Algebraic shifting and sequentially Cohen-Macaulay simplicial complexes

Art M. Duval

The electronic journal of combinatorics, Tome 3 (1996) no. 1

Résumé

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.

Détail

DOI : 10.37236/1245

Classification : 06A11, 52B05
Mots-clés : \(h\)-triangle, nonpure shellable complexes, Cohen-Macaulay conditions, algebraic shifting