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\listok{9}{Geometry 9: Serre-Swan theorem}

{\scriptsize
{\bf Rules:} You may choose to solve only 
``hard'' exercises (marked with !, * and **) 
or ``ordinary'' ones (marked with ! or unmarked),
or both, if you want to have extra problems.
To have a perfect score, a student must obtain
(in average) a score of 10 points per week.
It's up to you to ignore handouts entirely,
because passing tests in class and having
good scores at final exams could compensate
(at least, partially) for the points obtained 
by grading handouts.

Solutions for the problems are to be explained to the 
examiners orally in the class and marked in the score sheet. 
It's better to have a written version of your solution with 
you. It's OK to share your solutions with other students, and use
books, Google search and Wikipedia, we encourage it.

If you have got credit for 2/3 of ordinary problems
or 2/3 of ``hard'' problems, you receive  
$6t$ points, where $t$ is a number depending on the
date when it is done. Passing all ``hard'' 
or all ``ordinary'' problems
(except at most 2) brings you $10t$ points.
Solving of ``**'' (extra hard) problems is not
obligatory, but each such problem gives you a credit
for 2 ``*'' or ``!'' problems in the ``hard'' set.

The first 3 weeks after giving a handout, $t=1.5$,
between 21 and 35 days, $t=1$, and afterwards, $t=0.7$.
The scores are not cumulative, only the
best score for each handout counts.

Please keep your score sheets until the final
evaluation is given.
}



\def\cac{{\cal C}}
\def\Ob{\operatorname{{\cal O}b}}
\def\Mor{\operatorname{{\cal M}or}}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Vector bundles and Whitney theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\exercise
Let  $M\subset \R^n$ be a smooth submanifold of
$\R^n$, and $TM\subset \R^n \times \R^n$ the set of all pairs
$(v, x) \in M \times \R^n$, where $x\in M \times \R^n$
is a point of $M$, and $v\in \R^n$ a vector tangent to $M$ in
$m$, that is, satisfying
\[
\lim_{t \arrow 0}\frac{d(M, m+tv)}{t} \arrow 0.
\]
\enum
\ite Prove that the natural additive operation on 
$TM\subset M \times \R^n$ (addition of the second argument)
defines a structure of a (relative) topological group over $M$ on $TM$.
\ite Prove that a multiplication by real numbers defines on
$TM$ a structure of a relative vector space over $M$.
\ite Prove that $TM$ is a total space of a vector bundle.
\ite[!] Prove that this vector bundle is isomorphic to a
tangent bundle, that is, to the sheaf $\Der_\R (C^\infty M)$.
\ee
\ez

\definition
The tangent bundle of $M$, as well as its total space,
is denoted by $TM$.
\ed


\exercise
\label{_vlo_v_triv_Zadacha_}
Let $M$ be a metrizable manifold. Prove that 
the bundle $TM$ is a direct summand of a trivial bundle.
\ez

\hint
Apply Whitney's embedding theorem and use the previous exercise.
\eh

\exercise
\label{_kasa_k_tot_Zadacha_}
Let $B$ be a vector bundle on $M$, and
$\Tot B$ its total space. Consider the tangent bundle
 $T\Tot B$, and let $M\stackrel \phi \hookrightarrow \Tot B$
be an embedding corresponding to a zero section.
Prove that the pullback  $\phi^* T\Tot B$ is isomorphic
(as a bundle) to the direct sum $TM \oplus B$.
\ez

\exercise[!]
\label{_pryamoe_Zadacha_}
Prove that any vector bundle on a metrizable, connected
manifold is a direct summand of a trivial bundle.
\ez

\hint Use exercises
\ref{_kasa_k_tot_Zadacha_} 
and \ref{_vlo_v_triv_Zadacha_}.
\eh



\exercise
Show that the bundle $TS^1$ is trivial
\ez

\exercise[!]
Let $M$ be a manifold which is not orientable.
Prove that the bundle $TM$ is non-trivial.
\ez

%\exercise
%Prove that $TS^2$ is non-trivial.
%\ez

\exercise
Prove that any 1-dimensional bundle on 
a sphere $S^2$  is trivial.
\ez


%\exercise
%Prove that $TS^3$ is trivial.
%\ez

\exercise[*]
Let $T S^2 \oplus \R$ be a direct sum of
a tangent bundle $TS^2$ and a trivial 1-dimensional bundle.
Is the bundle $T S^2 \oplus \R$ trivial?
\ez

\exercise
Let $G$ be a topological group,
diffeomorphic to a manifold, with all group maps
smooth (such a group is called {\bf Lie group}).
Prove that the tangent bundle $TG$ is trivial. 
\ez

\exercise[*]
Find a non-trivial vector bundle on $S^3$, or prove 
that it does not exist.
\ez

\definition
{\bf Rank} of a bundle is the dimension of its fibers.
\ed

\definition
A {\bf line bundle} is a bundle of rank 1.
\ed

\exercise
Let $M$ be a simply connected manifold.
Prove that any real line bundle on $M$ is trivial.
\ez

\definition
Let $B$ be a vector bundle of rank $n$, and
$\Lambda^n B$ its top exterior product.
This bundle is called {\bf determinant bundle}
of $B$.
\ed

\definition
A real vector bundle is called {\bf orientable}
if its determinant bundle is trivial.
\ed

\exercise
\enum
\ite
Prove that a direct sum of orientable vector bundles
is orientable. 
\ite
Prove that a tensor product of orientable vector bundles is 
orientable.
\ee
\ez

\exercise[*]
Find a non-trivial, orientable 3-dimensional real vector bundle
on a 2-dimensional torus, or prove that it does not exist.
\ez

\exercise[*]
Let $B$ be a real vector bundle on $S^n$
of dimension $\geq n+1$. Prove that $B$ is trivial,
or find a counterexample. 
\ez

\exercise[**]
Consider a bundle $\Lambda^2 S^n$ of 2-forms
on an $n$-dimensional sphere. Find all $n$
for which this bundle is trivial.
\ez


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Projective modules}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
Let $V$ be an $R$-module, and $V'\subset V$ its submodule.
Assume that $V$ contains a submodule $V''$, not intersecting
$V'$, such that $V'$ together with $V''$ generate $V$.
In this case,  $V'$ and $V''$ are called
{\bf direct summands} of $V$, and $V$ -- {\bf a direct sum}
of $V'$ and $V''$. This is denoted $V=V' \oplus V''$.
\ed

\exercise
Consider a submodule $n \Z\subset Z$. 
Can it be realized as a direct summand of $\Z$?
\ez

\exercise
Let $R$ be a ring without zero divisors,
and $V=R$ a free module of rank 1.
Find all direct summands of $V$.
\ez

\exercise[*]
Consider a ring of truncated polynomials
 $R:=\R[t]/(t^k)$, and let $V=R$ be a one-dimensional
$R$-module. Find all direct summands of $V$.
\ez

\definition
An $R$-module is called {\bf projective}
if it is a direct summand of a free module $\bigoplus_I R$
(possibly of infinite rank).
\ed

\exercise
Prove that each $R$-module is a quotient of a free module.
\ez


\exercise
\label{_proj_lifting_Exercise_}
Let $V$ be an $R$-module, described below, and
$A\stackrel \pi\arrow B$ a surjective homomorphism of
$R$-modules. Prove that each 
$R$-module homomorphism $V \stackrel \phi\arrow B$
can be lifted to a morphism $V \stackrel \psi\arrow A$,
making the following diagram commutative.
\begin{diagram}
V & \rTo^\phi & A\\
&\rdTo~\psi &\dTo~{\pi}\\
& & B
\end{diagram}
\enum
\ite Prove it in assumption that $V$ is a free $R$-module
\ite[!] Prove it in assumption that $V$ is projective.
\ee
\ez

%\NewVedomost

\exercise[!]
Let $V$ be a module for which the 
statement of Exercise \ref{_proj_lifting_Exercise_}
holds true. Prove that $V$ is projective.
\ez

\hint
Consider as $A$ a free $R$-module, mapped
to $V$ surjectively, and let $B$ be $V$, and
$\pi$ an identity map.
\eh

\definition
Let $0\arrow A \hookrightarrow B \arrow C \arrow 0$ -- 
be an exact sequence of $R$-modules. Assume that for
some $C'\subset B$ one has
$B=A \oplus C'$. In this case it is said that the
exact sequence 
$0\arrow A \hookrightarrow B \arrow C \arrow 0$ {\bf splits}.
\ed


\exercise[!]
Let $C$ be an $R$-module. Prove that the
following conditions are equivalent. 
\begin{description}
\item[(i)] Every exact sequence
$0\arrow A \hookrightarrow B \arrow C \arrow 0$
splits
\item[(ii)] The module $C$ is projective.
\end{description}
\ez

\exercise
Let $V$ be a finitely generated
projective module over $R$. Prove that it is free, if
\enum
\ite[*] $R=\Z$.
\ite[*] $R$ is a polynomial ring $\C[t]$.
\ite[*] $R$ is a local ring.
\ee
\ez

\exercise[**]
A $\Z$-module $V$ is {\bf torsion-free} if the natural
map \\ $V \arrow V\otimes_\Z \Q$ is injective.
Prove that any torsion-free $\Z$-module is 
projective, or find a counterexample (``finitely generated'' is not
assumed here).
\ez

\exercise
Let ${\cal B}$ be a bundle over a metrizable manifold $M$,
and ${\cal B}(M)$ the space of smooth sections of $B$.
Prove that ${\cal B}(M)$ is a projective
$C^\infty M$-module.
\ez

\hint
Use the exercise \ref{_pryamoe_Zadacha_}.
\eh



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Лемма Накаямы}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%\definition
%Пусть $I\subset R$ -- идеал.
%$R$-модуль вида $R/I$ называется
%{\bf циклическим}.
%\ed
%
%\exercise
%Пусть $V$ -- циклический $R$-модуль вида
%$R/I$, а $I_1 \supset I$ - идеал в $R$.
%Докажите, что $V/I_1V= R/I_1 R$.
%\ez
%
%\exercise
%Пусть $V$ -- циклический модуль над локальным
%кольцом $A$, а ${\goth m}$ -- максимальный
%идеал. Предположим, что $V= {\goth m}V$. Докажите, что $V=0$.
%\ez
%
%\hint
%Докажите, что $I\subset {\goth m}$,
%и воспользуйтесь предыдущей задачей.
%\eh
%
%\exercise[!]
%(лемма Накаямы)
%Пусть $V$ -- конечно-порожденный модуль над локальным кольцом
%$A$. Предположим, что $V= {\goth m}V$. Докажите, что $V=0$.
%\ez
%
%\hint
%Воспользуйтесь индукцией, и примените
%предыдущую задачу.
%\eh
%
%\exercise[**]
%(лемма Накаямы в большей общности).
%Пусть $V$ - конечно-порожденный модуль над $R$,
%а $I\in R$ - идеал, такой, что $IV=V$.
%Докажите, что найдется $r\in R$, сравнимый
%с единицей по модулю $I$, и такой, что
%$rV=0$. 
%\ez
%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Categories and functors}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\definition
{\bf A category} $\cac$ is a collection of data
 (``set of objects of $\cac$", ``set of morphisms from an object to an object", 
``operation of composition on morphisms",
``identity morphism"), satisfying the following axioms
\begin{description}
\item[Objects:] The set $\Ob(\cac)$ of objects of $\cac$.

\item[Morphisms:] For each $X, Y \in \Ob(\cac)$, one is given
{\bf the set of morphisms from $X$ to $Y$}, denoted by $\Mor(X,Y)$.

\item[Composition of morphisms:] If
$\phi\in \Mor(X,Y), \psi \in \Mor(Y,Z)$, one is given
the morphism $\phi\circ \psi \in \Mor(X, Z)$,
called {\bf composition of $\phi$ and $\psi$}.

\item[Identity morphism:] For each $A\in \Ob(\cac)$
one has a distinguished morphism $\Id_A \in \Mor(A,A)$.
\end{description}

\noindent
These data satisfy the following axioms.
\begin{description}
\item[Associativity of composition:] 
$\phi_1\circ(\phi_2\circ\phi_3)=(\phi_1\circ\phi_2)\circ\phi_3$.
\item[Properties of identity morphism:]
For each morphism $\phi\in \Mor(X,Y)$, one has
$\Id_X\circ \phi = \phi = \phi\circ \Id_Y$.
\end{description}
\ed

\exercise
Prove that the following data define categories.
\enum
\ite Objects are groups, morphisms are group homomorphisms.
\ite Objects are vector spaces, morphisms are linear maps.
\ite Objects are vector spaces, morphisms are surjective
linear maps. 
\ite Objects are topological spaces, morphisms --
continuous maps.
\ite Objects are smooth manifolds, morphisms are
smooth maps.
\ite Objects -- vector bundles on $M$, morphisms 
are morphisms of vector bundles.
\ee
\ez

\definition
Let $\cac_1, \cac_2$ be categories. {\bf A covariant functor}
from $\cac_1$ to $\cac_2$ is the following collection of data.
\begin{description}
\item[(i)] A map $F:\; \Ob(\cac_1) \arrow \Ob(\cac_2)$.
\item[(ii)] A map $F:\; \Mor(X,Y) \arrow \Mor(F(X), F(Y))$,
defined for each $X, Y \in \Ob(\cac_1)$.
\end{description}
These data define {\bf a functor from
$\cac_1$ to $\cac_2$}, if $F(\phi) \circ F(\psi) =
F(\phi\circ\psi)$, and $F(\Id_X) = \Id_{F(X)}$.
\ed



\exercise
Let $\cac$ be a category of sheaves of 
modules over a ringed space $(M, {\cal F})$.
Prove that a correspondence ${\cal B}\arrow {\cal B}(M)$
defines a functor from $\cac$ to a category of ${\cal F}(M)$-modules.
\ez

\definition
Two functors $F, G:\;\cac_1\arrow \cac_2$ 
are called {\bf equivalent} if for each
$X \in \Ob(\cac_1)$ there exists an isomorphism $\Psi_X:\; F(X) \arrow
G(X)$, such that for each $\phi\in \Mor(X,Y)$ one has
\begin{equation}\label{_equi_fu_Equation_}
 F(\phi) \circ \Psi_Y= \Psi_X\circ G(\phi).
\end{equation}
\ed

\definition
A functor $F:\; \cac_1 \arrow \cac_2$ is called
{\bf equivalence of categories} if
there exist functors $G, G':\; \cac_2 \arrow \cac_1$
such that $F\circ G$ is equivalent to an identity
functor on $\cac_1$, and $G' \circ F$
is equivalent to identity functor on $\cac_2$.
\ed

\exercise
Prove that the following categories are not equivalent.
\enum
\ite[!] Category of vector spaces and category of groups.
\ite[!] Category of topological spaces and category of vector spaces.
\ite[!] Category of groups and category of topological spaces.
\ee
\ez


\exercise[*]
Let $M$ be a compact manifold. Prove that the
category of sheaves of $C^\infty M$-modules is equivalent
to the category of modules over $C^\infty M$,
or find a counterexample.
\ez

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Serre-Swan theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\definition
Let $x\in M$ be a point on a manifold.
{\bf A stalk} of a $C^\infty M$-module $V$ is a 
tensor product $C^\infty_x M\otimes_{C^\infty M} V$,
where $C^\infty_x M$ is a ring of germs of $C^\infty M$ in $x$.
We consider a stalk $V_x$ as a $C^\infty_x M$-module.
\ed

\definition
Recall that {\bf a stalk} of a sheaf $F$ at $x\in M$
is a space of germs of $F$ at $x$.
\ed

\exercise
Let $V$ be a free $C^\infty M$-module.
Prove that a stalk of the space of sections $V(M)$
in $x$ is a stalk of the sheaf $V$ in $x$. 
\ez


\definition
Let $x\in M$ be a point on a manifold.
Denote by ${\goth m}_x\subset C^\infty M$
the ideal of all functions vanishing in $x$.
Let ${\cal B}$ be a sheaf of $C^\infty M$-modules,
and $b$ a section of ${\cal B}$. We say that
$b$ {\bf nowhere vanishes} if its germ
$b_x$ does not lie in ${\goth m}_x {\cal B}$
for each $x\in M$.
\ed

\exercise[!]
\label{_determinant_Exercise_}
Let $R$ be a ring of smooth functions on a smooth manifold,
${\goth m}_z$ a maximal ideal of $z\in M$, and 
$B$ a free $R$-module, considered as a trivial vector
bundle on $M$ of rank $n$. Let $x_1,..., x_n\in B$ be 
a set of sections which are linearly independent
in $B/{\goth m}_{z_0}B$ and generate $B/{\goth m}_{z_0}B$,
for a fixed point $z_0\in M$. 
Let $\xi\in \Lambda^n B$, $\xi:= x_1 \wedge x_2 \wedge
... \wedge x_n$ be the determinant of $x_i$, considered
as a section of a line bundle $\det B$. Suppose that
$\xi$ nowhere vanishes on $U\subset M$. Prove that
$\left\{x_i\restrict U\right\}$ are free generators of $B\restrict U$.
\ez

\hint
Define a map $\nu:\; (C^\infty U)^n \arrow B\restrict U$
mapping generators $e_i \in (C^\infty U)^n$ to $x_i$.
To prove that $\nu$ is an isomorphism, use the inverse
function theorem.
\eh


\exercise[!]
\label{_local_decompo_Exercise_}
Let $R$ be a ring of germs of smooth functions on $\R^n$,
and $V$ a free $R$-module, 
and $V=V_1 \oplus V_2$ a direct sum decomposition.
Prove that $V_1$ and $V_2$ are also free modules.
\ez

\hint
Use the previous exercise.
\eh



\exercise
\label{_puchki_s_proe_Zadacha_}
Let $A$ be a free  $C^\infty M$-module, decomposed as
a direct sum of two projective modules: $A= B\oplus C$.
We identify $A$ with a space of sections of a 
trivial sheaf of $C^\infty M$-modules, denoted by ${\cal
  A}$. Let  ${\cal B}\subset {\cal A}$ be a subsheaf
consisting of all sections  $\gamma \in {\cal V}(U)$,
such that the germs of $\gamma$ at each $x\in M$ 
lie in the stalk $B_x$. Define
${\cal C}\subset {\cal A}$ in a similar fashion.
\enum
\ite
Prove that ${\cal B}$, ${\cal C}$ are sheaves of $C^\infty M$-modules.
\ite Prove that  ${\cal A} ={\cal B}\oplus {\cal C}$.
\ite Prove that the stalk $B_x$ of a $C^\infty M$-module $B$ in $x\in M$
is isomorphic to the stalk ${\cal B}_x$ of the corresponding
sheaf of modules.
\ee
\ez

\definition
Let $V$ be a projective $C^\infty M$-module, and
$\rk_x V:= \dim V / {\goth m}_x V$ dimension of its fiber
in $x$. This number is called
{\bf rank} of $V$ in $x$.
\ed

\exercise
Let $V$ be a projective $C^\infty M$-module, and $x\in M$.
Assume that $\rk_x V=n$.
Prove that the stalk $V_x$ at $x$ is a free $C^\infty_x M$-module of 
rank $n$.
\ez

\hint
Use exercise \ref{_local_decompo_Exercise_}.
\eh


\exercise
Prove that the rank of a projective $C^\infty M$-module
over a connected manifold $M$ is constant.
\ez

\hint 
Use exercise \ref{_determinant_Exercise_}.
\eh


\definition
Let $B$ be a projective $C^\infty M$-module, 
$x_1,..., x_n$ its sections such that
such that their determinant 
$x_1 \wedge x_2 \wedge ...\wedge x_k\in \Lambda^k{\cal B}$
is nowhere vanishing. Then $\{x_i\}$ are called {\bf
  linearly
independent}.
\ed

\exercise
Let ${\cal B}$ be a sheaf of $C^\infty M$-modules
generated by linearly independent sections
$x_1, ..., x_k$. Prove that ${\cal B}$ is free.
\ez

\exercise[!]
Let $A$ be a free  $C^\infty M$-module, $A= B\oplus C$
its decomposition, and ${\cal B}$, ${\cal C}$ the corresponding
sheaves of modules. Prove each point $x\in M$ has
a neighbourhood  $U$ such that the sheaf ${\cal B}\restrict U$
is generated by $k:= \rk_x B$ linearly independent
sections $\{x_1, ..., x_k\}$.
\ez

\hint
Use exercise \ref{_determinant_Exercise_}.
\eh


\exercise[!]
\label{_bundle_from_proje_Exercise_}
Let $B$ be a projective $C^\infty M$-module, and
${\cal B}$ a sheaf of modules, generated as in 
Exercise \ref{_puchki_s_proe_Zadacha_}. Prove that
this sheaf is locally trivial.
\ez

\hint Use the previous exercise.
\eh

\exercise[!]
Let $\cac_p$ be a category with objects projective
$C^\infty M$-modules, and morphisms homomorphism of
$C^\infty M$-modules with kernels and cokernels projective. 
Check that the axioms
of category are satisfied.
\ez

\remark
Recall that we defined morphisms of vector bundles
as morphisms of the corresponding sheaves of $C^\infty M$-modules
such that their kernels and cokernels are locally free
$C^\infty M$-modules.
\er

\exercise
(Serre-Swan theorem)
Let $\cac_b$ be a category of vector bundles 
on $M$.
\enum
\ite[*] 
Consider a map $\Psi$ making a vector bundle from
a projective  $C^\infty M$-module, as in Exercise
\ref{_bundle_from_proje_Exercise_}.
Prove that $\Psi(B)$ does not depend on a choice
of a free module $A\supset B$.

\ite[*] Prove that $\Psi$ defines a functor from
$\cac_p$ to $\cac_b$.

\ite[*] Show that this functor defines an equivalence
of categories.
\ee
\ez

\end{document}
