Multigraded Commutative Algebra of Graph Decompositions
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 nonzero 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 $K_{4}$minor free graphs, and (c) the recursive computation of primary decompositions of conditional independence ideals.
 Publication:

arXiv eprints
 Pub Date:
 February 2011
 arXiv:
 arXiv:1102.2601
 Bibcode:
 2011arXiv1102.2601E
 Keywords:

 Mathematics  Commutative Algebra;
 13C05;
 05C75;
 14M25
 EPrint:
 33 pages, v2:typos corrected, v3: Section 5 merged into Section 3 with Conj 5.1 now Thm. 3.6, v4: shortened and improved presentation, v5: final published version