Consecutive cancellations in Betti numbers of local rings
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- by Maria Evelina Rossi and Leila Sharifan PDF
- Proc. Amer. Math. Soc. 138 (2010), 61-73 Request permission
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.References
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Additional Information
- Maria Evelina Rossi
- Affiliation: Department of Mathematics, University of Genoa, Via Dodecaneso 35, 16146 Genoa, Italy
- MR Author ID: 150830
- ORCID: 0000-0001-7039-5296
- 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
- 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
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 61-73
- MSC (2000): Primary 13D02
- DOI: https://doi.org/10.1090/S0002-9939-09-10010-2
- MathSciNet review: 2550170