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            lecture 9 \hfil
  \tiny M. Verbitsky }}
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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Geometry of manifolds \\[15mm]
\small Lecture 9: Serre-Swan theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

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

{\tiny\bf Math in Moscow and HSE
\\[2mm]  April 15, 2013
}
\end{center}

\newpage

{\bf \blue Locally trivial fibrations}

\definition
A smooth map $f:\; X \arrow Y$ is called {\bf \blue a locally trivial
fibration} if each point $y\in Y$ has a neighbourhood
$U\ni y$ such that $f^{-1}(U)$ is diffeomorphic to
$U\times F$, and the map
$f:\; f^{-1}(U)= U\times F \arrow U$ is a projection.
In such situation, $F$ is called {\bf \blue the fiber}
of a locally trivial fibration.

\definition {\bf \blue A trivial fibration}
is a map $X \times Y \arrow Y$.

\definition
{\bf \blue A total space of a 
vector bundle} on $Y$ is a locally trivial fibration
$f:\; X\arrow Y$ with fiber $\R^n$, with each fiber 
$V:= f^{-1}(y)$ equipped with a structure of a vector
space, smoothly depending on $y\in Y$.

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

\remark
Let $\pi:\; B \arrow M$ be a total space of a vector bundle, 
$U\subset M$ open subset, and ${\cal B}(U)$ the space of all smooth sections
of $\pi^{-1}(U)\stackrel \pi \arrow U$. {\bf \purple Then
  ${\cal B}$ is a locally free sheaf of $C^\infty M$-modules}.

\remark {\bf \red
This construction is an ``equivalence of categories'';}
see below for a definition.

\newpage

{\bf \blue Categories}

\definition 
{\bf \blue A category} $\cac$ is a collection of data
called ``objects'' and ``morphisms between objects''
which satisfies the axioms below.

{\bf \green DATA.}\\
\phantom{XX} {\bf \red Objects:} The set $\Ob(\cac)$ of {\bf \blue objects}
of $\cac$. \\
\phantom{XX} {\bf \red Morphisms:} For each $X, Y\in \Ob(\cac)$,
one has a set $\Mor(X,Y)$ of {\bf \blue morphisms from $X$ to $Y$}.\\
\phantom{XX} {\bf \red Composition of morphisms:} For each
$\phi\in \Mor(X,Y), \psi \in \Mor(Y,Z)$ there exists
{\bf \blue the composition} $\phi\circ \psi \in \Mor(X, Z)$\\
\phantom{XX} {\bf \red Identity morphism:} For each 
$A\in \Ob(\cac)$ there exists a morphism $\Id_A \in \Mor(A,A)$.



{\bf \green AXIOMS.}\\
\phantom{XX} {\bf \red Associativity of composition:}
$\phi_1\circ(\phi_2\circ\phi_3)=(\phi_1\circ\phi_2)\circ\phi_3$.\\
\phantom{XX} {\bf \red Properties of identity morphism:}
For each $\phi\in \Mor(X,Y)$, one has
$\Id_x\circ \phi = \phi = \phi\circ \Id_Y$.

\newpage

{\bf \blue Functors and equivalence of categories}


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


\definition
Two functors $F, G:\;\cac_1\arrow \cac_2$ 
are called {\bf\blue 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
$F(\phi) \circ \Psi_Y= \Psi_X\circ G(\phi).$


\definition
A functor $F:\; \cac_1 \arrow \cac_2$ is called
{\bf\blue 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 $\cac_2$.


\example
Let $\cac$ be a category of finite-dimensional vector spaces
ovet $\R$ with a fixed basis (morphisms are linear maps), and
$\cac'$ a category with 
$\Ob(\cac')=\{\emptyset, \R, \R^2, \R^3, ...\}$, and morphisms 
also linear maps. Prove that the inclusion map
$\cac' \arrow \cac$ {\bf \red is an equivalence of categories,}
but {\bf \purple not an isomorphism}.

\newpage

{\bf \blue Total space of a vector bundle from its sheaf
  of sections}

\definition
{\bf\blue Category of vector bundles} $\cac_b$
is a category where objects are locally free $C^\infty M$-sheaves,
and morphisms are morphisms of $C^\infty M$-sheaves such that
all kernels and cokernels are locally free.

\exercise
{\bf \purple Prove that it is a category.}

\definition
{\bf\blue Category of total spaces of vector bundles} $\cac_t$
is a category where objects are total spaces of vector bundles,
and morphisms of total spaces over $M$ are maps $B_1 \arrow B_2$ 
compatible with projection to $M$, the multiplicative
structure, and of constant rank at each fiber.

\exercise
{\bf \purple Prove that it is a category.}

\theorem
Let $\pi:\; B \arrow M$ be a total space of a vector bundle, 
$U\subset M$ open subset, and ${\cal B}(U)$ the space of all smooth sections
of $\pi^{-1}(U)\stackrel \pi \arrow U$. Then this map {\bf \red
defines an equivalence of categories $\cac_b\tilde \arrow \cac_t$.}

\remark {\bf \purple The proof was given in the last lecture,}
using different language.

\exercise
{\bf \purple Produce a proof of this theorem.} 


\newpage

{\bf \blue Tensor product}


\definition
Let $V,V'$ be $R$-modules, $W$ a free abelian group
generated by $v\otimes v'$, with $v\in V, v'\in V'$,
and $W_1\subset W$ a subgroup generated by combinations
$rv \otimes v'-v\otimes rv'$, $(v_1+ v_2)\otimes v'-
v_1 \otimes v' - v_2 \otimes v'$ and $v\otimes (v'_1+ v'_2)-
v\otimes v'_1 - v\otimes v'_2$.
Define {\bf\blue the tensor product} $V \otimes_R V'$
as a quotient group $W/W_1$.

\exercise
Show that $r \cdot v\otimes v'\mapsto (rv)\otimes v'$
{\bf \purple defines an $R$-module structure on $V \otimes_R V'$.}

\remark
Let ${\cal F}$ be a sheaf of rings, and
${\cal B}_1$ and ${\cal B}_2$ be sheaves
of locally free $(M, {\cal F})$-modules.
{\bf \purple Then
\[ U \arrow {\cal B}_1(U)\otimes_{{\cal F}(U)} {\cal B}_2(U)
\]
is also a locally free sheaf of modules.}


\definition
{\bf \blue Tensor product} of vector bundles
is a tensor product of the corresponding sheaves of modules.



\newpage

{\bf \blue Dual bundle and bilinear forms}

\definition
Let $V$ be an $R$-module. {\bf\blue A dual $R$-module}
$V^*$ is $\Hom_R(V, R)$ with the $R$-module structure
defined as follows: $r\cdot h(\dots) \mapsto rh(\dots)$.

\claim
Let ${\cal B}$ be a vector bundle, that is,
a locally free sheaf of  $C^\infty M$-modules, and
$\Tot{\cal B}\stackrel \pi \arrow M$
its total space. Define ${\cal B}^*(U)$
as a space of smooth functions on $\pi^{-1}(U)$ 
linear in the fibers of $\pi$. 
{\bf \purple
Then  ${\cal B}^*(U)$ is a locally free sheaf over
$C^\infty(M)$.}

\definition
This sheaf is called {\bf \blue the dual vector bundle},
denoted by $B^*$. Its fibers are dual to the fibers 
of $B$.

\definition
{\bf \blue Bilinear form} on a bundle ${\cal B}$ 
is a section of $({\cal B}\otimes {\cal B})^*$.
A symmetric bilinear form on a real bundle ${\cal B}$
is called {\bf\blue positive definite} if it gives
a positive definite form on all fibers of 
${\cal B}$. Symmetric positive definite
form is also called {\bf\blue a metric}.
A skew-symmetric bilinear form on
${\cal B}$ is called {\bf\blue non-degenerate}
if it is non-degenerate on all fibers of ${\cal B}$.

\newpage

{\bf \blue Subbundles}

\definition
{\bf\blue A subbundle} ${\cal B}_1\subset {\cal B}$ 
is a subsheaf of modules which is also a vector bundle,
and such that the quotient ${\cal B}/{\cal B}_1$ is also
a vector bundle.

\definition
{\bf\blue Direct sum} $\oplus$ of vector bundles is a direct
sum of corresponding sheaves.

\example
Let ${\cal B}$ be a vector bundle equipped with a metric
(that is, a positive definite symmetric form),
and ${\cal B}_1 \subset {\cal B}$ a subbundle.
Consider a subset $\Tot {\cal B}_1^\bot\subset \Tot {\cal B}$,
consisting of all $v\in {\cal B}\restrict x$
orthogonal to ${\cal B}_1\restrict x \subset {\cal
B}\restrict x$. {\bf \purple Then $\Tot {\cal B}_1^\bot$ 
is a total space of a subbundle, denoted as ${\cal
B}_1^\bot\subset {\cal B}$,} and we have an isomorphism
${\cal B}= {\cal B}_1\oplus {\cal B}_1^\bot$.

\remark
A total space of a direct sum of vector
bundles ${\cal B}\oplus {\cal B}'$ {\bf \purple is homeomorphic to 
$\Tot {\cal B}\times_M \Tot {\cal B}'$.}

\exercise
Let ${\cal B}$ be a real vector bundle.
{\bf \red Prove that ${\cal B}$ admits a metric.}

\proposition
Let $A\subset B$ be a sub-bundle. {\bf \red Then $B\cong A\oplus C$.}

\proof {\bf \purple Find a positive definite metric on $B$,} and set
$C:=B^\bot$. \endproof


\newpage

{\bf \blue Tangent bundle}

\proposition
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
\[
\lim_{t \arrow 0}\frac{d(M, m+tv)}{t} \arrow 0.
\]
Then the natural additive operation on 
$TM\subset M \times \R^n$ (addition of the second argument)
and a multiplication by real numbers {\bf \blue defines on
$TM$ a structure of a relative vector space over $M$},
that is, makes $TM$ a total space of a vector bundle.
Moreover, this vector bundle is isomorphic to a
tangent bundle, that is, to the sheaf $\Der_\R (C^\infty M)$.

{\bf \green Proof. Step 1:}  
For each $z\in M$, we can choose coordinates in a neighbourhood
of $z$ in $\R^n$ in such a way that $M=\R^k \subset \R^n$.
Therefore, {\bf \purple
it would suffice to prove proposition when $M=\R^k\subset \R^n$.}

{\bf \green Proof. Step 2:} In this case, $TM=\R^k\times \R^k$ {\bf \purple is 
a total space of a vector bundle,} 
of the same dimension as the tangent bundle. It remains to construct
a sheaf morphism from the sheaf of
sections of $TM$ to $\Der_\R (C^\infty M)$, 
inducing an isomorphism.

\newpage

{\bf \blue Tangent bundle (cont.)}


{\bf \green Proof. Step 3:}
Let $\pi_x:\; \R^n \arrow T_x M$ be an orthogonal projection
map. By the inverse function theorem, {\bf \purple
$\pi_x\restrict M:\; M \arrow T_x M$ is a
diffeomorphism in a neighbourhood of $x\in M$.}
Let $U_x \subset T_x M$ be such an open neighbourhood
and $\pi_x^{-1}(U_x)\stackrel{\pi_x}\arrow U_x$ a diffeomphism.

{\bf \green Proof. Step 4:}
For each vector $v\in T_xM$, and $f\in C^\infty M$,
let $D_v(f)$ be the derivative of $\tilde f\in C^\infty U_x$
along $v$, where $\tilde f(z)= f(\pi_x^{-1}(z))$.
Then a section $\gamma \in TM(U)$ defines a derivation
$D_\gamma(f)(z):= D_{\gamma\restrict z}(f)$. {\bf \purple We obtained
a sheaf homomorphism $TM\stackrel \Psi \arrow \Der_\R (C^\infty M)$.}

{\bf \green Proof. Step 5:} The vector bundles
$TM$ and $\Der_\R (C^\infty M)$ have the same dimension,
and for each non-zero  vector $v\in T_xM$, {\bf \purple the corresponding
derivation is non-zero, hence $\ker \Psi=0$.}
\endproof


\definition
The tangent bundle of $M$, as well as its total space,
is denoted by $TM$. When one wants to distinguish the total
space and the tangent bundle, one writes $\Tot(TM)$.


\newpage

{\bf \blue Pullback}

\claim 
Let $M_1\stackrel \phi \arrow M$ be a smooth
map of manifolds, and $B \stackrel \pi \arrow M$ 
a total space of a vector bundle. {\bf \red Then $B \times_M M_1$
is a total space of a vector bundle on
$M_1$. }

{\bf\green Proof. Step 1:}
{\bf \purple $B \times_M M_1$ is obviously a relative vector space.}
Indeed, the fibers of projection $\pi_1:\; B \times_M M_1 \arrow M_1$
are vector spaces, $\pi_1^{-1}(m_1) = \pi^{-1}(\phi(m_1))$.
It remains only to show that it is locally trivial.

{\bf\green Step 2:}
Consider an open set $U\subset M$
that $B\restrict U = U \times \R^n$, and let $U_1:= \phi^{-1} U$.
Then $B \times_U U_1= U_1 \times \R^n$. {\bf \purple Since $M_1$ is covered by
such $U_1$, this implies that $\pi_1$ is a locally trivial
fibration,} and the additive structure smoothly depends on $m_1\in M_1$.
\endproof

\definition
The bundle $\pi_1:\; B \times_M M_1 \arrow M_1$
is denoted $\phi^* B$, and called
{\bf\blue inverse image}, or {\bf \blue a pullback} of $B$.


\newpage

{\bf \blue Pullback and the tangent bundle}


\claim Let $j:\; M\hookrightarrow N$ be a closed embedding
of smooth bundles. {\bf \red Then there is a natural injective
morphism of vector bundles $TM\hookrightarrow j^* TN$.}

\proof 
Using Whitney's theorem, we embed $N$ to $\R^n$.
Then $j^* TN\subset M\times \R^n$ is the set of pairs
$x\in M, v\in T_x N$. {\bf \purple The bundle $TM$ is embedded to 
$j^*TN$, because each tangent vector to $M$
is also tangent to $N$.}
\endproof

\exercise
Prove that the map $TM\hookrightarrow j^* TN$
{\bf \red is independent from the choice of embedding 
$N \subset \R^n$.}

\corollary
Let $M$ be a manifold, and $j:\; M\hookrightarrow \R^n$ a closed
embedding. {\bf \red Then $TM$ is a direct summand of a trivial bundle
$j^* T\R^n$.}

\newpage

{\bf \blue Any bundle is a direct summand of a trivial bundle}

\theorem
{\bf \red Any vector bundle on a metrizable manifold 
is a direct summand of a trivial bundle.}

{\bf \green Proof. Step 1:} 
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.
{\bf \purple Then the pullback $\phi^* T\Tot B$ is isomorphic
(as a bundle) to the direct sum $TM \oplus B$.}


{\bf \green Step 2:} Using Whitney's theorem, find a 
closed embedding $j:\; \Tot B\arrow \R^n$. {\bf \purple This gives 
injective morphisms of vector bundles
\[ B \hookrightarrow TM \oplus B = \phi^*(T\Tot B)\hookrightarrow 
(\phi j)^*T\R^n.\]} However, $(\phi j)^*T\R^n$ is trivial,
because the bundle $T\R^n$ is trivial.
\endproof


\newpage

{\bf \blue 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 \blue direct summands} of $V$, and $V$ -- {\bf\blue a direct sum}
of $V'$ and $V''$. This is denoted $V=V' \oplus V''$.


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

\corollary Let ${\cal B}$ be a vector bundle, and 
$B$ its space of sections, considered as a $C^\infty M$-module.
{\bf \red Then $B$ is projective.}

\theorem {\bf \blue (Serre-Swan theorem)}
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,
$\cac_b$ the category of vector bundles, and
$\Psi:\; \cac_b \arrow \cac_b$ a functor mapping
$B$ to its space of global sections. {\bf \red Then
$\Psi$ is an equivalence of categories.}

Proof later.

\newpage

{\bf \blue Determinant bundle}

\definition
{\bf \blue A line bundle} is a 1-dimensional vector bundle.

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

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

\remark It is a line bundle.

\remark
Let $M$ be an $n$-manifold, and $\Lambda^n TM$ a determinant
bundle of its tangent bundle. Prove that {\bf \purple $\Lambda^n TM$
is trivial if and only if $M$ is orientable.}

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

\newpage

{\bf \blue Trivializations and determinant}

\definition Recall that {\bf\blue a trivialization}
of a vector bundle $B$ over $U$ is a set of {\bf \blue
free generators} of $B$, that is, sections
$x_1,..., x_n\in B$ such that the map
$\nu:\; (C^\infty U)^n \arrow B\restrict U$
mapping generators $e_i \in (C^\infty U)^n$ to $x_i$
is an isomorphism.

\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\blue nowhere vanishes} on $U\subset M$ if its germ
$b_x$ does not lie in ${\goth m}_x {\cal B}$
for each $x\in U$.

\proposition
 Let $B$ be a vector bundle, and $x_1,..., x_n\in B$ 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 $U\subset M$. {\bf \purple Then
$\left\{x_i\restrict U\right\}$ are free generators of $B\restrict U$.}

\proof
Define a map $\nu:\; (C^\infty U)^n \arrow B\restrict U$
mapping generators $e_i \in (C^\infty U)^n$ to $x_i$.
{\bf \purple This map induces an isomorphism on each fiber, hence
bijective. } The inverse function theorem implies that
it is a diffeomorphism. \endproof


\newpage

{\bf \blue A stalk of a $C^\infty M$-module}


\definition
Let $x\in M$ be a point on a manifold.
{\bf\blue 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.

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

\claim
Let $A$ be a free  $C^\infty M$-module of rank $n$, 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 ${\cal
  A}$. {\bf \blue 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.
Then\\
\phantom{xx} (i) ${\cal B}$, ${\cal C}$ are sheaves of $C^\infty M$-modules.\\
\phantom{xx} (ii) ${\cal A} ={\cal B}\oplus {\cal C}$.\\
\phantom{xx} (iii) {\bf \red The sheaves ${\cal B}$, ${\cal C}$
are locally free.}

\proof
Next slide.


\newpage

{\bf \blue The proof of Serre-Swan theorem}

\claim
Let $A$ be a free  $C^\infty M$-module of rank $n$, 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 ${\cal
  A}$. {\bf \blue 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.
Then\\
\phantom{xx} (i) ${\cal B}$, ${\cal C}$ are sheaves of $C^\infty M$-modules.\\
\phantom{xx} (ii) ${\cal A} ={\cal B}\oplus {\cal C}$.\\
\phantom{xx} (iii) {\bf \red The sheaves ${\cal B}$, ${\cal C}$
are locally free.}

\proof
The first two claims are clear.



Fix $z\in M$.
Let $x_1, ..., x_k$ be sections of ${\cal B}$
generating ${\cal B}/{\goth m}_z {\cal B}$ and
$y_1, ..., y_l$ sections of ${\cal C}$
generating ${\cal C}/{\goth m}_z {\cal C}$.
Choose them to be linearly independent,
and let $U$ be an open neighbourhood of $z$ such that the section
$x_1\wedge x_2 \wedge ... \wedge x_k \wedge y_1\wedge ... \wedge y_l\in \Lambda^n B$
is nowhere degenerate on $U$. Then $\{x_i, y_j\}$ are 
free generators of ${\cal A}$, hence $\{x_i\}$ are free
generators of ${\cal B}$ and $\{y_j\}$ are free
generators of ${\cal C}$. {\bf \red We have shown that
these sheaves are locally free}. \endproof

\remark 
This gives a way of reconstructing a vector bundle
from a projective $C^\infty M$-module.
{\bf \purple The rest of the proof of Serre-Swan is left
as an exercise.}

\end{document}


