DOI: 10.1007/s10801-012-0381-1

Summary: There are two seemingly unrelated ideals associated with a simplicial complex $\Delta $: one is the Stanley-Reisner ideal I $_{ \Delta }$, the monomial ideal generated by minimal non-faces of $\Delta $, well-known in combinatorial commutative algebra; the other is the toric ideal I $_{ M(\Delta )}$ of the facet subring of $\Delta $, whose generators give a Markov basis for the hierarchical model defined by $\Delta $, playing a prominent role in algebraic statistics. In this note we show that the complexity of the generators of I $_{ M(\Delta )}$ is determined by the Betti numbers of I $_{ \Delta }$. The unexpected connection between the syzygies of the Stanley-Reisner ideal and degrees of minimal generators of the toric ideal provide a framework for further exploration of the connection between the model and its many relatives in algebra and combinatorics.

Stanley-resiner ideal, Betti numbers, Markov basis, Markov width, toric ideal generators