Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. I. M. Aberbach and F. Enescu, The structure of $F$-pure rings, preprint, 2003.
  • 2. M. Hochster, Cyclic purity versus purity in excellent Noetherian rings, Trans. Amer. Math. Soc. 231 (1977), no. 2, 463-488. MR 57:3111
  • 3. M. Hochster and C. Huneke, Infinite integral extensions and big Cohen-Macaulay algebras, Ann. of Math. (2) 135 (1992), no. 1, 53-89. MR 92m:13023
  • 4. M. Hochster and C. Huneke, $F$-regularity, test elements, and smooth base change, Trans. Amer. Math. Soc. 346 (1994), no. 1, 1-62. MR 95d:13007
  • 5. Craig Huneke, Tight closure and its applications, With an appendix by Melvin Hochster. CBMS Regional Conference Series in Mathematics, 88. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1996. x+137 pp. ISBN: 0-8218-0412-X. MR 96m:13001
  • 6. E. Kunz, Characterizations of regular local rings for characteristic $p$, Amer. J. Math. 91 (1969), 772-784. MR 40:5609
  • 7. E. Kunz, On Noetherian rings of characteristic $p$., Amer. J. Math. 98 (1976), no. 4, 999-1013. MR 55:5612
  • 8. H. Schoutens, On the vanishing of Tor for the absolute integral closure, J. Algebra 275 (2004), 567-574.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 13A35, 13H05

Retrieve articles in all journals with MSC (2000): 13A35, 13H05

Additional Information

Ian M. Aberbach
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211

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

American Mathematical Society