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 of I. Irena 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, n). To this purpose we have to consider more general kinds of cancellations. The 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.
Consecutive cancellations in Betti numbers of local rings
ROSSI, MARIA EVELINA;
2010-01-01
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 of I. Irena 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, n). To this purpose we have to consider more general kinds of cancellations. The 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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