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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Geometry of manifolds \\[15mm]
\small Lecture 6: Germs and duality}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

{\small Misha Verbitsky } 
\\[20mm]

{\tiny\bf Math in Moscow and HSE
\\[2mm]  March 11, 2013
}
\end{center}

\newpage

{\bf \blue Sheaves (reminder)}

\definition
{\bf\blue An open cover} of a topological space $X$ is a
family of open sets $\{U_i\}$ such that $\bigcup_iU_i=X$.


\remark The definition of a sheaf below is a
more abstract version of the notion of 
``sheaf of functions'' defined previously.

\definition
A {\bf\blue presheaf} on a
topological space $M$ is a collection of vector spaces
${\cal F}(U)$, for each open 
subset $U\subset M$,  together with {\bf \blue restriction maps}
$R_{UW}{\cal F}(U)\arrow {\cal F}(W)$ defined for each $W\subset U$,
such that for any three open sets $W\subset V\subset U$,
$\Psi_{UW}=\Psi_{UV}\circ \Psi_{VW}$. Elements of ${\cal F}(U)$
are called {\bf \blue sections of ${\cal F}$ over $U$},
and restriction map often denoted $f\restrict W$

\definition
A presheaf 
${\cal F}$ is called {\bf\blue a sheaf} 
if for any open set $U$ and any cover $U=\bigcup U_I$
the following two conditions are satisfied.\\
\phantom{xu} 1. Let $f\in {\cal F}(U)$ be a section of ${\cal F}$ on $U$
such that its restriction to each $U_i$ vanishes. {\bf \purple 
Then $f=0$.} \\
\phantom{xu}  2. Let $f_i\in{\cal F}(U_i)$ be a family of
sections compatible on the pairwise intersections:
$f_i|_{U_i\cap U_j}=f_j|_{U_i\cap U_j}$
for every pair of members of the cover. {\bf \purple Then there exists
$f\in{\cal F}(U)$ such that $f_i$ is the restriction of $f$ to $U_i$ for
all $i$.}


\newpage

{\bf \blue Sheaves and exact sequences (reminder)}



\definition
A sequence $A_1 \arrow A_2 \arrow A_3 \arrow ...$
of homomorphisms of abelian groups or vector spaces
is called {\bf\blue exact} if the image of each map is
the kernel of the next one.


\claim
A presheaf
${\cal F}$ is a sheaf if and only if for every cover $\{U_i\}$ of
an open subset $U\subset M$, {\bf \red the sequence of restriction maps
$$0\to{\cal F}(U)\to\prod\limits_i{\cal F}(U_i)\to\prod\limits_{i\ne
j}{\cal F}(U_i\cap U_j)$$
is exact,} with $\eta\in {\cal F}(U_i)$ mapped to $\eta\restrict{U_i\cap U_j}$
and $-\eta\restrict{U_j\cap U_i}$. 


\newpage

{\bf \blue Ringed spaces (reminder)}

\definition
{\bf\blue A sheaf of rings} is a sheaf ${\cal F}$ such that
all the spaces ${\cal F}(U)$ are rings, and all restriction
maps are ring homomorphisms. 

\definition
{\bf \blue A sheaf of functions} is a subsheaf
in a sheaf of all functions, closed under multiplication.

{\bf \red For simplicity, I assume now that a sheaf of rings
is a subsheaf in a sheaf of all functions}.

\definition
A {\bf\blue ringed space} $(M,{\cal F})$ is a
topological space equipped with a sheaf of
rings. A~{\bf \blue morphism}
$(M,{\cal F})\stackrel\Psi\longrightarrow(N,{\cal F}')$ of ringed spaces
is a continuous map $M\stackrel\Psi\longrightarrow N$ such that, for
every open subset $U\subset N$ and every function $f\in{\cal F}'(U)$, the
function $\psi^* f:=f\circ\Psi$ belongs to the ring
${\cal F}\big(\Psi^{-1}(U)\big)$. An {\bf\blue  isomorphism} of ringed spaces
is a homeomorphism $\Psi$ such that $\Psi$ and $\Psi^{-1}$ 
are morphisms of ringed spaces.


\newpage

{\bf \blue Smooth manifolds (reminder)}

\definition
Let $(M,{\cal F})$ be a topological manifold
equipped with a sheaf of functions. It is said to be a {\bf\blue smooth
manifold of class} $C^\infty$ or $C^i$ if every point in
$(M,{\cal F})$ has an open neighborhood isomorphic to the ringed space
$({\Bbb B}^n,{\cal F}')$, where ${\Bbb B}^n\subset \R^n$ 
is an open ball and ${\cal F}'$ is a ring of 
functions on an open ball ${\Bbb B}^n$ of this class.

\definition
{\bf\blue Diffeomorphism} of smooth manifolds
is a homeomorphism $\phi$ which induces an isomorphims
of ringed spaces, that is, $\phi$ and $\phi^{-1}$ map 
(locally defined) smooth functions to smooth functions.

{\bf \red Assume from now on that all manifolds are
  Hausdorff and of class $C^\infty$}.

\newpage


{\bf \blue Partition of unity (reminder)}

\definition
 Let $M$ be a smooth manifold and let $\{U_\alpha\}$
a locally finite cover of $M$. A {\bf \blue partition of unity}
subordinate to the cover $\{U_\alpha\}$ is a family of smooth functions
$f_i:M\to[0,1]$ with compact support indexed by the same indices as the
$U_i$'s and satisfying the following conditions.\\
(a) Every function $f_i$ vanishes outside $U_i$\\
(b) $\sum_if_i=1$


\theorem
Let $\{U_\alpha\}$ be a countable, locally finite cover of a manifold
$M$, with all $U_\alpha$ diffeomorphic to $\R^n$. {\bf
  \red Then
there exists a  partition of unity subordinate to $\{U_\alpha\}$.}


\definition
Let  $U\subset V$ be open subsets in $M$.
We write $U\Subset V$ if the closure of $U$ is contained in $V$.

\definition
Let $f\in {\cal F}(M)$ be a section of a sheaf ${\cal F}$
on $M$. A point $x\in M$ does not lie in the
{\bf\blue support} $\Sup(f)$  of $f$ if $f\restrict U=0$ for
some neighbourhood $U\ni x$.

\remark {\bf \purple Support of a section is obviously closed.}


\newpage


{\bf \blue Direct limits}


\definition
{\bf\blue Commutative diagram} of vector spaces
is given by the following data. There is
a directed graph (graph with arrows). For each
vertex of this graph we have a vector space, 
and each arrow corresponds to a homomorphism of the
associated vector spaces. {\bf \purple These 
homomorphism are compatible, in the 
following way.} Whenever there exist two ways of going from
one vertex to another, the compositions of the
corresponding arrows are equal.


\definition
Let ${\cal C}$ be a commutative diagram of vector spaces,
 $A, B$ -- vector spaces, corresponding to two vertices
of a diagram, and  $a\in A, b\in B$ elements of these
vector spaces. Write  $a\sim b$ if $a$ and $b$ are mapped
to the same element $d\in D$ by a composition of 
arrows from  ${\cal C}$. Let $\sim$ be an
equivalence relation generated by such $a\sim b$.
A quotient $\bigoplus_i C_i/E$ is called
{\bf\blue a direct limit} of a diagram $\{C_i\}$.
The same notion is also called {\bf\blue colimit}
and {\bf\blue inductive limit}. Direct limit is denoted
$\lim\limits_\rightarrow$.

\definition
Let ${\cal F}$ be a sheaf on $M$, $x\in M$ a point,
and $\{U_i\}$ the set of all neighbourhoods of
$x$. Consider a diagram with the set of vertices 
indexed by $\{U_i\}$, and arrows from $U_i$ to $U_j$
corresponding to inclusions $U_j \hookrightarrow U_i$. 
The {\bf\blue space of germs} of ${\cal F}$ in
$x$ is a direct limit $\lim\limits_\rightarrow{\cal F}(U_i)$
over this diagram. The space of germs is also called
{\bf \blue a stalk} of a sheaf.


\newpage


{\bf \blue Germs of functions}



\definition
A diagram ${\cal C}$ is called {\bf\blue filtered}
if for any two vertices $C_i, C_j$, there exists
a third vertex $C_k$, and sequences of arrows
leading from $C_i$ to $C_k$ and from $C_j$ to $C_k$.

\example
{\bf \red The diagram formed by all neighbourhoods of a point
is obviously filtered.}

\claim
Let ${\cal C}$ be a commutative diagram of vector spaces 
$C_i$, with all $C_i$ equipped with a ring structure,
and all arrows ring homomorphisms. Suppose that 
the diagram ${\cal C}$ is filtered. {\bf \purple Then there exists
a unique ring structure on $C:=\lim\limits_\rightarrow C_i$
such that all the maps $C_i \arrow C$ are ring
homomorphisms.}

\definition
Let $M, {\cal F}$ be a ringed space,
$x\in M$ its point, and $\{U_i\}$ the set of all
its neighbourhoods. Consider a commutative
diagram with vertices indexed by $\{U_i\}$,
and arrows from $U_i$ to $U_j$ 
corresponding to inclusions $U_j \hookrightarrow U_i$.
For each vertex $U_i$ we take a vector space
of sections  ${\cal F}(U_i)$, and for each
arrow the corresponding restriction map. 
The direct limit of this diagram is called
{\bf\blue the ring of germs of the sheaf ${\cal F}$
in $x$}.

\newpage

{\bf \blue Morphisms of sheaves}


\definition
Let ${\cal B}, {\cal B}'$ be sheaves on
$M$. {\bf\blue  A sheaf morphism} from ${\cal B}$ to ${\cal B}'$
is a collection of homomorphisms ${\cal B}(U)\arrow {\cal B}'(U)$,
defined for each open subset $U\subset M$,
and compatible with the restriction maps:
{\small \[
\begin{CD}
{\cal B}(U) @>>> {\cal B}'(U)\\
@VVV@VVV\\
{\cal B}(U_1) @>>> {\cal B}'(U_1)
\end{CD}
\]
}
\!\!\!\!\!\!\!\!\definition
A sheaf morphism is called {\bf\blue injective}, or {\bf \blue
  a monomorphism} if it is injective on stalks
and {\bf \blue surjective}, or {\bf \blue epimorphism}
if it is surjective on stalks.

\exercise
Let ${\cal B} \stackrel \phi \arrow {\cal B}'$ 
be an injective morphism of sheaves on $M$. {\bf \purple Prove that
$\phi$ induces an injective map ${\cal B}(U) \arrow {\cal B}'(U)$
for each $U$.}

\remark
A sheaf epimorphism ${\cal B} \stackrel \phi \arrow {\cal
  B}'$  {\bf \red does not necessarily induce a
surjective map ${\cal B}(U) \arrow {\cal B}'(U)$.}

\definition
{\bf\blue A sheaf isomorphism} is a homomorphism
$\Psi:\; {\cal F}_1 \arrow {\cal F}_2$, for which there exists
an homomorphism $\Phi:\; {\cal F}_2 \arrow {\cal F}_1$,
such that $\Phi\circ \Psi =\Id$ and 
$\Psi\circ \Phi =\Id$.

\exercise Show that
a morphism of sheaves $\Psi:\; {\cal F}_1 \arrow {\cal
  F}_2$ 
{\bf \red is an isomorphism if and only if it is epi and mono.}

\newpage

{\bf \blue Sheaves of modules}

\remark
Let  $A:\; \phi \arrow B$ be a ring homomorphism, and
$V$ a  $B$-module. {\bf \purple Then $V$ is equipped with a natural
$A$-module structure: $a v:= \phi(a) v$.}



\definition
Let  ${\cal F}$ be a sheaf of rings on
a topological space $M$, and 
 ${\cal B}$ another sheaf.
It is called {\bf\blue a sheaf of  ${\cal F}$-modules}
if for all $U\subset M$ the space of sections
${\cal B}(U)$ is equipped with a structure of ${\cal
  F}(U)$-module, and for all  $U'\subset U$, 
the restriction map 
${\cal B}(U) \stackrel{\phi_{U,U'}}\arrow {\cal B}(U')$
is a homomorphism of ${\cal F}(U)$-modules
(use the remark above
to obtain a structure of ${\cal F}(U)$-module
on  ${\cal B}(U')$).


\definition
A {\bf \blue free sheaf of modules}  ${\cal F}^n$ 
over a ring sheaf  ${\cal F}$ maps an open set $U$ to 
the space ${\cal F}(U)^n$. 


\definition
{\bf\blue Locally free sheaf of modules}
over a sheaf of rings  ${\cal F}$ is a sheaf 
of modules ${\cal B}$ satisfying the following
condition. For each $x\in M$ there exists a neighbourhood $U\ni x$
such that the restriction ${\cal B}\restrict U$ is free.

\definition
{\bf\blue A vector bundle} on a 
smooth manifold $M$ is a locally free sheaf of $C^\infty M$-modules.


\newpage

{\bf \blue Dual sheaves}

\claim
Let $U\subset V$ be open subsets of a Hausdorff space $M$.
{\bf \red A section $s\in {\cal F}(U)$ with compact support $Z\subset U$
can be uniquely extended to $\tilde s\in {\cal F}(U)$, also with support
in $Z$.}

{\bf \green Proof:} $Z$ is a closed subset of $U$, 
not intersecting $M\backslash V$. Let $U_1$ be an open neighbourhood
of $Z$ not intersecting $M\backslash V$, and 
$U_2:=M \backslash Z$. Then $\{U_1, U_2\}$
is a cover of $M$, and $s \restrict {U_1\cap U_2}=0$,
hence {\bf \purple $\tilde s$ can be glued from $s\in {\cal F}(U_1)$
and $0\in {\cal F}(U_2)$. }\endproof

\definition
Let ${\cal F}$ be a sheaf.
Denote the space of sections of ${\cal F}$
on $U$ with compact support by ${\cal F}_c(U)$.
Let ${\cal F}^*(U)$ map $U$ to the dual space
${\cal F}_c(U)^*$. Using the claim above, we
obtain a restriction map ${\cal F}^*(V)\arrow {\cal F}^*(U)$
for each open $V\supset U$. This gives {\bf \blue dual presheaf}
${\cal F}^*$

\exercise
Let $M$ be a manifold, and ${\cal F}$ a sheaf of modules
over $C^\infty M$. {\bf \red Prove that ${\cal F}^*$ is a sheaf.}

{\bf\green HINT:} Use partition of unity.


\newpage

{\bf \blue Rings and derivations (reminder)}

\remark
All rings in these handouts are assumed to be 
commutative and with unit. Algebras are associative,
but not necessarily commutative (such as the matrix algebra).
{\bf\blue Rings over a field $k$} are rings containing a field $k$.
We assume that $k$ has characteristic 0.

\definition
Let $R$ be a ring over a field $k$. A $k$-linear map
$D\; R \arrow R$ is called {\bf\blue a derivation}
if it satisfies {\bf\blue the Leibnitz equation}
$D(fg) = D(f) g + gD(f)$. The space of derivations
is denoted as  $\Der_k(R)$.

\example
$\frac{d}{dt}:\; \C[t]\arrow \C[t]$. \ \ 
$\frac{d}{dt}:\; C^\infty \R\arrow C^\infty\R$.


\theorem
Let  $x_1, ..., x_n$ be coordinate functions on $\R^n$, $R=C^\infty\R^n$,
and $\Der(R) \stackrel \Pi \arrow(C^\infty\R^n)^n$
map $D$ to $(D(x_1), D(x_2), ..., D(x_n))$.
{\bf \red Then \[ \Pi:\; \Der(C^\infty\R^n) \arrow R^n\] is an isomorphism.}


\newpage

{\bf \blue  Derivations of an algebra of compactly
  supported functions}

{\bf \green SUBLEMMA:}
Let $M$ be a smooth manifold, and $C^\infty_c(U)$
the space of compactly supported smooth functions,
equipped with natural multiplication. Consider
the space $\Der(M, C^\infty_c(M))$ of derivations
of $C^\infty_c(M)$ with values in $C^\infty(M)$.
Let $f$ be a function with support in $Z\subset M$.
{\bf \red Then $D(f)$ has support in $Z$ for each $D\in Der(M,
C^\infty_c(M))$.}

{\bf \green Proof:} For each $g$ with support outside of
$Z$, we have $0=D(fg) = f D(g) + g D(f)$,
hence for each $U\subset M\backslash Z$, we have
$0= g D(f)\restrict U$ whenever $\Sup(g)\subset U$. 
Then, $D(f)\restrict U=0$ as well.
\endproof



\lemma
{\bf \red Let $\phi$ be a smooth function with compact support
which is equal to 1 on $U\subset M$, and $D\in C^\infty_c(M)$.
Then $D(\phi f)\restrict U= D(f)\restrict U$.}

{\bf \green Proof. Step 1:}
$D(\phi f)= D(\phi) f - \phi D(f)$, hence {\bf \purple it would suffice
to prove that $D(\phi)\restrict U=0$.}

{\bf \green Step 2:}
Now, $\phi^2-\phi=0$ on $U$. By the Sublemma above,
$0=D(\phi^2-\phi)\restrict U= [2D(\phi)\phi
  -D(\phi)]\restrict U= D(\phi)\restrict U$.
\endproof


\newpage

{\bf \blue  Derivations with compact support and without}


\theorem
 Define the tautological map
$\Der(C^\infty(M))\stackrel \tau \arrow \Der(M, C^\infty_c(M))$,
taking $D$ to itself. {\bf \red 
Then $\tau$ is an isomorphism:
\[ \Der(M, C^\infty_c(M))=
  \Der(C^\infty(M)).\]}
\!\!\!{\bf \green Proof. Step 1:} For each $x\in M$, consider 
a smooth function $\psi_x$ with compact support
which is equal to 1 in some neighbourhood $U_x\ni x$.
Let $D\in \ker \tau$, and $f\in C^\infty M$. Then 
$0=D(\psi_x f)\restrict {U_x} = D(f)\restrict {U_x},$
hence $D(f)=0$ in a neighbourhood of any $x\in M$.
{\bf \purple This implies that $\ker \tau =0$.}

{\bf \green Step 2:} Consider $D\in \Der(M,
C^\infty_c(M))$, and let 
$D(f)\restrict {U_x}:=D(\psi_x f)$.
For any $U_x, U_y$, the Lemma above implies
\[ D(\psi_x f)\restrict{U_x\cap U_y}= D(\psi_y f)\restrict{U_x\cap U_y}
=D(f)\restrict{U_x\cap U_y}
\]
{\bf \purple Therefore, the sections $D(f)\restrict {U_x}$
agree on pairwise intersections, and define
a section $\tilde D(f)\in C^\infty M$.}

{\bf \green Step 3:} On germs, $\tilde D(f)=D(f)$,
hence it satisfies Leibnitz rule on each germ,
and therefore, {\bf \purple $\tilde D$ is a derivation}
{\bf \red (exercise: check this directly).}

{\bf \green Step 4:} By the Lemma above, 
$\tilde D(f)\restrict{U_x}=D(f)\restrict{U_x}$
for each $f$ with compact support. {\bf \purple Therefore,
$\tau(\tilde D)=D$, and $\tau$ is surjective.}
\endproof


\newpage

{\bf \blue  Derivations as a sheaf}

\definition
Let $U\subset V$ be open subsets of a smooth manifold $M$,
and $D\in \Der(V, C^\infty_c(V))$. For any $f\in C^\infty_c(U)$,
extend $f$ to $\tilde f\in C^\infty_c(V)$ with the same support,
(zero extension), and let $D\restrict U(f):= D(\tilde f)$.
{\bf \blue 
This defines a structure of presheaf $U\arrow C^\infty_c(U)$.}

\claim 
$U\arrow C^\infty_c(U)$ is a sheaf.

{\bf\green Proof. Step 1:} 
A vector field is uniquely determined by its restriction
to the germs of all sections, hence a derivation $D$ which vanishes
on all germs for all $x\in M$ vanishes everywhere. {\bf
  \purple This
takes care of the first sheaf axiom. }

{\bf\green Proof. Step 2:} Let $\{U_i\}$ be a cover of $M$. 
To glue a derivation
$D$ from its bits $D_i\in \Der(C^\infty(U_i))$,
consider a partition of unity $\psi_i$ subordinate to
$\{U_i\}$. {\bf \purple Then $D(f):= \sum D_i(\psi_i f)$ is a
derivation which restricts to all $D_i$.}
\endproof


\corollary 
{\bf \purple The sheaf of derivations is locally free, that is,
$\Der C^\infty M$ defines a vector bundle on $M$.}

\definition
It is called {\bf \blue the tangent bundle}, and denoted $TM$.


\newpage

{\bf \blue  NEXT TIME.}

{\bf \red \Huge Next time:\\[10mm] TEST ASSIGNMENT \# 2!}


\end{document}


