$ a $B$-module and $M$ an $A$-module.  Then we have maps
$$\alpha_M:M\longrightarrow {}_A(B\otimes_A M)$$
and
$$\beta_N:B\otimes_A({}_A N)\longrightarrow N$$
given by $m\mapsto (1\otimes m)$ and $b\otimes n\mapsto bn$ respectively.  It is straightforward
to check that these maps give the natural bijection
$$\Hom_B(B\otimes_A M,N)=\Hom_A(M,{}_A N).$$

\subsection{Examples of pushforward}

\begin{proposition}
Let $f:X\rightarrow Y$ be a continuous map of topological spaces and $\underline{\Sigma}_Y,\underline{\Sigma}_Y$ the constant
sheaf on $Y,X$ respectively, associated to the set $\Sigma$.
Then there exists a natural isomorphism $f^{-1}(\underline{\Sigma}_Y)\stackrel{\sim}{\longleftarrow}\underline{\Sigma}_X$.
\end{proposition}

Observe that the analog for pushforward is false as the preimage of a connected open set need not be connected.

\begin{proof}
	We have a map $$\Sigma\longrightarrow ``f^{-1}\underline{\Sigma}_Y"(X)\longrightarrow f^{-1}\underline{\Sigma}_Y(X)$$
	which, by the property that a map of $\Sigma$ to the global sections of any sheaf determines a map of $\underline{\Sigma}$
	to that sheaf, gives a unique
	$\underline{\Sigma}_X\stackrel{\sim}{\longrightarrow} f^{-1}\underline{\Sigma}_Y$ such that the induced map
	of stalks $\Sigma\simeq (\underline{\Sigma}_X)_x\longrightarrow (\underline{\Sigma}_Y)_{f(x)}\simeq \Sigma$
	is the identity mapping of $\Sigma$ (which is evident from the construction).
\end{proof}

Because of this Proposition, we often abuse notation and write $\underline{\Sigma}$ for the constant sheaf associated
to $\Sigma$ and omit the space on which it is considered.

As another example, let $f:X\rightarrow Y$ be a map of topological spaces, and $\pi:Y'\rightarrow Y$ a covering space.  Put $X'=X\times_{Y} Y'$,
that is, $X'=\{(x,y'): f(x)=\pi(y')\}$.  Let $\calF=\Gamma_{Y'/Y}$ be the sheaf of sections of $Y'$ over $Y$.
Then we have
$$f^{-1}\calF\simeq \Gamma_{X'/X}.$$

\begin{definition}
	Let $(X,\O_X)$ be a ringed space.  If $U$ is an open set we define the induced ring space structure on $U$ to 
	be $(U,\O_U)$ where $\O_U=\O_X\big|_U$.  A map $(f,f^{\#}):(X',\O_{X'})\rightarrow (X,\O_X)$
	is an {\em open immersion} if $f:X'\rightarrow X$ is a homeomorphism onto an open $U\subseteq X$
	and $f^{\#}:\O_X\rightarrow f_*\O_{X'}$ is adjoint to $f^{-1}\O_X\rightarrow \O_{X'}$.
\end{definition}

As an example, let $k$ be a field and consider the map
$f:\Spec k\hookrightarrow \Spec k[y]/y^2$ given by mapping the unique point of $\Spec k$
to the unique point of $\Spec k[y]/y^2$ and the surjective $k$-algebra map $f^{\#}:k[y]/y^2\twoheadrightarrow k$
given by $y\mapsto 0$.  This is {\em not} an open immersion.

\begin{definition}
	A {\em complex manifold} is a ringed space of $\C$-algebras $(X,\O_X)$
	such that there exists an open cover $(U_i,\O_{U_i})$ that are isomorphic
	(as ringed spaces) to opens in finite dimensional complex vector-spaces endowed with
	the sheaf of holomorphic functions.
 \end{definition}
 
 \section{Glueing of ringed spaces}
 
 Let $(X_i,\O_{X_i})$ be a collection of ringed spaces and $X_{ij}\subseteq X_i$ open sets,
 together with isomorphisms of ringed spaces $\varphi_{ij}:X_{ij}\simeq X_{ji}$
 such that
 \begin{enumerate}
 	\item $X_{ii}=X_i$ and $\varphi_{ii}=\id$.
	\item $\varphi(X_{ij}\cap X_{ik})=X_{jk}\cap X_{ji}$.
	\item $\varphi_{jk}\circ \varphi_{ij}=\varphi_{ik}$ on triple overlaps.
 \end{enumerate}
 
 Then there exists a unique ringed space $(X,\O_X)$ equipped with open immersions
 $$\psi_i:(X_i,\O_{X_i})\hookrightarrow (X,\O_X)$$ covering with overlaps ``$X_{ij}=X_{ji}$''
heving the correct universal property:

\begin{diagram}
	(X_i,\O_{X_i}) &\rInto & (X,\O_X) \\
			& \rdTo_{f_i} & \dDashto_{\exists !}\\
				&  & (Y,\O_Y)
\end{diagram}















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