# Rigidity of linear strands and generic initial ideals

Satoshi Murai, Pooja Singla

2 引用 （Scopus）

### 抜粋

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.

元の言語 English 35-61 27 Nagoya Mathematical Journal 190 https://doi.org/10.1017/S0027763000009557 Published - 2008 1 1 Yes

### ASJC Scopus subject areas

• Mathematics(all)