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Rigidity of Linear Strands and Generic Initial Ideals

Published online by Cambridge University Press:  11 January 2016

Satoshi Murai  and
Pooja Singla
Satoshi Murai
Affiliation:
Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka, 560-0043, Japan, s-murai@ist.osaka-u.ac.jp
Pooja Singla
Affiliation:
FB Mathematik, Universität Duisburg-Essen, Campus Essen, 45117 Essen, Germany, pooja.singla@uni-duisburg-essen.de
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Abstract

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Let K be a field, S a polynomial ring and E an exterior algebra over K, both in a finite set of variables. We study rigidity properties of the graded Betti numbers of graded ideals in S and E when passing to their generic initial ideals. First, we prove that if the graded Betti numbers for some i > 1 and k ≥ 0, then for all q ≥ i, where I ⊂ S is a graded ideal. Second, we show that if for some i > 1 and k ≥ 0, then for all q ≥ 1, where I ⊂ E is a graded ideal. In addition, it will be shown that the graded Betti numbers for all i ≥ 1 if and only if I(k) and I(k+1) have a linear resolution. Here I(d) is the ideal generated by all homogeneous elements in I of degree d, and R can be either the polynomial ring or the exterior algebra.

Keywords

13D02
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 190 , 2008 , pp. 35 - 61
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2008

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