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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Flat analytic extensions


Author: Ana M. D. Viola-Prioli
Journal: Trans. Amer. Math. Soc. 202 (1975), 385-404
MSC: Primary 13J05
MathSciNet review: 0389891
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Abstract: This paper is concerned, in the first place, with the conditions to be imposed on an ideal $ I$ of the power series ring in one indeterminate $ A[[x]]$ ($ A$ noetherian) in order that the analytic extension $ B = A[[x]]/I$ be a flat $ A$-module.

Also the relationship between the projectivity and finiteness of $ B$ is found when the content of $ I$ (the ideal of $ A$ generated by the coefficients of all power series in $ I$) equals $ A$. A generalization of this result to the power series ring in any finite number of indeterminates is obtained when $ A$ is local, noetherian of Krull $ \dim \geq 1$, and under certain restrictions on $ I$, for the global case but only for domains.

Finally, a contribution to the problem of the finiteness of $ I$ when $ A[[x]]/I$ is a flat analytic extension is given for $ A$ a local ring, not necessarily noetherian.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0389891-0
Keywords: Power series, flat module, projective module, content, invertible ideal, pure submodule, Krull dimension, coherency
Article copyright: © Copyright 1975 American Mathematical Society