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

 

A formula for the $ *$-core of an ideal


Authors: Louiza Fouli, Janet C. Vassilev and Adela N. Vraciu
Journal: Proc. Amer. Math. Soc. 139 (2011), 4235-4245
MSC (2010): Primary 13A30, 13A35, 13B22
Published electronically: April 27, 2011
MathSciNet review: 2823069
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Abstract: Expanding on the 2010 work of Fouli and Vassilev, we determine a formula for the $ *\textrm{-}\textrm{core}$ of an ideal in two different settings: (1) in a Cohen-Macaulay local ring of characteristic $ p>0$, with perfect residue field and test ideal of depth at least two, where the ideal has a minimal $ *$-reduction that is a parameter ideal, and (2) in a normal local domain of characteristic $ p>0$, with perfect residue field and $ \mathfrak{m}$-primary test ideal, where the ideal is a sufficiently high Frobenius power of an ideal. We also exhibit some examples where our formula fails if our hypotheses are not met.


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Additional Information

Louiza Fouli
Affiliation: Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003
Email: lfouli@math.nmsu.edu

Janet C. Vassilev
Affiliation: Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico 87131
Email: jvassil@math.unm.edu

Adela N. Vraciu
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
Email: vraciu@math.sc.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10858-X
PII: S 0002-9939(2011)10858-X
Keywords: Tight closure, reduction, core, *-independent, spread
Received by editor(s): October 23, 2009
Received by editor(s) in revised form: October 22, 2010
Published electronically: April 27, 2011
Additional Notes: The second author was partly supported by the NSA grant H98230-09-1-0057
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2011 American Mathematical Society