Algebraic shifting and graded betti numbers

Let S = K[x1, ⋯,xn] denote the polynomial ring in n variables over a field K with each deg xi = 1. Let Δ be a simplicial complex on [n] = {1, ⋯,n} and I. Δ S its Stanley-Reisner ideal. We write.e for the exterior algebraic shifted complex of Δ and.c for a combinatorial shifted complex of. Let ßii+j (I.) = dimK Tori(K, I.)i+j denote the graded Betti numbers of I. In the present paper it will be proved that (i) βii%+j (I.e) = βii%+j (I.c) for all i and j, where the base field is infinite, and (ii) βii%+j (I.) = βii%+j (I.c) for all i and j, where the base field is arbitrary. Thus in particular one has βii%+j (I.) = βii%+j (I.lex) for all i and j, where.lex is the unique lexsegment simplicial complex with the same f-vector as Δ and where the base field is arbitrary.