Betti splitting via componentwise linear ideals

A monomial ideal I admits a Betti splitting I = J + K if the Betti numbers of I can be determined in terms of the Betti numbers of the ideals J, K and J boolean AND K. Given a monomial ideal I, we prove that I = J + K is a Betti splitting of I, provided J and K are componentwise linear, generalizing a result of Francisco, Ha, and Van Tuyl. If I has a linear resolution, the converse also holds. We apply this result recursively to the Alexander dual of vertex-decomposable, shellable and constructible simplicial complexes. Moreover we determine the graded Betti numbers of the defining ideal of three general fat points in the projective space. (C) 2016 Elsevier Inc. All rights reserved.