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∩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 Francis...

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Bibliographic Details
Published in:	Journal of algebra Vol. 455; pp. 1 - 13
Main Author:	Bolognini, Davide
Format:	Journal Article
Language:	English
Published:	Elsevier Inc 01-06-2016
Subjects:	Betti numbers of monomial ideals Componentwise linear ideals Fat points Simplicial complexes secondary Componentwise linear ideals Betti numbers of monomial ideals Simplicial complexes Fat points primary
Online Access:	Get full text
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Summary:	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∩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, 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.
ISSN:	0021-8693 1090-266X
DOI:	10.1016/j.jalgebra.2016.02.003