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Abstract for Art M. Duval,
Algebraic Shifting and Sequentially Cohen-Macaulay Simplicial Complexes
Bj\"orner and Wachs generalized the definition of shellability by
dropping the assumption of purity; they also introduced the
{\sl $h$-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
{\sl 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.
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