Multigraded Commutative Algebra of Graph Decompositions

Details Multigraded Commutative Algebra of Graph Decompositions Multigraded Commutative Algebra of Graph Decompositions

2011-02-13

arxiv.org/abs/1102.2601v3

Mathematics

Commutative Algebra

38 pages, v2:typos corrected, v3: Section 5 merged into Section 3 with Conj 5.1 now Thm. 3.6

Scientific paper

The toric fiber product is a general procedure for gluing two ideals, homogeneous with respect to the same grading, to produce a new homogeneous ideal. Toric fiber products generalize familiar constructions in commutative algebra like adding monomial ideals and the Segre product of two arbitrary homogeneous ideals. We prove three new general results about toric fiber products and apply our results to analyze some families of ideals arising in algebraic statistics. In applications, we typically want to compute algebraic invariants of a family of ideals, each parametrized by a combinatorial object like a graph or a simplicial complex. If the graph or simplicial complex has a decomposition into simpler pieces, we can identify the associated ideal as the toric fiber product of two simpler ideals. In the best scenario, it is then possible to reduce questions on a family of ideals to a few simple base cases, which can be tackled by hand or using computer algebra. Some applications of our results include: (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.

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