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On the degree of Hilbert polynomials associated to the torsion functor


Authors: Daniel Katz and Emanoil Theodorescu
Journal: Proc. Amer. Math. Soc. 135 (2007), 3073-3082
MSC (2000): Primary 13A30, 13D02, 13D07, 13D40
DOI: https://doi.org/10.1090/S0002-9939-07-08879-X
Published electronically: May 14, 2007
MathSciNet review: 2322736
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Abstract: Let $ R$ be a local, Noetherian ring and $ I\subseteq R$ an ideal. A question of Kodiyalam asks whether for fixed $ i > 0$, the polynomial giving the $ i$th Betti number of $ I^n$ has degree equal to the analytic spread of $ I$ minus one. Under mild conditions on $ R$, we show that the answer is positive in a number of cases, including when $ I$ is divisible by $ \mathfrak{m}$ or $ I$ is an integrally closed $ \mathfrak{m}$-primary ideal.


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Additional Information

Daniel Katz
Affiliation: Department of Mathematics, University of Kansas, Lawrence, Kansas 66045
Email: dlk@math.ku.edu

Emanoil Theodorescu
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri, 65211
Address at time of publication: University of Iowa, Department of Actuarial Science and Statistics, 241 Schaeffer Hall, Iowa City, IA 52242
Email: theodore@math.missouri.edu, emanoil.theodorescu@g.mail.com

DOI: https://doi.org/10.1090/S0002-9939-07-08879-X
Keywords: Hilbert-Samuel polynomial, torsion functor, quasi-unmixed local ring
Received by editor(s): December 14, 2005
Received by editor(s) in revised form: June 28, 2006
Published electronically: May 14, 2007
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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