Rigidity of linear strands and generic initial ideals

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.