### Abstract

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 β _{ii+k}^{S}(S/I) = β_{ii+k}^{S}(S/Gin(I)) for some i > 1 and k ≥ 0, then β_{qq+k}^{S}(S/I) = β_{qq+k}^{S}(S/Gin(I)) for all q ≥ i, where I ⊂ S is a graded ideal. Second, we show that if β_{ii+k}^{E}(E/I) = β_{ii+k}^{E}(E/Gin(I)) for some i > 1 and k ≥ 0, then β_{qq+k}^{E}(E/I) = β_{qq+k} ^{E}(E/Gin(I)) for all q ≥ 1, where I ⊂ E is a graded ideal. In addition, it will be shown that the graded Betti numbers β _{ii+k}^{R}(R/I) = β_{ii+k}^{R}(R/Gin(I)) 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.

Original language | English |
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Pages (from-to) | 35-61 |

Number of pages | 27 |

Journal | Nagoya Mathematical Journal |

Volume | 190 |

DOIs | |

Publication status | Published - 2008 Jan 1 |

Externally published | Yes |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Nagoya Mathematical Journal*,

*190*, 35-61. https://doi.org/10.1017/S0027763000009557