Algebraic shifting and sequentially Cohen-Macaulay simplicial complexes
The electronic journal of combinatorics, Tome 3 (1996) no. 1
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.
DOI :
10.37236/1245
Classification :
06A11, 52B05
Mots-clés : \(h\)-triangle, nonpure shellable complexes, Cohen-Macaulay conditions, algebraic shifting
Mots-clés : \(h\)-triangle, nonpure shellable complexes, Cohen-Macaulay conditions, algebraic shifting
@article{10_37236_1245,
author = {Art M. Duval},
title = {Algebraic shifting and sequentially {Cohen-Macaulay} simplicial complexes},
journal = {The electronic journal of combinatorics},
year = {1996},
volume = {3},
number = {1},
doi = {10.37236/1245},
zbl = {0883.06003},
url = {http://geodesic.mathdoc.fr/articles/10.37236/1245/}
}
Art M. Duval. Algebraic shifting and sequentially Cohen-Macaulay simplicial complexes. The electronic journal of combinatorics, Tome 3 (1996) no. 1. doi: 10.37236/1245
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