Hilbert function and facet ideals of products of simplicial complexes

AbstractLet (R, ) be an analytically unramified local ring of positive prime characteristic p. For an ideal I, let I* denote its tight closure. We introduce the tight Hilbert function $$H_I^*\left( n \right) = \Im \left( {R/\left( {{I^n}} \right)*} \right)$$ and the corresponding tight Hilbert polynomial $$P_I^*\left( n \right)$$, where I is an m-primary ideal. It is proved that F-rationality can be detected by the vanishing of the first coefficient of $$P_I^*\left( n \right)$$. We find the tight Hilbert polynomial of certain parameter ideals in hypersurface rings and Stanley-Reisner rings of simplicial complexes.

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Face numbers of pseudomanifolds with isolated singularities

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-15204 ◽

2012 ◽

Vol 110 (2) ◽

pp. 198 ◽

Cited By ~ 7

Author(s):

Isabella Novik ◽

Ed Swartz

Keyword(s):

Simplicial Complex ◽

Upper Bound ◽

Hilbert Function ◽

Simplicial Complexes ◽

Topological Invariant ◽

Two Dimensional ◽

Isolated Singularities ◽

Face Ring ◽

The Face

We investigate the face numbers of simplicial complexes with Buchsbaum vertex links, especially pseudomanifolds with isolated singularities. This includes deriving Dehn-Sommerville relations for pseudomanifolds with isolated singularities and establishing lower and upper bound theorems when the singularities are also homologically isolated. We give formulas for the Hilbert function of a generic Artinian reduction of the face ring when the singularities are homologically isolated and for any pure two-dimensional complex. Some examples of spaces where the $f$-vector can be completely characterized are described. We also show that the Hilbert function of a generic Artinian reduction of the face ring of a simplicial complex $\Delta$ with isolated singularities minus the $h$-vector of $\Delta$ is a PL-topological invariant.

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Algebraic Shifting and Sequentially Cohen-Macaulay Simplicial Complexes

The Electronic Journal of Combinatorics ◽

10.37236/1245 ◽

1996 ◽

Vol 3 (1) ◽

Cited By ~ 22

Author(s):

Art M. Duval

Keyword(s):

Simplicial Complex ◽

Simplicial Complexes ◽

Pure Case ◽

Definition Of ◽

Algebraic Shifting

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.

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