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

 

The vanishing of $\operatorname{Tor}_1^R(R^+,k)$ implies that $R$ is regular


Author: Ian M. Aberbach
Journal: Proc. Amer. Math. Soc. 133 (2005), 27-29
MSC (2000): Primary 13A35; Secondary 13H05
Published electronically: June 23, 2004
MathSciNet review: 2085149
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Abstract: Let $(R,m,k)$ be an excellent local ring of positive prime characteristic. We show that if $\operatorname{Tor}_1^R(R^+,k) = 0$, then $R$ is regular. This improves a result of Schoutens, in which the additional hypothesis that $R$ was an isolated singularity was required for the proof.


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Additional Information

Ian M. Aberbach
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: aberbach@math.missouri.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07491-X
PII: S 0002-9939(04)07491-X
Keywords: Regular rings, characteristic $p$, absolute integral closure
Received by editor(s): July 15, 2003
Received by editor(s) in revised form: September 18, 2003
Published electronically: June 23, 2004
Additional Notes: The author was partially supported by the National Security Agency. He also wishes to thank the referee for a careful reading of this paper and several corrections.
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2004 American Mathematical Society