Betti splitting via componentwise linear ideals

TL;DR

This paper establishes conditions under which monomial ideals admit Betti splittings based on componentwise linearity, extending previous results and applying to various algebraic and combinatorial structures.

Contribution

It generalizes a known Betti splitting criterion to componentwise linear ideals and explores its implications for simplicial complexes and fat points.

Findings

Betti splitting characterized for componentwise linear ideals

Recursion applied to Alexander duals of specific complexes

Determination of Betti numbers for ideals of fat points

Abstract

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 $J \cap K$. 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, H\`a 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 and to determine the graded Betti numbers of the defining ideal of three general fat points in the projective space.