Generalized Hilbert-Kunz function of the Rees algebra of the face ring of a simplicial complex

Let R R be the face ring of a simplicial complex of dimension d − 1 d-1 and R ( n ) {\mathcal R}({\mathfrak {n}}) be the Rees algebra of the maximal homogeneous ideal n {\mathfrak {n}} of R . R. We show that the generalized Hilbert-Kunz function H K ( s ) = ℓ ( R ( n ) / ( n , n t ) [ s ] ) HK(s)=\ell ({\mathcal {R}}({\mathfrak {n}})/({\mathfrak {n}}, {\mathfrak {n}} t)^{[s]}) is given by a polynomial for all large s . s. We calculate it in many examples and also provide a Macaulay2 code for computing H K ( s ) . HK(s).

Spanning Simplicial Complex of Wheel Graph Wn

In this paper, we explore the spanning simplicial complex of wheel graph Wn on vertex set [n]. Combinatorial properties of the spanning simplicial complex of wheel graph are discussed, which are then used to compute the f-vector and Hilbert series of face ring k[Δs(Wn)] for the spanning simplicial complex Δs(Wn). Moreover, the associated primes of the facet ideal [Formula: see text] are also computed.

$k$-shellable simplicial complexes and graphs

In this paper we show that a $k$-shellable simplicial complex is the expansion of a shellable complex. We prove that the face ring of a pure $k$-shellable simplicial complex satisfies the Stanley conjecture. In this way, by applying an expansion functor to the face ring of a given pure shellable complex, we construct a large class of rings satisfying the Stanley conjecture.Also, by presenting some characterizations of $k$-shellable graphs, we extend some results due to Castrillón-Cruz, Cruz-Estrada and Van Tuyl-Villareal.

On algebraic characterization of SSC of the Jahangir’s graph 𝓙n,m

In this paper, some algebraic and combinatorial characterizations of the spanning simplicial complex Δs(𝓙n,m) of the Jahangir’s graph 𝓙n,m are explored. We show that Δs(𝓙n,m) is pure, present the formula for f-vectors associated to it and hence deduce a recipe for computing the Hilbert series of the Face ring k[Δs(𝓙n,m)]. Finally, we show that the face ring of Δs(𝓙n,m) is Cohen-Macaulay and give some open scopes of the current work.

Synthesis and crystal structures of (2E)-1,4-bis(4-chlorophenyl)but-2-ene-1,4-dione and (2E)-1,4-bis(4-bromophenyl)but-2-ene-1,4-dione

The molecular structure of (2E)-1,4-bis(4-chlorophenyl)but-2-ene-1,4-dione [C16H10Cl2O2, (1)] is composed of twop-chlorophenyl rings, each bonded on opposite ends to a near planar 1,4-transenedione moiety [–C(=O)—CH=CH—(C=O)–] [r.m.s. deviation = 0.003 (1) Å]. (2E)-1,4-Bis(4-bromophenyl)but-2-ene-1,4-dione [C16H10Br2O2, (2)] has a similar structure to (1), but with twop-bromophenyl rings and a less planar enedione group [r.m.s. deviation = 0.011 (1) Å]. Both molecules sit on a center of inversion, thusZ′ = 0.5. The dihedral angles between the ring and the enedione group are 16.61 (8) and 15.58 (11)° for (1) and (2), respectively. In the crystal, molecules of (1) exhibit C—Cl...Cl type I interactions, whereas molecules of (2) present C—Br...Br type II interactions. van der Waals-type interactions contribute to the packing of both molecules, and the packing reveals face-to-face ring stacking with similar interplanar distances of approximately 3.53 Å.

Face numbers of pseudomanifolds with isolated singularities

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.

Face Ring Multiplicity via CM-Connectivity Sequences

The multiplicity conjecture of Herzog, Huneke, and Srinivasan is verified for the face rings of the following classes of simplicial complexes: matroid complexes, complexes of dimension one and two, and Gorenstein complexes of dimension at most four. The lower bound part of this conjecture is also established for the face rings of all doubly Cohen–Macaulay complexes whose 1-skeleton's connectivity does not exceed the codimension plus one as well as for all (d−1)-dimensional d-Cohen– Macaulay complexes. The main ingredient of the proofs is a new interpretation of the minimal shifts in the resolution of the face ring via the Cohen–Macaulay connectivity of the skeletons of .

Gorenstein rings through face rings of manifolds

The face ring of a homology manifold (without boundary) modulo a generic system of parameters is studied. Its socle is computed and it is verified that a particular quotient of this ring is Gorenstein. This fact is used to prove that the algebraic g-conjecture for spheres implies all enumerative consequences of its far-reaching generalization (due to Kalai) to manifolds. A special case of Kalai’s conjecture is established for homology manifolds that have a codimension-two face whose link contains many vertices.

On Canonical Modules of Toric Face Rings

Generalizing the concepts of Stanley-Reisner and affine monoid algebras, one can associate to a rational pointed fan Σ in ℝd the ℤd-graded toric face ring K[Σ]. Assuming that K[Σ] is Cohen-Macaulay, the main result of this paper is to characterize the situation when its canonical module is isomorphic to a ℤd-graded ideal of K[Σ]. From this result several algebraic and combinatorial consequences are deduced. As an application, we give a relation between the cleanness of K[Σ] and the shellability of Σ.