Geodesic
Multigraded commutative algebra of graph decompositions
Journal of Algebraic Combinatorics, Tome 39 (2014) no. 2, pp. 335-372.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

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 $K_{4}$-minor free graphs, and (c) the recursive computation of primary decompositions of conditional independence ideals.
Classification : 13P25, 05E40, 13P10
Keywords: toric fiber product, toric ideal, multigraded ideals, Segre product, algebraic statistics, Markov basis, hierarchical model, conditional independence ideal, primary decomposition, normality of ideals 
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     author = {Engstr\"om, Alexander and Kahle, Thomas and Sullivant, Seth},
     title = {Multigraded commutative algebra of graph decompositions},
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Engström, Alexander; Kahle, Thomas; Sullivant, Seth. Multigraded commutative algebra of graph decompositions. Journal of Algebraic Combinatorics, Tome 39 (2014) no. 2, pp. 335-372. http://geodesic.mathdoc.fr/item/JAC_2014__39_2_a5/