Betti splitting via componentwise linear ideals

A monomial ideal I   admits a Betti splitting I=J+KI=J+K if the Betti numbers of I can be determined in terms of the Betti numbers of the ideals J, K   and J∩KJ∩K. Given a monomial ideal I  , we prove that I=J+KI=J+K is a Betti splitting of I, provided J and K are componentwise linear, generalizing a result of Francisco, Hà 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.