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

 

Detecting completeness from Ext-vanishing


Authors: Anders J. Frankild and Sean Sather-Wagstaff
Journal: Proc. Amer. Math. Soc. 136 (2008), 2303-2312
MSC (2000): Primary 13B35, 13D07, 13D25, 13D45, 13J10
Published electronically: February 28, 2008
MathSciNet review: 2390496
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Abstract | References | Similar Articles | Additional Information

Abstract: Motivated by work of C. U. Jensen, R.-O. Buchweitz, and H. Flenner, we prove the following result. Let $ R$ be a commutative noetherian ring and $ \mathfrak{a}$ an ideal in the Jacobson radical of $ R$. Let $ \widehat{\mathfrak{Ra}}$ be the $ \mathfrak{a}$-adic completion of $ R$. If $ M$ is a finitely generated $ R$-module such that $ \operatorname{Ext}^i_R(\,M)=0$ for all $ i\neq 0$, then $ M$ is $ \mathfrak{a}$-adically complete.


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Additional Information

Anders J. Frankild
Affiliation: Department of Mathematics, Institute for Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 København, Denmark

Sean Sather-Wagstaff
Affiliation: Department of Mathematics, California State University, Dominguez Hills, 1000 E. Victoria Street, Carson, California 90747
Address at time of publication: Department of Mathematics, North Dakota State University, 300 Minard Hall, Fargo, North Dakota 58105-5075
Email: sean.sather-wagstaff@ndsu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09199-5
PII: S 0002-9939(08)09199-5
Keywords: Completions, completeness, ext, local cohomology, local homology
Received by editor(s): June 28, 2006
Received by editor(s) in revised form: March 28, 2007
Published electronically: February 28, 2008
Additional Notes: This research was conducted while the first author had a Steno Stipend from the Danish Research Council.
Dedicated: Dedicated to Lex Remington
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.