Serre–Swan theorem

<h2 id="differential-geometry">Differential geometry</h2>
Suppose M is a <a href="/facts/Smooth_manifold/aSnbXKcV">smooth manifold</a> (not necessarily compact), and E is a <a href="/facts/Smooth_vector_bundle/cXACAa1I">smooth vector bundle</a> over M. Then Γ(E), the space of <a href="/facts/Section_(fiber_bundle)/DLnpXS5a">smooth sections</a> of E, is a <a href="/facts/Module_(mathematics)/jP0LIhFL">module</a> over C∞(M) (the commutative algebra of smooth real-valued functions on M). Swan's theorem states that this module is <a href="/facts/Finitely-generated_module/6Njcpgzr">finitely generated</a> and <a href="/facts/Projective_module/lHymqhNC">projective</a> over C∞(M). In other words, every vector bundle is a direct summand of some trivial bundle: 
 
 
 
 M
 ×
 
 
 R
 
 
 k
 
 
 
 
 {\displaystyle M\times \mathbb {R} ^{k}}
 
 for some k. The theorem can be proved by constructing a bundle epimorphism from a trivial bundle 
 
 
 
 M
 ×
 
 
 R
 
 
 k
 
 
 →
 E
 .
 
 
 {\displaystyle M\times \mathbb {R} ^{k}\to E.}
 
 This can be done by, for instance, exhibiting sections s1...sk with the property that for each point p, {si(p)} span the fiber over p. 
When M is <a href="/facts/Connected_space/5CUTxfoA">connected</a>, the converse is also true: every <a href="/facts/Finitely_generated_projective_module/lHymqhNC">finitely generated projective module</a> over C∞(M) arises in this way from some smooth vector bundle on M. Such a module can be viewed as a smooth function f on M with values in the n × n idempotent matrices for some n. The fiber of the corresponding vector bundle over x is then the range of f(x). If M is not connected, the converse does not hold unless one allows for vector bundles of non-constant rank (which means admitting manifolds of non-constant dimension). For example, if M is a zero-dimensional 2-point manifold, the module 
 
 
 
 
 R
 
 ⊕
 0
 
 
 {\displaystyle \mathbb {R} \oplus 0}
 
 is finitely-generated and projective over 
 
 
 
 
 C
 
 ∞
 
 
 (
 M
 )
 ≅
 
 R
 
 ×
 
 R
 
 
 
 {\displaystyle C^{\infty }(M)\cong \mathbb {R} \times \mathbb {R} }
 
 but is not <a href="/facts/Free_module/o6rC6yoJ">free</a>, and so cannot correspond to the sections of any (constant-rank) vector bundle over M (all of which are trivial).
Another way of stating the above is that for any connected smooth manifold M, the section <a href="/facts/Functor/l2u3rIBE">functor</a> Γ from the <a href="/facts/Category_theory/6zS3DXPa">category</a> of smooth vector bundles over M to the category of finitely generated, projective C∞(M)-modules is <a href="/facts/Full_functor/bMM75v7e">full</a>, <a href="/facts/Faithful_functor/bMM75v7e">faithful</a>, and <a href="/facts/Essentially_surjective_functor/DgtGpTCe">essentially surjective</a>. Therefore the category of smooth vector bundles on M is <a href="/facts/Equivalence_of_categories/k6uzqAcy">equivalent</a> to the category of finitely generated, projective C∞(M)-modules. Details may be found in (Nestruev 2003).

<h2 id="topology">Topology</h2>
Suppose X is a compact <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff space</a>, and C(X) is the ring of <a href="/facts/Continuous_function_(topology)/sKbl02pB">continuous</a> real-valued functions on X. Analogous to the result above, the category of real vector bundles on X is equivalent to the category of finitely generated projective modules over C(X). The same result holds if one replaces "real-valued" by "complex-valued" and "real vector bundle" by "complex vector bundle", but it does not hold if one replace the field by a <a href="/facts/Totally_disconnected/lTu5Rvyw">totally disconnected</a> field like the <a href="/facts/Rational_number/VJQmyY05">rational numbers</a>.
In detail, let Vec(X) be the <a href="/facts/Category_theory/6zS3DXPa">category</a> of <a href="/facts/Complex_vector_bundle/0nE8TEIn">complex vector bundles</a> over X, and let ProjMod(C(X)) be the category of <a href="/facts/Finitely_generated_module/6Njcpgzr">finitely generated</a> projective modules over the <a href="/facts/Cstar_algebra/B3tnGHIo">C*-algebra</a> C(X). There is a <a href="/facts/Functor/l2u3rIBE">functor</a> Γ : Vec(X) → ProjMod(C(X)) which sends each complex vector bundle E over X to the C(X)-module Γ(X, E) of <a href="/facts/Section_(fiber_bundle)/DLnpXS5a">sections</a>. If 
 
 
 
 τ
 :
 (
 
 E
 
 1
 
 
 ,
 
 π
 
 1
 
 
 )
 →
 (
 
 E
 
 2
 
 
 ,
 
 π
 
 2
 
 
 )
 
 
 {\displaystyle \tau :(E_{1},\pi _{1})\to (E_{2},\pi _{2})}
 
 is a <a href="/facts/Vector_bundle/cXACAa1I">morphism of vector bundles over X</a> then 
 
 
 
 
 π
 
 2
 
 
 ∘
 τ
 =
 
 π
 
 1
 
 
 
 
 {\displaystyle \pi _{2}\circ \tau =\pi _{1}}
 
 and it follows that

∀
        s
        ∈
        Γ
        (
        X
        ,
        
          E
          
            1
          
        
        )
        
        
          π
          
            2
          
        
        ∘
        τ
        ∘
        s
        =
        
          π
          
            1
          
        
        ∘
        s
        =
        
          
            id
          
          
            X
          
        
        ,
      
    
    {\displaystyle \forall s\in \Gamma (X,E_{1})\quad \pi _{2}\circ \tau \circ s=\pi _{1}\circ s={\text{id}}_{X},}

giving the map

{
            
              
                
                  Γ
                  τ
                  :
                  Γ
                  (
                  X
                  ,
                  
                    E
                    
                      1
                    
                  
                  )
                  →
                  Γ
                  (
                  X
                  ,
                  
                    E
                    
                      2
                    
                  
                  )
                
              
              
                
                  s
                  ↦
                  τ
                  ∘
                  s
                
              
            
            
          
        
      
    
    {\displaystyle {\begin{cases}\Gamma \tau :\Gamma (X,E_{1})\to \Gamma (X,E_{2})\\s\mapsto \tau \circ s\end{cases}}}

which respects the module structure (Várilly, 97). Swan's theorem asserts that the functor Γ is an <a href="/facts/Equivalence_of_categories/k6uzqAcy">equivalence of categories</a>.

<h2 id="algebraic-geometry">Algebraic geometry</h2>
The analogous result in <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, due to Serre (1955, §50) applies to vector bundles in the category of <a href="/facts/Affine_variety/WVynz2Kn">affine varieties</a>. Let X be an affine variety with structure sheaf 
 
 
 
 
 
 
 O
 
 
 
 X
 
 
 ,
 
 
 {\displaystyle {\mathcal {O}}_{X},}
 
 and 
 
 
 
 
 
 F
 
 
 
 
 {\displaystyle {\mathcal {F}}}
 
 a <a href="/facts/Coherent_sheaf/h2NoUN7e">coherent sheaf</a> of 
 
 
 
 
 
 
 O
 
 
 
 X
 
 
 
 
 {\displaystyle {\mathcal {O}}_{X}}
 
 -modules on X. Then 
 
 
 
 
 
 F
 
 
 
 
 {\displaystyle {\mathcal {F}}}
 
 is the sheaf of germs of a finite-dimensional vector bundle if and only if 
 
 
 
 Γ
 (
 
 
 F
 
 
 ,
 X
 )
 ,
 
 
 {\displaystyle \Gamma ({\mathcal {F}},X),}
 
 the space of sections of 
 
 
 
 
 
 F
 
 
 ,
 
 
 {\displaystyle {\mathcal {F}},}
 
 is a projective module over the commutative ring 
 
 
 
 A
 =
 Γ
 (
 
 
 
 O
 
 
 
 X
 
 
 ,
 X
 )
 .
 
 
 {\displaystyle A=\Gamma ({\mathcal {O}}_{X},X).}

<ul><li><a href="/facts/Max_Karoubi/emR1J5TL">Karoubi, Max</a> (1978), K-theory: An introduction, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-08090-1</li>
<li>Manoharan, Palanivel (1995), "Generalized Swan's theorem and its application", <a href="/facts/Proceedings_of_the_American_Mathematical_Society/W3wRF3MV">Proceedings of the American Mathematical Society</a>, 123 (10): 3219–3223, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1090%2FS0002-9939-1995-1264823-X">10.1090/S0002-9939-1995-1264823-X</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2160685">2160685</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1264823">1264823</a>.</li>
<li><a href="/facts/Jean-Pierre_Serre/1xSr7BCh">Serre, Jean-Pierre</a> (1955), "Faisceaux algébriques cohérents", <a href="/facts/Annals_of_Mathematics/CUYktUqD">Annals of Mathematics</a>, 61 (2): 197–278, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1969915">10.2307/1969915</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1969915">1969915</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0068874">0068874</a>.</li>
<li><a href="/facts/Richard_Swan/TKcBLTSN">Swan, Richard G.</a> (1962), "Vector Bundles and Projective Modules", <a href="/facts/Transactions_of_the_American_Mathematical_Society/JmKtyCwK">Transactions of the American Mathematical Society</a>, 105 (2): 264–277, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1993627">10.2307/1993627</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1993627">1993627</a>.</li>
<li>Nestruev, Jet (2003), Smooth manifolds and observables, Graduate texts in mathematics, vol. 220, Springer-Verlag, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-95543-7</li>
<li>Giachetta, G.; Mangiarotti, L.; <a href="/facts/Gennadi_Sardanashvily/EDrGkc9r">Sardanashvily, Gennadi</a> (2005), Geometric and Algebraic Topological Methods in Quantum Mechanics, World Scientific, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 981-256-129-3.</li></ul>
This article incorporates material from Serre-Swan theorem on <a href="/facts/PlanetMath/wphPCtLf">PlanetMath</a>, which is licensed under the Creative Commons Attribution/Share-Alike License.

Serre–Swan theorem open-in-new

Serre–Swan theorem