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\begin{document}

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\section*{Serre-Swan theorem}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{statement}{Statement}\dotfill \pageref*{statement} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{to_ktheory}{To K-theory}\dotfill \pageref*{to_ktheory} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \emph{Serre-Swan theorem} identifies suitable [[modules]] over an [[algebra of functions]] on some [[space]] with the modules of [[sections]] of [[vector bundles]] over that space and thereby identifies these modules with vector bundles themselves.

Together with theorems like \emph{[[Gelfand duality]]}, the Serre-Swan theorem is a central part of the [[Isbell duality|general duality]] between [[geometry]] and [[algebra]]. In particular it may serve to generalize the notion of vector bundle from standard [[geometry]] to more exotic forms of geometry, such as [[noncommutative geometry]].

\hypertarget{statement}{}\subsection*{{Statement}}\label{statement}

There are two different original theorems of the same intuitive spirit which are usually jointly called the \textbf{Serre-Swan theorem}, the first one is in [[algebraic geometry]], the second in [[topology]]:

1) \textbf{Serre's theorem} (\hyperlink{Serre55}{Serre 55}): let $R$ be a commmutative unital [[Noetherian ring]] (in particular, the [[structure sheaf|coordinate ring]] of an [[affine variety]] over a [[field]]), then the [[category]] of finitely-generated \emph{[[projective module|projective]] $R$-[[modules]] is [[equivalence of categories|equivalent]] to the category of [[algebraic vector bundles]] (= [[covering|locally]] [[free module|free]] [[sheaves]] of [[structure sheaf]]-[[modules]] of constant finite [[rank]]) on $Spec R$.}

2) \textbf{Swan's theorem} (\hyperlink{Swan}{Swan 62}): Given a [[Hausdorff space|Hausdorff]] [[compact space]] $X$, the [[category]] of [[finitely generated module|finitely generated]] [[projective modules]] over the [[continuous function|continuous]]-[[function algebra]] $C(X)$ is [[equivalence of categories|equivalent]] to the category of finite-[[rank]] [[vector bundles]] on $X$, where the equivalence is established by sending a vector bundle to the its module of continuous [[sections]].

But there are also various variations of these theorems, for instance to [[differential geometry]]:

3) \textbf{[[smooth Serre-Swan theorem]]} (\hyperlink{Nestruev03}{Nestruev 03, 11.33}) For $X$ a [[smooth manifold]] with $\mathbb{R}$-algebra of [[smooth functions]] $C^\infty(X)$ there is an [[equivalence of categories]] between that of finite [[rank]] smooth [[vector bundles]] over $X$ and [[finitely generated objects|finitely generated]] [[projective modules]] over $C^\infty(X)$.

A general statement of the Serre-Swan theorems over [[ringed spaces]] is in (\hyperlink{Morye}{Morye}).

If one drops the condition that the [[sheaf of modules]] over the [[structure sheaf]] of a [[ringed space]] is [[covering|locally]] [[free module|free]], and allows it instad to be just \emph{locally [[presentable module|presentable]]}, then one arrives at the notion of [[quasicoherent sheaf of modules]]. Here the Serre-Swan theorem serves to clarify in which sense precisely these are generalizations of [[vector bundles]].

The condition that the modules be [[projective module|projective]] can also naturally be relaxed. In [[higher geometry]] the Serre-Swan theorem becomes not only more general but also conceptually simpler: if instead of [[modules]] one considers [[chain complexes]] of modules ([[(∞,1)-modules]]) then under mild assumptions (see at \emph{[[projective resolution]]}) every chain complex of modules is [[equivalence|equivalent]] ([[quasi-isomorphism|quasi-isomorphic]]) to a chain complex of [[projective modules]], and hence this condition in the statement of the traditional Serre-Swan theorem becomes automatic. Or in other words, the non-projective modules also do correspond to [[vector bundles]], but to [[chain complexes]] of vector bundles (only that the [[chain homology]] of the complex is not itself a vector bundle again in this case). See at \emph{[[(∞,1)-vector bundle]]} for more on this.

\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

\hypertarget{to_ktheory}{}\subsubsection*{{To K-theory}}\label{to_ktheory}

The Serre-Swan theorem serves to relate [[topological K-theory]] with [[algebraic K-theory]]. (\ldots{})

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Gelfand duality]]


\item [[operator K-theory]]


\item [[Quillen-Suslin theorem]]



\end{itemize}
[[!include Isbell duality - table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

The two original articles are

\begin{itemize}%
\item [[Jean-Pierre Serre]], \emph{Faisceaux algebriques coherents}, Annals of Mathematics 61 (2): 197--278 (1955)


\item [[Richard Swan]], \emph{Vector bundles and projective modules}, Trans. AMS \textbf{105} (2): 264--277 (1962)



\end{itemize}
A textbook account in the context of [[differential geometry]] is in

\begin{itemize}%
\item [[Jet Nestruev]], \emph{Smooth manifolds and observables}, Graduate texts in mathematics, 220, Springer-Verlag, ISBN 0-387-95543-7 (2003)

\end{itemize}
A general account of Serre-Swan-type theorems over [[ringed spaces]] is in

\begin{itemize}%
\item [[Archana Morye]], \emph{Note on the Serre-Swan Theorem} (\href{http://arxiv.org/abs/0905.0319}{arXiv:0905.0319})

\end{itemize}
A textbook account on the use of the theorem in [[K-theory]] is for instance

\begin{itemize}%
\item [[Max Karoubi]], \emph{$K$-theory. An introduction}, Grundlehren der Mathematischen Wissenschaften, Band \textbf{226}, Springer 1978. xviii+308 pp.

\end{itemize}
[[!redirects Serre?Swan theorem]] [[!redirects Serre--Swan theorem]]



\end{document}