Almost regular sequences and Betti numbers

Abstract

Let Δ be a simplicial complex, and Δ s the shifted simplicial complex of Δ, as defined by Kalai. It is shown that the Stanley-Reisner ideals I Δ and I Δ s have the same extremal Betti numbers. This theorem is the squarefree 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 Δ and Δ s have the same simplicial homology, and Δ is Cohen-Macaulay if and only if its shifted complex is Cohen-Macaulay.