Algebraic shifting and sequentially Cohen-Macaulay simplicial complexes. (English) Zbl 0883.06003

Duval, Art M.

Electron. J. Comb. 3, No. 1, Research paper R21, 14 p. (1996); printed version J. Comb. 3, No. 1, 297-310 (1996).

Summary: Björner and Wachs generalized the definition of shellability by dropping the assumption of purity; they also introduced the \(h\)-triangle, a doubly-indexed generalization of the \(h\)-vector which is combinatorially significant for nonpure shellable complexes. Stanley subsequently defined a nonpure simplicial complex to be sequentially Cohen-Macaulay if it satisfies algebraic conditions that generalize the Cohen-Macaulay conditions for pure complexes, so that a nonpure shellable complex is sequentially Cohen-Macaulay. We show that algebraic shifting preserves the \(h\)-triangle of a simplicial complex \(K\) if and only if \(K\) is sequentially Cohen-Macaulay. This generalizes a result of Kalai’s for the pure case. Immediate consequences include that nonpure shellable complexes and sequentially Cohen-Macaulay complexes have the same set of possible \(h\)-triangles.

MSC:

06A11	Algebraic aspects of posets
52B05	Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)

Keywords:

\(h\)-triangle; nonpure shellable complexes; Cohen-Macaulay conditions; algebraic shifting