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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Generalized Swan's theorem and its application


Author: P. Manoharan
Journal: Proc. Amer. Math. Soc. 123 (1995), 3219-3223
MSC: Primary 58D15; Secondary 13C10, 55R10
MathSciNet review: 1264823
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Abstract: Swan's theorem verifies the equivalence between finitely generated projective modules over function algebras and smooth vector bundles. We define $ {A^{(r)}}$-maps that correspond to usual non-linear differential operators of degree r under the equivalence of Swan's theorem and thus generalize Swan's theorem to include non-linear differential operators as morphisms. An $ {A^{(r)}}$-manifold structure is introduced on the space of sections of a fiber bundle through charts with $ {A^{(r)}}$-maps as transition homeomorphisms. A characterization for all the smooth maps between the spaces of sections of vector bundles, whose kth derivatives are linear differential operators of degree r in each variable, is given in terms of $ {A^{(r)}}$-maps.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1264823-X
PII: S 0002-9939(1995)1264823-X
Keywords: Non-linear differential operators, $ {A^{(r)}}$-map, $ {A^{(r)}}$-manifold
Article copyright: © Copyright 1995 American Mathematical Society