A slice algorithm for koszul simplicial complexes on the lcm lattice of monomial ideals

The set of Cohen-Macaulay monomial ideals with a given radical contains the so-called Cohen-Macaulay modifications. Not all Cohen-Macaulay squarefree monomial ideals admit nontrivial Cohen-Macaulay modifications. We present classes of Cohen-Macaulay squarefree monomial ideals with infinitely many nontrivial Cohen-Macaulay modifications.

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Homological invariants of the Stanley–Reisner ring of a k-decomposable simplicial complex

Asian-European Journal of Mathematics ◽

10.1142/s1793557117500619 ◽

2017 ◽

Vol 10 (03) ◽

pp. 1750061

Author(s):

Somayeh Moradi

Keyword(s):

Simplicial Complex ◽

Upper Bound ◽

Betti Numbers ◽

Recursive Formula ◽

Projective Dimension ◽

Simplicial Complexes ◽

Monomial Ideals ◽

Edge Ideals ◽

Graded Betti Numbers ◽

Shellable Simplicial Complex

In this paper, we study the regularity and the projective dimension of the Stanley–Reisner ring of a [Formula: see text]-decomposable simplicial complex and explain these invariants with a recursive formula. To this aim, the graded Betti numbers of decomposable monomial ideals which is the dual concept for [Formula: see text]-decomposable simplicial complexes are studied and an inductive formula for the Betti numbers is given. As a corollary, for a shellable simplicial complex [Formula: see text], a formula for the regularity of the Stanley–Reisner ring of [Formula: see text] is presented. Finally, for a chordal clutter [Formula: see text], an upper bound for [Formula: see text] is given in terms of the regularities of edge ideals of some chordal clutters which are minors of [Formula: see text].

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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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