Almost regular sequences and Betti numbers

Let Delta be a simplicial complex, and Delta(S) the shifted simplicial complex of Delta, as defined by Kalai. It is shown that the Stanley-Reisner ideals I-Delta and I-Delta' have the same extremal Betti numbers. This theorem is the squareface analogue of the corresponding result by Bayer, Charalambous and Popescu which asserts that a graded ideal and its generic ideal have the same extremal Betti numbers. Our theorem implies well-known results by Kalai according to which Delta and Delta(S) have the same simplicial homology, and Delta is Cohen-Macaulay if and only if its shifted complex is Cohen-Macaulay.