Betti numbers of Stanley-Reisner rings determine hierarchical Markov degrees

Details Betti numbers of Stanley-Reisner rings determine hierarchical Markov degrees Betti numbers of Stanley-Reisner rings determine hierarchical Markov degrees

2009-10-08

arxiv.org/abs/0910.1610v3

Mathematics

Commutative Algebra

Major revision: proved Conjecture from previous version for all simplicial complexes

Scientific paper

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.

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