A connectedness result in positive characteristic
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- by Anurag K. Singh and Uli Walther PDF
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Abstract:
Let $(R,\mathfrak {m})$ be a complete local ring of dimension at least two, which contains a separably closed coefficient field of positive characteristic. Using a vanishing theorem of Peskine-Szpiro, Lyubeznik proved that the local cohomology module $H^1_{\mathfrak {m}}(R)$ is Frobenius-torsion if and only if the punctured spectrum of $R$ is connected in the Zariski topology. We give a simple proof of this theorem and, more generally, a formula for the number of connected components in terms of the Frobenius action on $H^1_{\mathfrak {m}}(R)$.References
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Additional Information
- Anurag K. Singh
- Affiliation: Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, Utah 84112
- Email: singh@math.utah.edu
- Uli Walther
- Affiliation: Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, Indiana 47907
- Email: walther@math.purdue.edu
- Received by editor(s): March 26, 2006
- Published electronically: January 8, 2008
- Additional Notes: The first author was supported by NSF grants DMS 0300600 and DMS 0600819
The second author was supported by NSF grants DMS 0100509 and DMS 0555319, and by NSA grant H98230-06-1-0012. We are grateful to Gennady Lyubeznik for useful discussions and comments. - © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 360 (2008), 3107-3119
- MSC (2000): Primary 13D45; Secondary 13A35
- DOI: https://doi.org/10.1090/S0002-9947-08-04427-9
- MathSciNet review: 2379789
Dedicated: Dedicated to Professor Paul Roberts on the occasion of his sixtieth birthday