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