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Hilbert-Samuel functions of modules over Cohen-Macaulay rings

Authors: Srikanth Iyengar and Tony J. Puthenpurakal
Journal: Proc. Amer. Math. Soc. 135 (2007), 637-648
MSC (2000): Primary 13D40; Secondary 13D02, 13D07
Published electronically: August 28, 2006
MathSciNet review: 2262858
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Abstract | References | Similar Articles | Additional Information

Abstract: For a finitely generated, non-free module $ M$ over a CM local ring $ (R,\mathfrak{m},k)$, it is proved that for $ n\gg 0$ the length of $ \operatorname{Tor}_1^R(M,R/\mathfrak{m}^{n+1})$ is given by a polynomial of degree $ \dim R-1$. The vanishing of $ \operatorname{Tor}_ i^R(M,N/\mathfrak{m}^{n+1}N)$ is studied, with a view towards answering the question: If there exists a finitely generated $ R$-module $ N$ with $ \dim N\ge 1$ such that the projective dimension or the injective dimension of $ N/\mathfrak{m}^{n+1}N$ is finite, then is $ R$ regular? Upper bounds are provided for $ n$ beyond which the question has an affirmative answer.

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Additional Information

Srikanth Iyengar
Affiliation: Department of Mathematics, University of Nebraska, Lincoln, Nebraska 68588

Tony J. Puthenpurakal
Affiliation: Department of Mathematics, IIT Bombay, Powai, Mumbai 400 076, India

Keywords: Hilbert-Samuel functions, growth and vanishing of derived functors
Received by editor(s): November 18, 2004
Received by editor(s) in revised form: September 22, 2005
Published electronically: August 28, 2006
Additional Notes: The first author was partly supported by NSF grant DMS 0442242
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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