Hypergraphs and the Regularity of Square-free Monomial Ideals

Hypergraphs and the Regularity of Square-free Monomial Ideals

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Jason G McCullough , MSRI, Rider University
Fine Hall 214

This talk represents joint work with Kuei-Nuan Lin.  Fix a polynomial ring R over a field and consider a homogeneous ideal I.  The Castelnuovo-Mumford regularity of I is a measure of the complexity of the minimal graded free resolution of I.  Regularity was originally defined in the context of coherent sheaves and has connections to Hilbert functions and Groebner bases.  There has been a lot of interest in finding combinatorial ways of computing or bounding the regularity of a square-free monomial ideal.  Traditional methods involve associating some combinatorial object to a square-free monomial ideal, e.g. a simplicial complex, graph or clutter.  In this talk, I will present a new combinatorial object associated to any square-free monomial ideal that we call a labeled hypergraph.  We give several simple upper bounds on the regularity of square-free monomial ideals and prove that these upper bounds are sharp for large classes of ideals.  While this talk is more combinatorial commutative algebra than algebraic topology, I will briefly recall Stanley-Reisner theory and Hochster's formula relating simplicial homology to the Betti numbers of the associated Stanley-Reisner ideal.