math-ph0102016.tex
\documentstyle[12pt,fleqn]{article}

\textheight=7.8in
\textwidth=5.9in
\oddsidemargin .5cm
\evensidemargin .5cm
\topmargin= 1cm
\headsep 30pt
\flushbottom\textheight=7.8in
\renewcommand\baselinestretch{1.1}

\def\labelenumi{(\roman{enumi})}

\let\Large=\large

\def\op#1{\mathop{{\it\fam0} #1}\limits}
\newcommand{\id}{{\rm Id\,}}
\newcommand{\pr}{{\rm pr}}
\newcommand{\df}{{\rm def}}
\newcommand{\di}{{\rm dim\,}}
\newcommand{\Id}{{\rm Id}}
\newcommand{\Ker}{{\rm Ker\,}}
\newcommand{\im}{{\rm Im\, }}
\newcommand{\re}{{\rm Re\, }}
\newcommand{\hm}{{\rm Hom\,}}
\newcommand{\nm}[1]{\mid {#1}\mid}
\newcommand{\sU}{{\{U\}}}
\newcommand{\lng}{\langle}
\newcommand{\rng}{\rangle}

\newcommand{\beq}{\begin{equation}}
\newcommand{\eeq}{\end{equation}}
\newcommand{\ben}{\begin{eqnarray}}
\newcommand{\een}{\end{eqnarray}}
\newcommand{\be}{\begin{eqnarray*}}
\newcommand{\ee}{\end{eqnarray*}}
\newcommand{\bea}{\begin{eqalph}}
\newcommand{\eea}{\end{eqalph}}

\newcommand{\cB}{{\cal B}}
\newcommand{\gU}{{\got U}}
\newcommand{\cO}{{\got O}}
\newcommand{\cA}{{\cal A}}
\newcommand{\gD}{{\got D}}
\newcommand{\gG}{{\got G}}
\newcommand{\cG}{{\got g}}
\newcommand{\gd}{{\got d}}
\newcommand{\gj}{{\got J}}
\newcommand{\gE}{{\got E}}
\newcommand{\gS}{{\got S}}
\newcommand{\gT}{{\got T}}

\newcommand{\cT}{{\cal T}}
\newcommand{\cP}{{\cal P}}
\newcommand{\cR}{{\cal R}}
\newcommand{\cL}{{\cal L}}
\newcommand{\cV}{{\cal V}}
\newcommand{\cU}{{\cal U}}
\newcommand{\cQ}{{\cal Q}}
\newcommand{\cE}{{\cal E}}
\newcommand{\cH}{{\cal H}}
\newcommand{\cF}{{\cal F}}
\newcommand{\cC}{{\cal C}}
\newcommand{\cZ}{{\cal Z}}
\newcommand{\cI}{{\cal I}}

\newcommand{\cK}{{\cal K}}
\newcommand{\cM}{{\cal M}}
\newcommand{\cN}{{\cal N}}
\newcommand{\ccG}{{\cal G}}
\newcommand{\ccO}{{\cal O}}

\newcommand{\cS}{{\cal S}}
\newcommand{\cD}{{\cal D}}
\newcommand{\bpi}{{\bf\Pi}}
\newcommand{\bL}{{\bf L}}
\newcommand{\bb}{{\bf 1}}
\newcommand{\bH}{{\bf H}}
\newcommand{\bA}{{\bf A}}
\newcommand{\br}{{\bf r}}
\newcommand{\bk}{{\bf k}}
\newcommand{\bp}{{\bf p}}
\newcommand{\rL}{{\rm L}}
\newcommand{\rs}{{\rm s}}
\newcommand{\rrq}{{\ol q}}
\newcommand{\rp}{{\ol p}}
\newcommand{\al}{\alpha}
\newcommand{\bt}{\beta}
\newcommand{\dl}{\delta}
\newcommand{\la}{\lambda}
\newcommand{\La}{\Lambda}
\newcommand{\f}{\phi}
\newcommand{\vf}{\varphi}
\newcommand{\F}{\Phi}
\newcommand{\p}{\pi}
\newcommand{\s}{\psi}
\newcommand{\x}{\xi}
\newcommand{\om}{\omega}
\newcommand{\Om}{\Omega}
\newcommand{\m}{\mu}
\newcommand{\n}{\nu}
\newcommand{\g}{\gamma}
\newcommand{\G}{\Gamma}
\newcommand{\e}{\epsilon}
\newcommand{\ve}{\varepsilon}
\newcommand{\th}{\theta}
\newcommand{\Th}{\Theta}
\newcommand{\r}{\rho}
\newcommand{\vr}{\varrho}
\newcommand{\up}{\upsilon}
\newcommand{\vt}{\vartheta}
\newcommand{\kp}{\kappa}

\newcommand{\wL}{\wh\cL}

\newcommand{\si}{\sigma}
\newcommand{\Si}{\Sigma}
\newcommand{\z}{\zeta}

\newcommand{\bla}{{\bf\Lambda}}
\newcommand{\bom}{{\bf\Omega}}
\newcommand{\bth}{{\bf\Theta}}
\newcommand{\bT}{{\bf T}}
\newcommand{\bS}{{\bf S}}
\newcommand{\w}{\wedge}
\newcommand{\wt}{\widetilde}
\newcommand{\wh}{\widehat}
\newcommand{\ol}{\overline}
\newcommand{\ul}{\underline}
\newcommand{\dr}{\partial}

\newcommand{\lto}{{\leftarrow}}
\newcommand{\va}{\longmapsto}
\newcommand{\lla}{\op\longleftarrow}
\newcommand{\ar}{\op\longrightarrow}
\newcommand{\har}{\op\hookrightarrow}
\newcommand{\llra}{\longleftrightarrow}
\newcommand{\dar}{\downarrow}
\newcommand{\uar}{\uparrow}
\newcommand{\ot}{\otimes}
\newcommand{\ap}{\approx}

\newcommand{\bs}{{\bf s}}


\renewcommand{\theequation}{\thesection.\arabic{equation}}

\newcounter{eqalph}
\newcounter{equationa}
\newcounter{example}
\newcounter{remark}
\newcounter{theorem}
\newcounter{proposition}
\newcounter{lemma}
\newcounter{corollary}
\newcounter{definition}

\setcounter{example}{0}
\setcounter{remark}{0}
\setcounter{theorem}{0}
\setcounter{proposition}{0}
\setcounter{lemma}{0}
\setcounter{corollary}{0}
\setcounter{definition}{0}

\def\theremark{\arabic{definition}}
\def\thetheorem{\arabic{definition}}
\def\theproposition{\arabic{definition}}
\def\thelemma{\arabic{definition}}
\def\thecorollary{\arabic{definition}}
\def\thedefinition{\arabic{definition}}

\newenvironment{proof}{\noindent{\it Proof.}}{\hfill $\Box$
\medskip }
\newenvironment{rem}{\refstepcounter{definition} \medskip \noindent{\bf Remark
\thedefinition}.}{ \medskip }
\newenvironment{theo}{\refstepcounter{definition} \medskip\noindent{\bf
Theorem \thedefinition.}\sl}{\medskip }
\newenvironment{prop}{\refstepcounter{definition} \medskip\noindent{\bf
Proposition \thedefinition.}\sl}{\medskip }
\newenvironment{lem}{\refstepcounter{definition} \medskip\noindent{\bf  Lemma
\thedefinition.}\sl }{\medskip }
\newenvironment{cor}{\refstepcounter{definition} \medskip\noindent{\bf 
Corollary \thedefinition.}\sl }{\medskip }
\newenvironment{defi}{\refstepcounter{definition} \medskip\noindent{\bf 
Definition \thedefinition.}\sl }{\medskip }

\def\theequation{\arabic{equation}}
\newenvironment{eqalph}{\stepcounter{equation}
\setcounter{equationa}{\value{equation}}
\setcounter{equation}{0}
\def\theequation{\arabic{equationa}\alph{equation}}
\begin{eqnarray}}{\end{eqnarray}\setcounter{equation}{\value{equationa}}}


\hyphenation{ma-ni-fold La-gran-gi-ans di-men-si-o-nal
-di-men-si-o-nal La-gran-gi-an Ha-mil-to-ni-an}

\begin{document}

\begin{center}

{\large \bf Remark on the Serre--Swan theorem for non-compact manifolds}
\bigskip

{\sc G.Sardanashvily}
\medskip

{\small Department of Theoretical Physics, Moscow State University, 117234
Moscow, Russia

E-mail: sard@grav.phys.msu.su}

\end{center}

\bigskip

{\small
\centerline{\bf Abstract}
\medskip

The Serre--Swan theorem provides the link between projective modules of
finite rank and vector bundles over compact 
manifolds, and plays a prominent role in non-commutative geometry. Its
extension to non-compact manifolds is discussed.
}

\bigskip
\bigskip

The Serre--Swan theorem had been established for continuous functions
and sections \cite{swan}, but then it was extended to the smooth case
(see Proposition 4.1 in Ref. \cite{land} and Proposition 3.1,
Theorem 3.2 in Ref. \cite{var}). 

Recall that smooth manifolds are assumed to be real,
finite-dimensional, Hausdorff,  
second-countable, and connected. Vector bundles are finite-dimensional.

\begin{theo} \label{sp60} (Serre--Swan). 
(i) The $C^\infty(X)$-module of global sections of a smooth
vector bundle
$E\to X$ over a compact manifold $X$ is a projective module of finite rank.

(ii) Any projective $C^\infty(X)$-module of finite rank is isomorphic
to the module of global sections of some smooth vector bundle over $X$.
\end{theo}

In non-commutative geometry, one therefore thinks of a finite
projective $*$-module over a dense unital $*$-subalgebra of a
$C^*$-algebra as being a non-commutative vector bundle.

Note that item (i) of Theorem \ref{sp60} is equivalent to the
following well-known theorem (see Theorem 6.5 in Ref. 
\cite{karo} in the continuous case).

\begin{theo} \label{ss1} 
Let $E$ be a vector bundle over a compact manifold $X$. There exists a
vector bundle $E'\to X$ such that the Whitney sum $E\oplus E'$ is a
trivial vector bundle.
\end{theo}

The key point of the proof of item (i) of Theorem \ref{sp60} is that any
vector bundle 
over a compact manifold $X$ admits a bundle atlas over a finite covering
of $X$. Proposition IX in Ref. \cite{greub} generalizes this assertion
to non-compact manifolds as follows.

\begin{prop} \label{sp2} 
A smooth fibre bundle $Y\to X$ over an arbitrary manifold $X$ 
admits a bundle atlas over a finite
covering of $X$.
\end{prop}

Its proof is based on the fact that, 
for any covering of a manifold $X$, there exists a refinement
$\{U_{ij}\}$, where $j$ and $i$ run through a countable set and a
finite set, respectively, such that 
$U_{ij}\cap U_{ik}=\emptyset$, $j\neq k$. 
Let $\{U_\xi,\psi_\xi\}$ be a bundle atlas of a fibre bundle $Y\to X$ over a
covering $\{U_\xi\}$ of $X$. Let $\{U_{ij}\}$ be the above mentioned
refinement of this covering and 
$\{(U_{ij}, \psi_{ij})\}$ the corresponding bundle atlas of 
$Y\to X$. Then $Y\to X$ has the finite bundle atlas 
\beq
U_i\op=^\df\op\cup_j U_{ij}, \qquad \psi_i(x)\op=^\df\psi_{ij}(x), \qquad x\in
U_{ij}\subset U_i. \label{ss2}
\eeq

It is readily observed that, if $Y\to X$ is a vector bundle,
the atlas (\ref{ss2}) is an atlas of a vector bundle. It follows that every
smooth vector bundle admits a finite atlas, and
the proof in Refs. \cite{land,var} of item (i) of Theorem
\ref{sp60} can be generalized straightforwardly to
non-compact manifolds. Similarly, the above
mentioned Theorem 6.5 in \cite{karo} can be extended to non-compact
finite-dimensional topological manifolds. The proof of item (ii) in Theorem
\ref{sp60} does not imply the compactness of $X$.



\begin{thebibliography}{ederf}


\bibitem{greub} W.Greub, S.Halperin and R.Vanstone, {\it Connections,
Curvature, and Cohomology}, Vol. 1,  (Academic Press, N.Y., 1972).

\bibitem{karo} M.Karoubi, {\it $K$-Theory. An Introduction} (Springer-Verlag,
Berlin, 1978).

\bibitem{land} G.Landi, An introduction to non-commutative spaces and
their geometry, E-print arXiv: hep-th/9701078.

\bibitem{swan} R.Swan, Vector bundles and projective modules, {\it
Trans. Amer. Math. Soc.} {\bf 105} (1962) 264.

\bibitem{var} J.V\'arilly and J.Grasia-Bondia, Connes' noncommutative
differential geometry and the standard model, {\it J. Geom. Phys.} {\bf
12} (1993) 223.

\end{thebibliography}







\end{document}
math-ph0102016.extracted_bib
\begin{thebibliography}{ederf}


\bibitem{greub} W.Greub, S.Halperin and R.Vanstone, {\it Connections,
Curvature, and Cohomology}, Vol. 1,  (Academic Press, N.Y., 1972).

\bibitem{karo} M.Karoubi, {\it $K$-Theory. An Introduction} (Springer-Verlag,
Berlin, 1978).

\bibitem{land} G.Landi, An introduction to non-commutative spaces and
their geometry, E-print arXiv: hep-th/9701078.

\bibitem{swan} R.Swan, Vector bundles and projective modules, {\it
Trans. Amer. Math. Soc.} {\bf 105} (1962) 264.

\bibitem{var} J.V\'arilly and J.Grasia-Bondia, Connes' noncommutative
differential geometry and the standard model, {\it J. Geom. Phys.} {\bf
12} (1993) 223.

\end{thebibliography}