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arXiv · 0904.1086

Consecutive cancellations in Betti numbers of local rings

Maria Evelina Rossi·Leila Sharifan
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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=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,m) 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, several applications are presented. 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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Maria Evelina Rossi, Leila Sharifan. 2009-04-07. Consecutive cancellations in Betti numbers of local rings. https://arxiv.org/abs/0904.1086

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