Consecutive cancellations in Betti numbers of local rings

Rossi, Maria Evelina; Sharifan, Leila

doi:10.1090/S0002-9939-09-10010-2

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.

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