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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A connectedness result in positive characteristic


Authors: Anurag K. Singh and Uli Walther
Journal: Trans. Amer. Math. Soc. 360 (2008), 3107-3119
MSC (2000): Primary 13D45; Secondary 13A35.
Published electronically: January 8, 2008
MathSciNet review: 2379789
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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)$.


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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

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04427-9
PII: S 0002-9947(08)04427-9
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.
Dedicated: Dedicated to Professor Paul Roberts on the occasion of his sixtieth birthday
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.