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Descent of projectivity for locally free modules


Author: Roger Wiegand
Journal: Proc. Amer. Math. Soc. 41 (1973), 342-348
MSC: Primary 13C10
DOI: https://doi.org/10.1090/S0002-9939-1973-0327737-0
MathSciNet review: 0327737
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Abstract: Let $ R \to \hat R$ be the natural homomorphism from the commutative ring $ R$ into its associated von Neumann regular ring, and let $ M$ be a locally free $ R$-module such that $ \hat R \otimes M$ is a projective $ \hat R$-module. We show that if $ M$ is either countably generated or locally finitely generated, then $ M$ is projective, and we deduce that the trace of any projective ideal is projective. These results are a consequence of a more general theorem on the descent of the Mittag-Leffler condition. The ``locally free'' hypothesis may be weakened to ``flat'' if and only if $ R$ is locally perfect.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0327737-0
Keywords: Projective module, locally free module, Mittag-Leffler module, von Neumann regular ring, locally perfect ring
Article copyright: © Copyright 1973 American Mathematical Society

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