# zbMATH — the first resource for mathematics

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