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A non-linear version of Swan’s theorem. (English) Zbl 0760.58001
Swan’s theorem [R. G. Swan, Trans. Am. Math. Soc. 105, 264-277 (1962; Zbl 0109.416)]asserts that there is an equivalence between the category of smooth vector bundles over a compact connected smooth finite dimensional manifold $$K$$ with the bundle morphism and the category of finitely generated projective modules over the algebra $$\mathcal A$$ ($$=C^ \infty(K)$$) of smooth functions on $$K$$ with morphisms $$\mathcal A$$-linear maps.
The author defines $$\mathcal A$$-maps between finitely generated projective $$\mathcal A$$-modules to be smooth maps whose derivatives are $$\mathcal A$$-linear and shows the equivalence between the category of smooth maps over $$K$$ with fibre preserving smooth maps and the category of finitely generated projective $$\mathcal A$$-modules with $$\mathcal A$$-maps. The author then defines the notion of $$\mathcal A$$-manifolds which represent the nonlinear correspondents of the finitely generated projective $$\mathcal A$$-modules and establishes the equivalence between the category of deformations over $$K$$ and a full subcategory of $$\mathcal A$$-manifolds with $$\mathcal A$$-maps.

##### MSC:
 58A05 Differentiable manifolds, foundations 57R19 Algebraic topology on manifolds and differential topology 57R22 Topology of vector bundles and fiber bundles 58D29 Moduli problems for topological structures
##### Keywords:
Swan’s theorem; projective modules; bundle morphism
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##### References:
 [1] [H] Hamilton, R.S.: The inverse function theorem of Nash and Moser. Bull. Am. Math. Soc.7, 65–222 (1982) · Zbl 0499.58003 [2] [S] Swan, R.G.: Vector bundles and projective modules. Trans. Am. Math. Soc.105, 264–277 (1962) · Zbl 0109.41601
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