Multigraded commutative algebra of graph decompositions

Authors:
Alexander Engström;Thomas Kahle;Seth Sullivant
Affiliations:
Department of Mathematics and Systems Analysis, Aalto University, Helsinki, Finland;Zentrum Mathematik, TU München, Garching b. München, Germany 85747;Department of Mathematics, North Carolina State University, Raleigh, USA 27695
Venue:
Journal of Algebraic Combinatorics: An International Journal
Year:
2014

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Abstract

The toric fiber product is a general procedure for gluing two ideals, homogeneous with respect to the same multigrading, to produce a new homogeneous ideal. Toric fiber products generalize familiar constructions in commutative algebra like adding monomial ideals and the Segre product. We describe how to obtain generating sets of toric fiber products in non-zero codimension and discuss persistence of normality and primary decompositions under toric fiber products.Several applications are discussed, including (a) the construction of Markov bases of hierarchical models in many new cases, (b) a new proof of the quartic generation of binary graph models associated to K4-minor free graphs, and (c) the recursive computation of primary decompositions of conditional independence ideals.