Depths and Stanley depths of path ideals of spines. (English) Zbl 1328.05078

Campos, Daniel; Gunderson, Ryan; Morey, Susan; Paulsen, Chelsey; Polstra, Thomas

Involve 9, No. 1, 155-170 (2016).

Summary: For a special class of trees, namely trees that are themselves a path, a precise formula is given for the depth of an ideal generated by all (undirected) paths of a fixed length. The dimension of these ideals is also computed, which is used to classify which such ideals are Cohen-Macaulay. The techniques of the proofs are shown to extend to provide a lower bound on the Stanley depth of these ideals. Combining these results gives a new class of ideals for which the Stanley conjecture holds.

MSC:

05C25	Graphs and abstract algebra (groups, rings, fields, etc.)
05E40	Combinatorial aspects of commutative algebra
13C14	Cohen-Macaulay modules
13F55	Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
13A15	Ideals and multiplicative ideal theory in commutative rings
05C65	Hypergraphs
05C05	Trees

Keywords:

edge ideal; depth; path ideal; Cohen-Macaulay; monomial ideal