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 | References | Similar Articles | Additional Information

Abstract: Let be a homogeneous ideal in a polynomial ring over a field. By Macaulay's Theorem there exists a lexicographic ideal with the same Hilbert function as Peeva has proved that the Betti numbers of can be obtained from the graded Betti numbers of by a suitable sequence of consecutive cancellations. We extend this result to any ideal in a regular local ring 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

Email:
rossim@dima.unige.it

**Leila Sharifan**

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

Email:
leila-sharifan@aut.ac.ir

DOI:
https://doi.org/10.1090/S0002-9939-09-10010-2

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.