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Consecutive cancellations in Betti numbers of local rings

Authors: Maria Evelina Rossi and Leila Sharifan
Journal: Proc. Amer. Math. Soc. 138 (2010), 61-73
MSC (2000): Primary 13D02
Published electronically: August 28, 2009
MathSciNet review: 2550170
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Abstract: Let $ I$ be a homogeneous ideal in a polynomial ring $ P$ over a field. By Macaulay's Theorem there exists a lexicographic ideal $ L=\operatorname{Lex}(I)$ with the same Hilbert function as $ I. $ Peeva has proved that the Betti numbers of $ P/I $ can be obtained from the graded Betti numbers of $ P/L$ by a suitable sequence of consecutive cancellations. We extend this result to any ideal $ I$ in a regular local ring $ (R,\mathfrak{n}) $ by passing through the associated graded ring. To this purpose it will be necessary to enlarge the list of the allowed cancellations. Taking advantage of Eliahou-Kervaire's construction, we present several applications. This connection between the graded perspective and the local one is a new viewpoint, and we hope it will be useful for studying the numerical invariants of classes of local rings.

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Additional Information

Maria Evelina Rossi
Affiliation: Department of Mathematics, University of Genoa, Via Dodecaneso 35, 16146 Genoa, Italy

Leila Sharifan
Affiliation: Faculty of Mathematics and Computer Science, Amirkabir University of Technology, 424, Hafez Avenue, 15914 Tehran, Iran

Keywords: Minimal free resolution, filtered module, associated graded module, Betti numbers, lexicographic ideal, standard bases.
Received by editor(s): February 11, 2009
Received by editor(s) in revised form: April 17, 2009
Published electronically: August 28, 2009
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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