### 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 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Nagoya Mathematical Journal*,

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

**Rigidity of linear strands and generic initial ideals.** / Murai, Satoshi; Singla, Pooja.

Research output: Contribution to journal › Article

*Nagoya Mathematical Journal*, vol. 190, pp. 35-61. https://doi.org/10.1017/S0027763000009557

}

TY - JOUR

T1 - Rigidity of linear strands and generic initial ideals

AU - Murai, Satoshi

AU - Singla, Pooja

PY - 2008/1/1

Y1 - 2008/1/1

N2 - 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) = βii+k S(S/Gin(I)) for some i > 1 and k ≥ 0, then βqq+k S(S/I) = βqq+k S(S/Gin(I)) for all q ≥ i, where I ⊂ S is a graded ideal. Second, we show that if βii+k E(E/I) = βii+k E(E/Gin(I)) for some i > 1 and k ≥ 0, then βqq+k E(E/I) = βqq+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) = βii+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.

AB - 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) = βii+k S(S/Gin(I)) for some i > 1 and k ≥ 0, then βqq+k S(S/I) = βqq+k S(S/Gin(I)) for all q ≥ i, where I ⊂ S is a graded ideal. Second, we show that if βii+k E(E/I) = βii+k E(E/Gin(I)) for some i > 1 and k ≥ 0, then βqq+k E(E/I) = βqq+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) = βii+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.

UR - http://www.scopus.com/inward/record.url?scp=45849124153&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=45849124153&partnerID=8YFLogxK

U2 - 10.1017/S0027763000009557

DO - 10.1017/S0027763000009557

M3 - Article

AN - SCOPUS:45849124153

VL - 190

SP - 35

EP - 61

JO - Nagoya Mathematical Journal

JF - Nagoya Mathematical Journal

SN - 0027-7630

ER -